TEACHER HANDBOOK LEAVING CERTIFICATE - Project Maths
TEACHER HANDBOOK LEAVING CERTIFICATE - Project Maths
TEACHER HANDBOOK LEAVING CERTIFICATE - Project Maths
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<strong>TEACHER</strong> <strong>HANDBOOK</strong><br />
<strong>LEAVING</strong> <strong>CERTIFICATE</strong><br />
HIGHER<br />
PROBABILITY & STATISTICS<br />
(Oct 2008 Syllabus)
Exploring concepts of probability and data analysis<br />
(Draft for October Syllabus 2008)<br />
Leaving Certificate Higher Level<br />
“When we are not sure, we are alive”<br />
Graham Greene<br />
Section 1: Suggested sequence of lessons<br />
Section 2: Notes and Terms<br />
Section 3: Appendices<br />
0.7<br />
f(x)<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
3 5 7 9 11 13 15 17<br />
x<br />
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0.3<br />
0.2<br />
0.1<br />
p<br />
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17<br />
r<br />
Section 1: Suggested sequence of lessons<br />
LCH Strand 1 Statistics and Probability<br />
Prior Knowledge (for students who have not studied <strong>Project</strong> <strong>Maths</strong> for the Junior Cert)<br />
Collecting and recording data, tabulating data, drawing and interpreting bar charts,<br />
pie chart, trend graphs, discrete array expressed as a frequency table, drawing and<br />
interpreting histograms, mean and mode, mean of a grouped frequency distribution,<br />
cumulative frequency, ogive, median, interquartile range.<br />
Recurring themes in this section<br />
Recognising everyday examples of the use of statistics in relevant applications,<br />
evaluating reliability and quality of data and data sources, critically evaluating claims<br />
and inferences made based on statistics, drawing conclusions from graphical and<br />
numerical summaries of data, recognising assumptions and limitations, relevance of<br />
sample size<br />
Note 1: Items in square brackets refer to suitable reference document.<br />
e.g. [Census at school how to] refers to a document on the website,<br />
[Appendix 3 Conditional Probability] refers to Appendix 3 in this document<br />
Note 2: The CD icon<br />
is used throughout this document to indicate that there is a resource on<br />
the Student Disc, which is relevant to the topic in the title of the lesson.<br />
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Lesson 1/Lesson 2<br />
Title:<br />
References:<br />
Content:<br />
Data Handling Cycle<br />
[Data Handling Cycle] document, [Census at school how to] document<br />
Complete Census at School (CAS) Questionnaire (possibly homework)<br />
www.censusatschool.ntu.ac.uk<br />
Complete Census at School (CAS) Questionnaire (possibly homework)<br />
www.censusatschool.ntu.ac.uk<br />
Data handling cycle (Pose a question, collect data, analyse data, interpret the result<br />
and revisit the original question to check if it was suitable)<br />
Primary and secondary sources of data<br />
Designing Questionnaires<br />
Analyse Census at School questionnaire<br />
(data types and suitable graphical representations, open/closed questions)<br />
Analyse spreadsheet from Census at School<br />
Types of data – category vs. ordered, discrete vs. continuous<br />
Match CAS data to data types.<br />
Using prior knowledge of graphical representations decide on suitable graphical<br />
representations for category vs. ordered data. Interpret data. Find the mode.<br />
(Bar charts and pie charts are suitable data representations for these types of data.)<br />
Time series data and trend graph e.g. met office data, temperature over period of<br />
time, points accumulated at end of season over a number of years.<br />
Lesson 3<br />
Title:<br />
References:<br />
Representing and interpreting data<br />
[Data Handling Cycle] document,<br />
[Limitations of measures of central tendency] document,<br />
[Let’s investigate] document,<br />
Single variable numeric data discrete v continuous,<br />
Content:<br />
(Ranking, range, tallying, grouping and use of different class intervals,<br />
frequency table)<br />
Mean, median and mode of raw data, grouping to facilitate calculation of these<br />
Mean of raw data v mean of grouped data<br />
Median identified after ranking and median identified from grouped data<br />
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Mode identified<br />
Choice of suitable data representation - Discrete data – bar charts and pie charts<br />
Continuous data - histograms, frequency polygons, frequency curves<br />
Describe the distributions in terms of their shape, centre, and spread, and note any<br />
gaps and/or outliers.<br />
Subgroups of the class could work on different data, e.g. height, foot size, arm span<br />
from CAS.<br />
Lesson 4<br />
Title:<br />
Weighted mean<br />
References:<br />
Content:<br />
Weighted mean and applications<br />
Lessons 5, 6, 7<br />
Title:<br />
References:<br />
Stem Plots<br />
[Data Handling Cycle] document, [Spreadsheet practice] document,<br />
[Let’s investigate] document,<br />
Content:<br />
Stem Plots<br />
Lessons 8, 9, 10<br />
Students can use CAS data or generate their own data e.g. by guessing the number<br />
of sweets in a jar, the length of a line segment drawn on the board etc.<br />
For what types of data are stem plots suitable?<br />
One-sided, back-to-back<br />
Find median, range, maximum and minimum values, look for clustering of data,<br />
outliers. What are the advantages and disadvantages of stem plots?<br />
Title:<br />
Introduction to hypotheses testing and the Tukey quick test<br />
References:<br />
[Data Handling Cycle] document, [Tukey] document,<br />
Content:<br />
Sampling v Census, size of sample, randomness of sample<br />
Concept of hypothesis testing (Inferences made about populations based on<br />
samples)<br />
Null hypothesis H 0 , Alternate hypothesis H 1<br />
Tukey (Linked to back-to-back stem plot)<br />
Decision-making and Type 1 and Type 11 errors<br />
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Lessons 11, 12<br />
Title:<br />
References:<br />
Content:<br />
Cumulative frequency<br />
[Let’s investigate] document<br />
Cumulative frequency polygon, cumulative frequency curve, quartiles, interquartile<br />
range, percentiles<br />
Lessons 13, 14<br />
Title:<br />
Measures of dispersion<br />
References:<br />
Content:<br />
[Limitations of measures of central tendency] document,<br />
Measures of dispersion – Range then standard deviation and limitations thereof<br />
(effect of outliers)<br />
Lessons 15, 16, 17<br />
Shifting and rescaling data – adding or subtracting a constant and multiplying or<br />
dividing by a constant (see exercise on Student disc).<br />
Title:<br />
References:<br />
Paired data and correlation<br />
[Data Handling Cycle] document, [Correlation Coefficient] documents,<br />
[Appendix 1 Correlation coefficient by eye and by calculator],<br />
Content:<br />
Paired data, concept of correlation (positive, negative, strong, weak) scatter graph,<br />
line of best fit (by eye and by calculator), correlation coefficient (by calculator),<br />
correlation vs. causality, relevance of sample size<br />
Lessons 18, 19<br />
Title:<br />
Fundamental Principle of Counting and permutations<br />
References: [Permutations and Combinations LCO] ,<br />
Content:<br />
Fundamental Principle of Counting and permutations<br />
Lessons 20, 21, 22<br />
Title:<br />
References:<br />
Content:<br />
Combinations<br />
[Permutations and Combinations LCO] document<br />
Combinations<br />
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Lessons 23, 24<br />
Title:<br />
Introduction to probability<br />
Content:<br />
Lessons 25, 26<br />
Language of probability, sample space(S), event, estimate likelihood of an event on a<br />
scale of 0% to 100%, 0 to 1, outcomes of simple random processes using practical<br />
experiments, listing all possible outcomes, relative frequency, probability as long<br />
term relative frequency<br />
The probability of an event E,<br />
number of times E occurs<br />
P( E)<br />
Lim<br />
n<br />
number of times the experiment is performed<br />
Title:<br />
References:<br />
Content:<br />
Using combinations to evaluate probabilities<br />
[Appendix 2 Probability and Statistics Practice Sheet 1] in this document<br />
How to use combinations to evaluate probabilities<br />
Lessons 27, 28<br />
Title:<br />
Content:<br />
Axioms of probability<br />
Formally list rules/axioms of probability + complement +General Addition Rule<br />
Lesson 29<br />
Title:<br />
Content:<br />
Multiplication Rule<br />
Multiplication Rule for Independent events<br />
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Lessons 30, 31, 32<br />
Title:<br />
References:<br />
Content:<br />
Investigating conditional probability<br />
[Appendix 3 Conditional Probability] in this document<br />
Conditional probability - Use of tables, tree diagrams, and set theory<br />
P( A and B) P( A<br />
B)<br />
P( B | A)<br />
<br />
P( A) P( A)<br />
Lead onto the General Multiplication Rule of probabilities, which does not require<br />
the events to be independent.<br />
P( A B) P( A) P( B | A) or<br />
P(A B)=P(B) P(A|B) (Either set may be called A or B).<br />
Formal definition of independence: A and B are independent if P( B | A) P( B)<br />
Lessons 33, 34<br />
Title:<br />
Content:<br />
Consolidation of probability concepts<br />
Overview of probability using all rules and methods<br />
Lessons 35, 36, 37<br />
Title:<br />
References:<br />
Content:<br />
Bernoulli Trials<br />
[Binomial] T&L, [Appendix 4 Binomial Activity document] in this document<br />
Bernoulli Trials<br />
Lessons 38, 39<br />
Title:<br />
References:<br />
Random variables and expected value<br />
[Expected Value notes NCCA] document, [Expected value activities] document,<br />
[Let’s investigate] document<br />
Content:<br />
Frequency distributions grouped and ungrouped represented by histograms, uniform<br />
distribution, Random variables, discrete and continuous which lead to discrete and<br />
continuous probability distributions<br />
Expected value E(X) of Probability distributions<br />
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Lessons 40, 41<br />
Title:<br />
References:<br />
Content:<br />
Binomial Distribution<br />
http://demonstrations.wolfram.com/BinomialDistribution/<br />
Binomial Distribution as a discrete probability distribution<br />
The number of successes r from n trials is a random variable.<br />
Mean (expected value) of r and standard deviation for the binomial distribution,<br />
representing the binomial distribution graphically, r on the horizontal axis and P(r)<br />
on the vertical axis (Autograph)<br />
Lessons 42, 43, 44, 45<br />
Title: Normal Distribution and Standard Normal<br />
References: [Appendix 5], [Appendix 6]<br />
3 rules for success:<br />
1. Draw a diagram<br />
2. Draw a diagram<br />
3. Draw a diagram<br />
Content:<br />
Continuous Probability distributions – the Normal Distribution and the Standard<br />
Normal Distribution, probabilities of (or proportion of the population in) particular<br />
ranges in normal distributions using a standardising transformation<br />
Lessons 46, 47<br />
Title:<br />
References:<br />
Normal approximation to the binomial distribution<br />
[See page 16 Notes and Terms section in this document]<br />
Content: Normal approximation to the binomial ( np 5, nq 5 )<br />
Lessons 48, 49, 50<br />
Title:<br />
Sampling distribution of the mean<br />
References: [Appendix 7], [Appendix 8]<br />
Content:<br />
Sampling distribution of the mean, the role of the normal distribution, standard<br />
error of the mean<br />
<br />
x<br />
Lessons 51, 52<br />
Title:<br />
Confidence intervals<br />
References: [Appendix 9]<br />
Content:<br />
95% confidence interval for population mean from a sample<br />
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Lessons 53, 54, 55<br />
Title:<br />
Hypothesis testing<br />
References: single sample z –test http://en.wikipedia.org/wiki/Z-test ,<br />
Hypothesis testing http://en.wikipedia.org/wiki/Hypothesis_testing<br />
Content:<br />
Hypothesis testing of specified population mean, hypothesis testing for binomial<br />
distributions using one- and two-tailed tests at a 5% level of significance, assuming a<br />
normal distribution applies.<br />
Types of hypothesis testing<br />
There are 3 types of hypothesis testing on the LCH course<br />
Population Mean<br />
(one sample z –test)<br />
1.96 z 1.96<br />
x <br />
<br />
<br />
n<br />
1<br />
1.96 1.96<br />
Hypothesis Testing<br />
Binomial Distribution<br />
1.96 z 1.96<br />
x <br />
1.96 1.96<br />
<br />
Tukey Quick Test<br />
Significance level 5% 1% 0.1% Significance level<br />
Critical value of tail - 7 10 13 Critical value of<br />
count<br />
tail - count<br />
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Addition Rule (general addition rule)<br />
SECTION 2 NOTES AND TERMS<br />
For any 2 events E and F, the probability of E or F is<br />
P( E F) P( E) P( F) P( E F)<br />
.<br />
Bernoulli trial<br />
A Bernoulli trial is an experiment with only two possible outcomes, success and<br />
failure. If p is the probability of success then the probability of failure q = 1 – p. If the<br />
same Bernoulli trial is repeated n times in succession, then the probability of any set<br />
of outcomes can be obtained from the binomial expansion of (p + q) n .<br />
Binomial Experiment (Bernoulli experiment) – essential features of<br />
<br />
<br />
<br />
<br />
<br />
There are a fixed number of trials denoted by n<br />
The n trials are independent and are repeated under identical conditions<br />
Each trial has only 2 outcomes: success and failure<br />
For each trial, the probability of success (p) is the same. The probability of failure is<br />
denoted by q. Since each trial results in either success or failure , p + = 1 and q = 1-p<br />
In a binomial experiment we are trying to find the probability of r successes in n<br />
n<br />
r<br />
trials P( X r)<br />
p q<br />
r<br />
<br />
nr<br />
Binomial Probability Model<br />
A binomial probability model is appropriate for a random variable that counts the<br />
number of successes r in n (a fixed number) of Bernoulli trials.<br />
Binomial Probability Distribution<br />
When a Bernoulli trial is carried out n times, it gives a binomial probability<br />
distribution. The mean and the standard deviation of a binomial probability<br />
distribution are given by:<br />
np<br />
<br />
npq<br />
The binomial distribution is used when a researcher is interested in the occurrence<br />
of an event, not in its magnitude. For instance, in a clinical trial, a patient may<br />
survive or die. The researcher studies the number of survivors, and not how long the<br />
patient survives after treatment.<br />
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Category type data -<br />
Category data -ordered<br />
Can be identified by names or categories and cannot be organised according to any<br />
natural order e.g. colour of hair, make of car, pets<br />
These are identified by categories which can be ordered in some way e.g. months of<br />
the year, days of the week, schoolwork pressure – a lot, some, a little, none<br />
Complement Rule (useful consequence of the Axioms of Probability)<br />
Conditional Probability<br />
P( E ') 1 P( E)<br />
Conditional probability is a probability that takes into account a given condition e.g.<br />
the probability of owning a games console given that you are between the ages<br />
12 – 25 (not the same as the probability of owning a games console.) P( B | A ) reads<br />
as the probability of B given A.<br />
P( A<br />
B)<br />
P( B | A)<br />
<br />
PA ( )<br />
Correlation<br />
A statistical measure referring to the relationship between two random variables<br />
A positive correlation exists when each variable tends to increase or decrease as the<br />
other does, and a negative or inverse correlation if one tends to increase as the<br />
other decreases.<br />
Correlation Coefficient -<br />
A numerical value (between +1 and -1) which identifies the strength of the linear<br />
relationship between variables<br />
A value of +1 indicates an exact positive relationship, -1 indicates an exact inverse<br />
relationship, and 0 indicates no predictable relationship between the variables.<br />
Distribution<br />
The distribution of a variable gives the possible values of the variable and the<br />
relative frequency of each value.<br />
Distribution – uniform distribution<br />
A histogram which doesn’t appear to have any mode and in which all the bars are<br />
approximately the same height is said to be uniform. Hence, a distribution that is<br />
roughly flat is said to be uniform.<br />
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Discrete Probability Distribution<br />
Expected Value<br />
If X is a discrete random variable, then for each possible value of X, i.e. X=x, we can<br />
work out the probability P(X = x) i.e. P(x). The set of all values of x with their<br />
associated probabilities is called a discrete probability distribution.<br />
The expected value of a random variable is its theoretical long – run average value.<br />
As it is an average value, it need not be a point of the sample space.<br />
E( X ) xP( x)<br />
<br />
Event (E)<br />
Histogram<br />
Independent Events<br />
The name given to an outcome or a particular set of outcomes when an experiment<br />
is conducted in probability<br />
A bar graph such that the area over each class interval is proportional to the relative<br />
frequency of data within this interval<br />
Two events are independent if the occurrence of one has no effect on the probability<br />
that the other will occur e.g. two rolls of a fair die. Two events A and B are<br />
independent if any one of the following is true<br />
P( A| B) P( A)<br />
P( B | A) P( B)<br />
P( A AND B) P( A) P( B)<br />
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Measures of Central Tendency<br />
Mean<br />
Advantages:<br />
1. It is easy to define and understand<br />
2. It is easy to calculate<br />
3. It takes into account every value in the distribution<br />
4. It is often used for other statistical processing. e.g.<br />
standard deviation, comparing averages. The mean<br />
facilitates this type of further analysis better than the<br />
median<br />
5. It is used when sampling and can give reliable<br />
information about a population<br />
6. It is preferred to the median if the distribution is<br />
relatively even<br />
Disadvantages:<br />
1. If there are extreme values (outliers) in a distribution,<br />
it may give a distorted view of the distribution.<br />
2. It can give an unrealistic value for discrete data. E.g.<br />
average car passengers 3.2<br />
Median<br />
Advantages:<br />
1. It is more suitable than the mean<br />
if there are outliers. It gives a<br />
more accurate view of central<br />
tendency than the mean if there<br />
are extreme values.<br />
2. It is usually a value from within<br />
the distribution, as opposed to<br />
the mean which may not be a<br />
value within the distribution at all<br />
3. It is easy to identify, especially<br />
using a stem and leaf diagram<br />
Disadvantages:<br />
1. Difficult to use it in any further<br />
statistical processing<br />
2. It is not widely known nor used<br />
Multiplication Rule<br />
For independent events the probability that<br />
both A and B occur is<br />
P( A and B) P( A B) P( A) P( B)<br />
(General) Multiplication Rule<br />
P( A and B) P( AB) P( A) P( B | A)<br />
Mutually exclusive or disjoint events<br />
Numeric Data<br />
Mutually exclusive or disjoint events have no outcomes in common. They cannot<br />
occur at the same time/in a single outcome. They cannot both be the result of a<br />
single experiment e.g. when a card is removed from a deck of cards it cannot be<br />
both black and red (mutually exclusive) but it can be both black and a king (not<br />
mutually exclusive).<br />
Data represented by real numbers<br />
Numeric Data – continuous numeric data<br />
Data which can take assume an infinite number of values between any two given<br />
values e.g. height, arm span, foot length<br />
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Numeric data – discrete numeric data<br />
Normal Distribution<br />
Normal Curve<br />
Data that can only have a finite number of numeric values e.g. the number of peas in<br />
a pod, age in years, and the number of persons in each household<br />
This is one of the most important continuous probability distributions. It was<br />
studied by De Moivre and Gauss and is sometimes called Gaussian. It turns up in<br />
very many real situations including nearly all measured physical characteristics of<br />
living things. It is very important not only because it can be applied to a wide variety<br />
of situations but also because other distributions tend to become normal under<br />
certain conditions.<br />
A normal curve is the graph of a normal distribution. It has a shape very like the<br />
cross section of a pile of dry sand. It is called a bell shaped curve because<br />
blacksmiths would sometimes use a pile of dry sand in the construction of the mould<br />
for a bell.<br />
About 68% of all values fall within 1 standard deviation of the mean, about 95% fall<br />
within 2 standard deviations of the mean and about 99.7% fall within 3 standard<br />
deviations of the mean.<br />
Normal Curve – characteristics of<br />
<br />
The curve is bell shaped with its highest point over the mean on the x – axis.<br />
It is symmetrical about a vertical line through .<br />
<br />
<br />
<br />
The curve approaches the horizontal axis but never touches it or crosses it<br />
The transition points between being concave upwards and concave downwards<br />
are at + and -.<br />
The parameters governing the shape of a normal curve are (locates the<br />
balance point) and (the extent of the spread)<br />
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Normal Model as an approximation to a Binomial Model<br />
For a normal model to be a good approximation to a Binomial model, we must<br />
expect at least 5 successes and 5 failures i.e. np 5 and nq 5 .<br />
(i) np 5<br />
Stick graph of a Binomial distribution with n = 10 and p=0.2 written as B (10,<br />
0.2). The overall shape does not have the characteristics of a normal<br />
distribution.<br />
0.4<br />
p<br />
0.3<br />
0.2<br />
0.1<br />
r<br />
Figure 1<br />
(ii)<br />
−2 −1 1 2 3 4 5 6 7 8 9 10<br />
np 5<br />
“Stick graph” of a Binomial distribution with n = 16 and p=0.5 written as B (16, 0.5)<br />
(np = 8). Although it is a discrete distribution, its shape is very close to that of a<br />
normal.<br />
0.3<br />
p<br />
0.2<br />
0.1<br />
Figure 2<br />
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17<br />
r<br />
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Fitting a normal curve with the same mean and standard deviation as a binomial distribution to a<br />
binomial distribution<br />
Normal curve with mean = 8 and<br />
standard deviation =2<br />
0.3<br />
p<br />
“Stick graph” of a Binomial<br />
distribution B(16,0.5) i.e. mean<br />
np =8 and standard deviation<br />
0.2<br />
npq =2<br />
0.1<br />
r<br />
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17<br />
Figure 3<br />
As n increases, the Binomial distribution becomes closer to a normal distribution<br />
0.1<br />
p<br />
0.08<br />
B(200,0.5)<br />
0.06<br />
0.04<br />
0.02<br />
20 40 60 80 100 120 140 160 180 200<br />
r<br />
Figure 4<br />
Normal as an appromimation to the Binomial – continuity correction<br />
As the normal distribution is a continuous distribution the binomial needs to be<br />
“made continuous” in order to compare it with the normal. We can do this by<br />
treating each individual x as an interval, from x - 0.5 to x + 0.5. Hence 8 becomes<br />
the interval 7.5 to 8.5.<br />
-0.5 and +0.5 are known as continuity corrections.<br />
In Figure 2 above, the stick whose height represents the probability of 8 successes in<br />
16 trials is replaced by a bar whose area represents this probability and the stick<br />
graph becomes a histogram when using the continuity correction. When using the<br />
normal distribution to approximate the binomial we are working out the area under<br />
the bars.<br />
As seen from figure 4, for large values of n this correction become negligible. In<br />
practice we only use a continuity correction for small n and if specifically asked.<br />
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Outcome<br />
The outcome of a trial is the value measured or observed for an individual instance<br />
of that trial.<br />
Outcomes – discrete outcomes -<br />
All of the outcomes can be listed<br />
Outcomes – continuous outcomes<br />
Probability density function<br />
Probability Axioms<br />
There are an infinite number of outcomes and therefore they cannot be listed.<br />
Continuous probability distributions are commonly represented with a curve,<br />
y f ( x)<br />
called a probability density function, such that the area under the curve<br />
between a and b is equal to the probability that the outcome of the experiment is<br />
in the interval a x b . Since area under the curve is equal to probability, the<br />
total area under the curve is 1.<br />
See Appendix 5 on area under a normal curve using Autograph.<br />
1.0 P( E) 1,<br />
E S<br />
2. PS ( ) 1; P( )=0<br />
3. Addition Rule: P( E F) P( E) P( F) if E F=<br />
<br />
i.e E and F are mutually exclusive events.<br />
E<br />
F<br />
Random phenomenon:<br />
Random Variable<br />
We call a phenomenon random if we know what outcomes could occur but not what<br />
outcomes will occur.<br />
If all the possible outcomes of an experiment can be assigned different numeric<br />
values and we use a variable to label the outcomes, that variable is a random<br />
variable labelled X. We say that the quantitative variable X is a random variable<br />
because the value it takes on in an experiment is a random outcome.<br />
A particular value that X can have is denoted by the corresponding lower case letter<br />
x.<br />
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Random Variable – Discrete<br />
If the random variable X can take on only distinct discrete values e.g. the number of<br />
heads I get when I toss 4 coins, then it is a discrete random variable. A discrete<br />
random variable gives rise to a discrete probability distribution. e.g. the binomial<br />
probability distribution<br />
Random Variable - continuous<br />
Sample Space(S) -<br />
Scatter plot<br />
If the random variable X can take on any real value in an interval, then X is a<br />
continuous random variable e.g. If x is the weight of a person chosen at random,<br />
then x is a continuous random variable. Continuous random variables lead to<br />
continuous probability distributions.<br />
The set of all possible outcomes when an experiment is conducted<br />
A graphical representation of the distribution of two random variables as a set of<br />
points whose coordinates represent their observed paired values<br />
Standard Deviation<br />
<br />
N<br />
<br />
i1<br />
( x )<br />
i<br />
N<br />
2<br />
The standard deviation is the square root of the average squared deviations of the<br />
data values from the mean. Conceptually the standard deviation measures the<br />
“spread of the data”. A small standard deviation means that most data items are<br />
closely clustered about the mean while a large standard deviation means that the<br />
data items vary a lot from the mean.<br />
Probability<br />
density<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
f(x)<br />
Small <br />
Large <br />
For a normal distribution<br />
approximately 2/3 (68%) of the data<br />
lies within one standard deviation of<br />
the mean.<br />
−40 −30 −20 −10 10 20 30 40 50<br />
x<br />
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Standard Normal Distribution<br />
Standard Units z<br />
This is the most frequently used normal distribution where the random variable is<br />
denoted z (standardised normal variable) and the mean = 0 and standard deviation<br />
= 1. (See standard units).<br />
As there are many different normal distributions with different means and standard<br />
deviations, and as their probability density functions are not integrable, we would<br />
need tables to work out 1 2<br />
P( x x x ). As it is not practical to have tables for<br />
every type of normal distribution we convert x values to z values i.e. standard units.<br />
z <br />
x <br />
<br />
Stem plots – advantages of<br />
When we standardize, we shift by the mean and rescale by the standard deviation.<br />
As the original mean was , when we take the mean of the data from every value<br />
the mean itself becomes zero. Shifting the data alone does not change the standard<br />
deviation.<br />
When we divide each of the shifted values by the standard deviation, the standard<br />
deviation is divided by as well and since it was to start with the new standard<br />
deviation becomes 1. A z score tells us how unusual a value is as it tells us how far<br />
away it is from the mean – the larger it is the more unusual it is. Any z score of 3 or<br />
more is rare and a z score of 6 or 7 is way out!<br />
<br />
<br />
<br />
<br />
<br />
A stem plot displays each separate data value<br />
Stem plots give a quick clear picture of the shape of a distribution and make<br />
it easy to identify clustering of data from the lengths of the branches.<br />
When the values on each branch are ordered, it is very easy to pick out the<br />
median, quartiles, max and min values and to identify any outliers.<br />
Stem plots can be used to display discrete or continuous type data<br />
Two data sets can be compared with a back-to-back stem plot.<br />
Stem plots - disadvantage of<br />
<br />
<br />
<br />
They are unsuitable for very large data sets<br />
Small data sets with a large range can be difficult to display on the same<br />
stem plot without rounding e.g. 432, 507,534,581,609,626,671,712,719. If<br />
these data points are rounded to 430, 500, 510,530, 580, 610,630,710,720<br />
they can easily be displayed with the hundreds digits as the stem and the<br />
tens digits as the leaves<br />
Stem plots cannot be used for category type data.<br />
Trial<br />
A single experiment in probability<br />
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Sampling and Hypothesis Testing<br />
http://www.stats.gla.ac.uk/steps/glossary/hypothesis_testing.html#pvalue<br />
Census<br />
In a census, every member of the population is included.<br />
Population<br />
The entire group of individuals or instances about whom we hope to learn<br />
Sample<br />
Symbols used<br />
A sample is a subset of a population, from which we actually collect information in<br />
the hope of making inferences about the population, as most of the time the<br />
population mean and standard deviation are impossible to determine exactly.<br />
The sample should be representative of the population.<br />
Population parameters are written in Greek letters (e.g. for mean and for<br />
standard deviation, while sample statistics are written in Roman letters (e.g. x for<br />
mean and s for standard deviation). The mean of the sample means is written as<br />
x<br />
and the standard error of the distribution of the mean (see below) is written as<br />
<br />
x .<br />
Sample statistics<br />
These are numbers computed from a sample, such as sample size n,<br />
sample mean x , and sample standard deviation s.<br />
Sample Means –Distribution of Sample means<br />
If I take a random sample of 100 female students and measure their height – I would<br />
expect that the mean height of this sample would be equal to or very close to the<br />
mean height of the population of female students from which the sample was<br />
drawn. Another random sample of 100 students might give a different mean and<br />
again I would expect it to be close to the population mean. The means of all possible<br />
samples of size 100 would form a distribution of sample means, which would have<br />
its own mean and its own standard deviation , called the standard error of the<br />
x<br />
mean.<br />
http://www.censusatschool.org/international/rds/index.asp?country=&lang=en<br />
x<br />
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Central Limit Theorem<br />
Let x denote the mean of a random sample of size n from a population having mean<br />
and standard deviation .<br />
<br />
x = mean of all the sample means of size n<br />
<br />
x = standard error of the mean<br />
Then<br />
<br />
<br />
<br />
x<br />
x<br />
<br />
<br />
<br />
n<br />
When the population is normally distributed, the distribution of the sample<br />
means is normal for any n.<br />
For 30<br />
regardless of the population distribution.<br />
n the distribution of the sample means, x is approximately normal<br />
Statistical Inference<br />
Inferring certain facts about a population from results found in a random sample<br />
from the population e.g. we may wish to draw conclusions about the heights of<br />
15,000 adult female students (the population) by examining only 100 students (a<br />
sample) from this population.<br />
Standard Error of the Mean<br />
<br />
x<br />
This is a good estimate of the standard deviation of the sample means, assuming<br />
that the sample is sufficiently large, n 30 .<br />
(Note standard error is not an “error” and it is not “standard” but it is a lot simpler to<br />
say than the “estimated standard deviation of the sample means”.)<br />
To convert sample means x in a normal distribution of sample means to standard normal z values<br />
use<br />
x x<br />
x <br />
z (one sample z – test)<br />
<br />
x<br />
n<br />
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Sampling distributions of x and of x<br />
0.7<br />
f(x)<br />
0.6<br />
0.5<br />
0.4<br />
Probability distribution of sample<br />
means (n= 5) taken from the same<br />
population with <br />
x = =10.2 and<br />
1.4<br />
x 0.63<br />
n 5<br />
0.3<br />
0.2<br />
Probability distribution of random<br />
variable x taken for a population<br />
with =10.2 and = 1.4<br />
0.1<br />
x<br />
0<br />
3 5 7 9 11 13 15 17<br />
One of the graphs above represents the probability distribution of a random variable X, with mean<br />
=10.2 and = 1.4<br />
The second graph represents the distribution of the sample means from the same population. Since<br />
the original population is normally distributed, the distribution of the sample means is also normal.<br />
Then mean of the population and the mean of the sample means x<br />
are the same i.e. 10.2. The<br />
difference is in the standard deviation for x and x . The samples are of size n = 5, hence the standard<br />
deviation of the x distribution is much smaller than that of the population. This has the effect of<br />
squashing much more of the total probability into the region above the mean.<br />
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95% Confidence interval for a population mean from a sample<br />
The interval in which the population mean will lie with 95% certainty, given a sample<br />
mean, the population standard deviation and the sample size n. If the population<br />
standard deviation is not known then the 1 sample standard deviation s may be used<br />
provided the sample size n is large.<br />
In the sampling distribution of the means 95% of all sample means lie within 1.96<br />
of the population mean as the sample means are normally distributed.<br />
<br />
x<br />
Normal Distribution of sample means<br />
-1.96<br />
<br />
x<br />
<br />
+1.96<br />
<br />
x<br />
1.96 z 1.96<br />
x <br />
<br />
<br />
1<br />
1.96 1.96<br />
x<br />
1.96 x 1.96<br />
x<br />
1<br />
x<br />
1.96 x x 1.96 and<br />
x<br />
1 1<br />
x 1.96 x 1.96 <br />
1 x 1<br />
x<br />
x<br />
Hence the interval between 1<br />
x 1.96 x 1.96<br />
1 x<br />
1<br />
x 1.96<br />
x and x1 1.96<br />
x<br />
x<br />
will contain the population mean with<br />
95% certainty. This is known as the 95% confidence interval for the population mean.<br />
1 Calculating the sample standard deviation: Strictly speaking we should use n-1 instead of n in the denominator of the formula for s.<br />
s <br />
<br />
( x<br />
x)<br />
n 1<br />
2<br />
If n is large however this makes very little difference.<br />
http://www.stats.gla.ac.uk/steps/glossary/confidence_intervals.html#confinterval<br />
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Statistical decisions<br />
Hypotheses<br />
Very often we wish to make a decision on a population based on a sample e.g.<br />
whether a new drug really works or whether one method of teaching a topic is<br />
better than another or whether a new machine is producing higher quality<br />
components than another one or whether one hair colour lasts longer than another.<br />
These decisions are called statistical decisions.<br />
In order to make statistical decisions it is useful to make some assumptions, which<br />
may or may not be true, about the populations. These assumptions are called<br />
hypotheses. We might assume that the new drug is no better than the old one (no<br />
change) or that both methods of teaching are equally effective (no difference) or<br />
that the new machine is no better than the old one or that the new hair colorant<br />
does not last any longer than the old one – such hypotheses as these are called null<br />
hypotheses.<br />
Null Hypothesis H 0<br />
Hypothesis Testing<br />
This is the term used for any hypothesis which is set up primarily for the purpose of<br />
seeing whether it can be rejected. Usually it is a statement of “no effect”, “no<br />
change”, “no difference”, other than that which would be caused by chance<br />
variations alone. We are assuming that the outcome is not “unusual”. If the null<br />
hypothesis cannot be rejected this does not mean that it is true beyond all doubt,<br />
but that under our method of testing it cannot be rejected.<br />
(Courtroom analogy – the defendant is initially assumed innocent – innocence is the<br />
null hypothesis)<br />
Hypothesis testing is the statistical method of testing the truth or otherwise of the<br />
null hypothesis.<br />
(Courtroom analogy – the trial)<br />
Alternative Hypothesis H 1<br />
This is the hypothesis to be accepted when the null hypothesis must be rejected.<br />
(Courtroom analogy – the defendant is guilty)<br />
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Examples of null and alternative hypotheses<br />
H 0 : The new machine is the same as the old one - no change<br />
Depending on the decision we are asked to make then H 1 could be:<br />
H 1 : The new machine is better than the old one (gives rise to a one-tailed test -see<br />
below)<br />
H 1 : The new machine is worse than the old one (gives rise to a one-tailed test -see<br />
below)<br />
H 1 : The new machine is different from the old one (gives rise to a two-tailed test -<br />
see below)<br />
The level of significance of a hypothesis test<br />
This is the probability of rejecting H 0 when it is in fact true.<br />
(e.g. in a courtroom finding a defendant guilty when they are in fact innocent, or<br />
diagnosing a patient with a disease when they don’t actually have it)<br />
An outcome is said to be significant (“unusual”) at the 5% level of significance if the<br />
probability of it or a more unusual one occurring is less than 5%.<br />
Two tailed hypothesis test at a 5% level of significance<br />
Region of acceptance of the null<br />
hypothesis – it represents 95% of<br />
all results.<br />
Regions of rejection of the null<br />
hypothesis (0.025% of all results in<br />
each tail)<br />
0<br />
Reject the null hypothesis H 0 if z 1.96 or z 1.96<br />
-1.96 and 1.96 are the critical values of z for this test and the regions of rejection as<br />
shown are the critical regions.<br />
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One tailed hypothesis test at a 5% level of significance<br />
(i)<br />
Right tailed test<br />
Reject the null hypothesis H 0 if z 1.645<br />
The entire critical region goes to one side of the distribution.<br />
0<br />
1.645<br />
The critical value of z for this test is z = 1.645 and the region of rejection is the critical<br />
region.<br />
(ii)<br />
Left tailed test<br />
-1.645<br />
0<br />
Reject the null hypothesis H 0 if z 1.645<br />
The critical value of z for this test is z = -1.645 and the region of rejection is the<br />
critical region.<br />
Deciding whether a test is one or two tailed is determined by the way the question<br />
is asked and not by the data.<br />
Phrases leading to two tailed tests<br />
<br />
<br />
<br />
<br />
“is different from”<br />
“is biased”<br />
“is fair”<br />
“is consistent with”<br />
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Phrases leading to one-tailed tests<br />
<br />
<br />
<br />
<br />
<br />
“is better than”<br />
“is worse than”<br />
“is greater than”<br />
“is less than”<br />
“is biased towards”<br />
“helps to cure “<br />
Key question we ask in hypothesis testing<br />
Could these data have happened by chance if the null hypothesis were true? If they<br />
were very unlikely to have occurred if the null hypothesis were true, this raises<br />
doubt about the null hypothesis. (Medical analogy – you would be very unlikely to<br />
have these symptoms if you were healthy (null hypothesis.)<br />
Procedure for a hypothesis test<br />
Type l error<br />
Type ll error<br />
P –value<br />
Write down the null hypothesis H 0<br />
Decide whether it’s a one tailed or a two tailed test,<br />
<br />
<br />
<br />
Specify the significance level and hence write down the critical value/values<br />
of z. (5% level of significance only required at L. C.)<br />
Convert the observed result into z - units.<br />
Reject the null hypothesis if z is in the critical region and accept the<br />
alternative hypothesis H 1 , else fail to reject H 0 .<br />
Rejecting the null hypothesis when it is in fact true<br />
(Courtroom analogy – finding the defendant guilty when they are in fact innocent,<br />
in medicine diagnosing a person with a disease (rejecting the null hypothesis that<br />
they are healthy) when they don’t in fact have the disease i.e. a false positive)<br />
Failing to reject the null hypothesis when it is in fact false<br />
(Courtroom analogy – finding the defendant innocent when they are in fact guilty, or<br />
in medicine diagnosing a person as healthy (accepting the null hypothesis that they<br />
are healthy) when in fact they have the disease i.e. a false negative.)<br />
The probability of finding data like these or even less likely, if the null hypothesis<br />
were true is the P-value. If the P-value is high these data are not unusual, they<br />
happen a lot, are consistent with the null hypothesis and we have no reason to<br />
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eject the null hypothesis. (Medical analogy -you have a high probability of having<br />
these symptoms when you are healthy – the null hypothesis - so no reason to reject<br />
the assumption that you are healthy.) If the P-value is low then it means that these<br />
data would be very unlikely if the null hypothesis were true, so we are inclined to<br />
reject the null hypothesis but of course, we cannot be sure.<br />
Tukey quick test (compact form)<br />
Significance level 5% 1% 0.1% Significance level<br />
Critical value of 7 10 13 Critical value of<br />
tail - count<br />
tail - count<br />
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Appendices<br />
Appendix 1: Correlation coefficient by calculator<br />
Appendix 2: Probability and Statistics Practice Sheet 1<br />
Appendix 3: Conditional probability<br />
Appendix 4: Binomial Activity<br />
Appendix 5: Investigate area under a normal curve using Autograph<br />
Appendix 6: Student activity on the normal distribution<br />
Answers to student activity on the normal distribution<br />
Appendix 7: Student activity to investigate the distribution of Sample Means<br />
Answers to Student activity to investigate the distribution of Sample<br />
Means<br />
Appendix 8: Using Autograph to investigate the distribution of Sample Means<br />
Appendix 9: Student Activity on confidence intervals<br />
Answers to student activity on confidence intervals<br />
Appendix 10: Files from Student Disc on Strand 1 JC and LC<br />
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Appendix 1<br />
Correlation coefficient and equation of line of best fit using Casio fx-83ES, Natural Display<br />
Input the following data on fat grams and total calories in fast food<br />
Total Fat (g)<br />
Total Calories<br />
1 Hamburger 9 260<br />
2 Cheeseburger 13 320<br />
3 Quarter Pounder 21 420<br />
4 Quarter Pounder with 30 530<br />
Cheese<br />
5 Big Mac 31 560<br />
6 Sandwich Special 31 550<br />
7 Sandwich Special with 34 590<br />
Bacon<br />
8 Crispy Chicken 25 500<br />
9 Fish Fillet 28 560<br />
10 Grilled Chicken 20 440<br />
11 Grilled Chicken Light 5 300<br />
1. Number each row of data if not already done to make it less likely to miss a row as data is<br />
being input into the calculator.<br />
2. MODE, 2(STAT ), 2(A+BX)<br />
3. Input the data into columns x and y.( Press = after inputting each data item)<br />
4. When they are all entered press SHIFT and 1(STAT)<br />
5. Choose 7( Reg i.e. regression)<br />
6. Choose 3 (r i.e. correlation coefficient), press =<br />
Gives correlation coefficient of 0 .9746<br />
To find the equation of the line of best fit y = A+Bx<br />
We are looking for the values of A (intercept on the y-axis) and B (slope of the line) from step 2<br />
above.<br />
1. The correlation coefficient, which has been calculated in Step 6 above, will have gone into<br />
row 12, column Y so it must be deleted, as it is not one of the original data items. Move the<br />
up arrow to row 12, column Y and press the DEL button.<br />
2. Press SHIFT, 1(STAT), 7(REG), 1(A) followed by = which gives A=193.85. (This appears in row<br />
12, column Y)<br />
3. The intercept, A, which has been calculated, will have gone into row 12 so it must be<br />
deleted, as it is not one of the original data items. Move the up arrow to row 12, column Y<br />
and press the DEL button.<br />
4. Press SHIFT, 1, 7, 2 followed by = which gives B = 11.731. This appears in row 12 column Y.<br />
The equation of the line of best fit is: y = 193.85+11.731x<br />
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Appendix 2<br />
LCH Probability and Statistics<br />
Practice Sheet 1<br />
1) A bag contains 6 pens and 4 pencils. 2 items are drawn at random from the bag.<br />
What is the probability of getting: (a) 2 pens (b) 2 pencils (c) 1 pen and 1 pencil<br />
2) 2 cards are drawn at random from a pack of 52. What is the probability of getting:<br />
(a) 2 clubs (b) 2 red cards (c) 2 face cards (d) 2 aces<br />
3) A bag contains 7 red discs and 4 blue discs. Three discs are drawn at random. What is<br />
the probability of getting (a) 3 red discs (b) 3 blue discs (c) 2 red discs and 1 blue<br />
disc<br />
4) A person is dealt 5 cards from a pack of 52. (This is called a ‘hand’ in cards). What is<br />
the probability that this hand will be the 10, Jack, Queen, King and Ace of Hearts ?<br />
5) A person is dealt 5 cards from a pack of 52.<br />
(a)What is the probability that all of the 5 cards will be Diamonds?<br />
(b)What is the probability that none of the 5 cards is a Diamond?<br />
(Without using the P(S) = 1- P(F) rule)<br />
6) 25 women and 10 men attend a PTA meeting. At the meeting, a committee of 4<br />
people is formed to raise funds for the school. What is the probability that: (a) the<br />
committee is all female? (b) The committee is all male? (c) If the chairperson of<br />
the PTA (who is one of the 25 women) must be on the committee what is the<br />
probability that the committee will have a gender balance? (i.e. equal numbers of<br />
men and women)<br />
7) A bag contains five blue and six red discs. Three discs are drawn at random from the<br />
bag. Find the probability that at least one red disc is drawn.<br />
8) Verify the odds of winning each of the following combinations given in the box<br />
below.<br />
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Appendix 3<br />
Conditional Probability<br />
U<br />
A bag contains 9 identical discs, numbered from 1 to 9.<br />
One disc is drawn from the bag.<br />
Let A = the event that ‘an odd number is drawn.’<br />
Let B = the event that ‘a number less than 5 is drawn’<br />
(i)<br />
(ii)<br />
What is (B|A)?<br />
Are the events A and B independent?<br />
A<br />
.5<br />
.7<br />
.9<br />
.1<br />
.3<br />
.2<br />
.4<br />
B<br />
.6<br />
.8<br />
(i) P(B|A) means the probability that the disc drawn is less than 5 given that it is odd.<br />
#( A<br />
B) Probability that the number is less than 5 and odd 2<br />
P( B | A) = = =<br />
# A<br />
Probability that the number is odd 5<br />
More generally we can<br />
say<br />
#( AB) P( AB)<br />
P( B | A)<br />
= <br />
# A P(<br />
A)<br />
2<br />
9 2<br />
= 5 5<br />
9<br />
(ii) No the events A and B are not independent.<br />
# B 4<br />
PB ( ) = # U 9<br />
2<br />
P( B | A) = 5<br />
Since P( B) P( B | A) the two events are not independent.<br />
i.e. the fact that the event A happened changed the probability of the event B happening.<br />
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Appendix 4<br />
Binomial Activity<br />
Task 1:<br />
(a) If you were to roll a fair die 30 times how many times would you expect to get a 6?<br />
(b) Roll a die 30 times and record the results in the Table 1 below.<br />
Prediction __________<br />
Table 1<br />
1<br />
Number<br />
which<br />
appears on<br />
die<br />
(outcome of<br />
trial)<br />
2<br />
How many<br />
times did this<br />
happen<br />
(It may be helpful to use<br />
tally marks)<br />
3<br />
Frequency<br />
4<br />
Relative<br />
Frequency<br />
5<br />
% of total scores<br />
6<br />
Success<br />
1, 2, 3, 4, 5<br />
Fail<br />
Totals<br />
Task 2:<br />
(a) If we call getting a 6 a “Success” and getting any other number a “Fail”, calculate the relative<br />
frequencies of success and failure and complete columns 4 and 5 in the table above.<br />
(b) Compare your prediction to the actual data recorded in Table 1.<br />
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Task 3:<br />
(a) Fill in the Master Table below by recording and totalling the data collected by the other groups in<br />
the class.<br />
(b) Compare the relative frequencies from the Master Table below with those in the Table 1 above<br />
and comment on these with respect to your initial prediction.<br />
Master Table<br />
Task 4:<br />
(a) The bar chart and corresponding table 2 below show the data recorded on a simulation of<br />
throwing a die 300 times.<br />
Using this data calculate the relative frequencies of success and failure as defined in Task 2(a) above.<br />
(b) Comment on the relative frequencies of success and failure, comparing these with those<br />
calculated using Table 1 above.<br />
Table 2<br />
1 2 3 4 5 6<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 35 of 66
Appendix 5<br />
Investigate area under a normal curve using Autograph.<br />
Click on File, New 1D Statistics Page<br />
Click Data, Enter Probability Distribution<br />
Select Normal, OK<br />
Select = 0 and = 1<br />
Right click on the curve and select Probability Calculations<br />
Input lower and upper x – values to find the probability of finding x in a given<br />
interval. Read the probability on the task bar at the bottom of the screen<br />
Drag the yellow boxes to vary the limits<br />
0.5<br />
f(x)<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
−3 Draft 01 −2 −1 © <strong>Project</strong> <strong>Maths</strong> Development 1 Team 2 Page 36 3of 66<br />
x
Appendix 6<br />
Student Activity on the Normal Distribution and Probability<br />
1. If the sales of a DVD in a store follow the normal distribution with mean of 300 and<br />
standard distribution of 50<br />
a) What is the probability that the sales on a certain day will be less than or equal to<br />
320?<br />
b) What is the probability that the number of sales on a certain day will be greater than<br />
or equal to 320?<br />
c) What is the probability that the number of sales on a certain day will be less than or<br />
equal to 280?<br />
d) What is the probability that the number of sales on a certain day will be between<br />
280 and 320 inclusive?<br />
e) What is the probability that the number of sales on a certain day will be between<br />
320 and 340?<br />
2) A certain type of battery has a life span that follows the normal distribution curve with a<br />
mean life of 180 hours and a standard deviation of 20 hours.<br />
a) What is the probability that a battery of this type selected at random will last more<br />
than 230 hours?<br />
b) What is the probability that a battery bought form this batch will last between 160<br />
and 200 hours inclusive?<br />
c) If 80% of the batteries last less than or equal to x hours, find the value of x.<br />
d) If 10% of these batteries last more than x hours, find the value of x.<br />
3) The time taken to undertake a particular task follows the normal distribution with a<br />
mean of 20 minutes and a standard deviation of 5 minutes.<br />
a) Estimate the probability that it takes 26 minutes or longer to complete the task.<br />
b) Estimate the probability that it takes less than or equal to 12 minutes.<br />
c) Estimate the probability that it takes between 17 and 24 minutes inclusive.<br />
4) A fair die is thrown 200 times, what is the probability that a 5 turns up 36 times or<br />
fewer?<br />
5) A coin is tossed 200 times, what is the probability of getting heads fewer than 80<br />
times?<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 37 of 66
6) If 1000 copies of a particular newspaper advertising a competition were sold on a<br />
particular day and it is estimated that the probability of someone replying to the<br />
competition who buys the newspaper is ∙08.<br />
a) What is the probability of getting 65 or fewer entries?<br />
b) What is the probability of getting 85 or more entries?<br />
c) What is the probability of getting between 65 and 85 entries inclusive?<br />
7) The probability of apple in a batch being rotten is ∙02. If there are 10,000 in the batch,<br />
what is the expected number of rotten apples and what is the standard deviation?<br />
8) If 5% of a large consignment of apples are rotten and out of this consignment I select<br />
500 at random. What is the mean and standard deviation of the distribution of rotten<br />
apples in my sample?<br />
9) Among 10,000 random digits, find the probability that the digit 6 appears at most 950<br />
times.<br />
10) The weight of jam in a glass jar labelled 200g is normally distributed with mean 205g and<br />
standard deviation 10g. The weight of an empty glass jar is normally distributed with<br />
mean 250g and standard deviation of 5g. The weight of a glass jar is independent of the<br />
weight of the jam it contains. Find the probability that a randomly selected jar weighs<br />
less than or equal to 254 g and contains less than or equal to 202g jam.<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 38 of 66
Answers for Student Activity on the Normal Distribution and<br />
Probability<br />
2. If the sales of a DVD in a store follow the normal distribution with mean of 300 and<br />
standard distribution of 50<br />
a) What is the probability that the sales on a certain day will be less than or equal to<br />
320?<br />
320-300 20<br />
Z= = = 4<br />
50 50<br />
Pr( x 320) Pr(z 4) 6554<br />
b) What is the probability that the number of sales on a certain day will be greater than<br />
or equal to 320?<br />
Pr(x 320)=Pr(z 4) 1 Pr(z 4)=1 6554= 3446<br />
c) What is the probability that the number of sales on a certain day will be less than or<br />
equal to 280?<br />
280-300<br />
Pr(x 280)=Pr(z )=Pr(z 4)=1 Pr(z 4)<br />
50<br />
=16554= 3446<br />
d) What is the probability that the number of sales on a certain day will be between<br />
280 and 320 inclusive? From above:<br />
Pr(280 x 320)<br />
Pr( x 320) 6554<br />
Pr( x 280) 3446<br />
Pr(280 x 320)= 6554 3446 3108<br />
e) What is the probability that the number of sales on a certain day will be between<br />
320 and 340?<br />
Pr(x 320)= 6554<br />
340 300 40<br />
Pr( x 340) Pr( z ) Pr( z ) Pr( z 8) 7881<br />
50 50<br />
Pr(320 x 340) 78816554= 1327<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 39 of 66
11) A certain type of battery has a life span that follows the normal distribution curve with a<br />
mean life of 180 hours and a standard deviation of 20 hours.<br />
a) What is the probability that a battery of this type selected at random will last more<br />
than 230 hours?<br />
230 180 50<br />
Z 25 <br />
20 20<br />
Pr( x 230) Pr( z 25) 1 Pr( z 2.5) 19938 0062<br />
b) What is the probability that a battery bought form this batch will last between 160<br />
and 200 hours inclusive?<br />
160 180<br />
For x 160 z= 1<br />
20<br />
Pr( x 160) Pr( z 1) 1 Pr( z 1) 18413 1587<br />
For x 200<br />
200-180<br />
Z= =1<br />
20<br />
Pr(x 200)=Pr(z 1)= 8413<br />
The probability of the battery lasting between 200 and 160 hours is<br />
8413 1587= 6828 .<br />
c) If 80% of the batteries last less than or equal to x hours, find the value of x.<br />
z<br />
Pr( z<br />
z ) 8<br />
1<br />
1<br />
84 84<br />
<br />
84 20 180<br />
x<br />
x 1968<br />
x 180<br />
20<br />
d) If 10% of these batteries last more than x hours, find the value of x.<br />
10% lasting more than x hours is the same as saying 90% last x hours or less.<br />
Pr( z z ) 0.9<br />
z<br />
1<br />
1<br />
128<br />
x 180 z 1 28<br />
20<br />
z 205.6<br />
Hence 10% of the batteries last more than 205.6 hours.<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 40 of 66
12) The time taken to undertake a particular task follows the normal distribution with a<br />
mean of 20 minutes and a standard deviation of 5 minutes.<br />
a) Estimate the probability that it takes 26 minutes or longer to complete the task.<br />
26 20<br />
z 12 <br />
5<br />
Pr( x 26) Pr( z 12) 1 Pr( z 1.2) 18894 1151<br />
b) Estimate the probability that it takes less than or equal to 12 minutes.<br />
12 20 8<br />
z 16<br />
5 5<br />
Pr( x 12) Pr( z 16) 1 Pr( z 16) 19452 0548<br />
c) Estimate the probability that it takes between 17 and 24 minutes inclusive.<br />
For 24<br />
24 20<br />
z= 8<br />
5<br />
Pr( x 24) Pr( z 8) 7881<br />
For x 17<br />
17 20<br />
z= 6<br />
5<br />
Pr( z 6) 1 Pr( z 6) 17257 2743<br />
Pr(17 x 24)= 7881- 2743= 5138<br />
13) A fair die is thrown 200 times, what is the probability that a 5 turns up 36 times or<br />
fewer?<br />
n <br />
1 5<br />
200 Pr(of a 5)=p= q=Pr(of not a 5)=<br />
6 6<br />
1 5<br />
np 200 5 nq=200 5 Hence binomial distribution<br />
6 6<br />
will approximate to the normal distribution.<br />
1<br />
=200 3333<br />
6<br />
1 5<br />
200 27.778 5.27<br />
6 6<br />
36 3333<br />
z 50664<br />
527<br />
Pr( x 36) Pr( z 5066) 6950<br />
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14) A coin is tossed 200 times, what is the probability of getting heads fewer than 80<br />
times?<br />
1 1 1<br />
200( ) 100 = 200( )( ) 7 071<br />
2 2 2<br />
80 100<br />
z 2828<br />
7071<br />
Pr( z 2828) 1 Pr( z 2828) 199760 0024<br />
15) If a 1000 copies of a particular newspaper advertising a competition were sold on a<br />
particular day and it is estimated that the probability of someone replying to the<br />
competition who buys the newspaper is ∙08.<br />
a) What is the probability of getting 65 or fewer entries?<br />
p 08<br />
1000( 08) 80 = 1000( 08)( 92) 858<br />
65 80<br />
Z 17483<br />
858<br />
Pr( x 65) Pr( z 17483) 1 Pr( z 17483) 19599 04<br />
b) What is the probability of getting 85 or more entries?<br />
85 80<br />
z 5828<br />
858<br />
Pr( z 5828) 1 Pr( z 5828) 17190 281<br />
c) What is the probability of getting between 65 and 85 entries inclusive?<br />
Pr( x 85) Pr( x 65) 7190 04 679<br />
16) The probability of apple in a batch being rotten is ∙02. If there are 10,000 in the batch,<br />
what is the expected number of rotten apples and what is the standard deviation?<br />
Expected number =10,000(∙02) =200<br />
10000( 02)( 98) 14<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 42 of 66
17) If 5% of a large consignment of apples are rotten and out of this consignment I select<br />
500 at random. What is the mean and standard deviation of the distribution of rotten<br />
apples in my sample?<br />
p 05 q= 95<br />
Mean =500( 05)=25<br />
Standard Deviation = 500( 05)( 95) 4873<br />
18) Among 10,000 random digits, find the probability that the digit 6 appears at most 950<br />
times.<br />
p <br />
1 9<br />
q=<br />
10 10<br />
1 1 9<br />
10000( ) 1000 = 10000( )( ) 900 30<br />
10 10 10<br />
950 1000 50<br />
z 167<br />
30 30<br />
Pr( x 950) Pr( z 167) 1 Pr( z 1.67) 19525 00475<br />
19) The weight of jam in a glass jar labelled 200g is normally distributed with mean 205g and<br />
standard deviation 10g. The weight of an empty glass jar is normally distributed with<br />
mean 250g and standard deviation of 5g. The weight of a glass jar is independent of the<br />
weight of the jam it contains. Find the probability that a randomly selected jar weighs<br />
less than or equal to 254 g and contains less than or equal to 202g jam.<br />
For more questions see<br />
254 250<br />
Pr( glass 254) Pr( z ) Pr( z 08) 7881<br />
5<br />
202 205<br />
Pr( jam 202) Pr( z ) Pr( z 03) 16179 3821<br />
10<br />
Pr( glass 254) and Pr( jam 202) 7881( 3821) 3011<br />
1. Questions on the normal and binomial distribution<br />
http://www.ltcconline.net/greenl/courses/201/probdist/NormalAreaBinomial.htm<br />
2. http://www.ltcconline.net/greenl/Courses/201/probdist/normal_distribution_and_c<br />
ontrol_.htm<br />
3. http://business.clayton.edu/arjomand/business/l6.html<br />
4. www.examinations.ie previous Leaving Cert Honours Option questions<br />
5. Ideas for questions http://www.intmath.com/Counting-probability/14_Normalprobability-distribution.php<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 43 of 66
Appendix 7<br />
Student activity to establish the relationship between the mean of a<br />
population and the mean of the sample means and to investigate the<br />
distribution of sample means<br />
We are usually unable to collect information about a total population. The aim of sampling is to<br />
draw inferences about a population based on information obtained from a random sample taken<br />
from that population. We need to be able to interpret the information from the sample and state<br />
the degree of confidence we have in our results.<br />
Consider a population for which we can see the big picture i.e. a population of 100 fifteen year olds,<br />
whose heights in centimetres are given below in Figure 1.<br />
Figure 1 The heights of 100 fifteen year olds<br />
The mean of this population is = 164.7cm and the standard deviation is = 10.2.<br />
1) Write down the formulae for calculating mean and standard deviation.<br />
= _____________________<br />
=_____________________<br />
2) Verify the values given for the mean of the population and the standard deviation of the<br />
population using a calculator.<br />
= _____________________<br />
=_____________________<br />
The data in Figure 1 can be represented on a histogram with a class interval of 1, where the area of<br />
each rectangle represents the frequency of each data point and since the class interval is 1 the<br />
height represents the frequency.<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 44 of 66
Figure 2: Histogram for the heights of a population of 100 fifteen year olds<br />
10<br />
f<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
x<br />
0<br />
130 135 140 145 150 155 160 165 170 175 180 185 190 195 200<br />
Questions and activities on the histogram in Figure 2<br />
1) Draw the frequency polygon.<br />
2) Are the heights normally distributed?________<br />
3) Explain your answer_________________________________________________________<br />
4) Draw a vertical line on the histogram, which indicates the mean of the histogram for the<br />
population.<br />
5) In which interval will the median lie? __________<br />
6) What is the mode of the distribution? ___________<br />
This is the big picture of the population. If we collected data from a sample of this population,<br />
would the sample reflect the population? Would different samples have a histogram looking the<br />
same as the histogram of the population? Would the samples have the same mean as the<br />
population?<br />
In order to get answers to these questions we take a large number of different samples (e.g. 100)<br />
from the list of heights in the table in Figure 3.<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 45 of 66
The class divides into pairs.<br />
Each pair takes 10 random samples of 5 heights from the table of heights, Figure 3, which shows<br />
all the heights from Figure 1 but they are now numbered from 1 to 100.<br />
How to generate a random sample by using the random number generator<br />
RAND# on the calculator to generate random numbers from 1 to 100<br />
a. Shift, Setup, 2( LINEIO)<br />
b. Shift, Setup 6, Fix,0<br />
c. Shift, RAND#x100=<br />
d. Keep pressing = to generate random numbers from 1 to 100<br />
e. For each number generated record the item of data (height) associated with<br />
this number from Figure 3 into a table (Figure 4).<br />
f. Calculate the mean for each random sample of 5 heights.<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 46 of 66
Figure 3: The heights of 100 fifteen yr olds numbered from 1 to 100 each height numbered to facilitate random sampling.<br />
(If number 41 comes up on the calculator the height associated with it is 176 cm)<br />
h/cm h/cm h/cm h/cm h/cm h/cm h/cm h/cm h/cm h/cm<br />
1 165 11 161 21 170 31 182 41 176 51 185 61 180 71 155 81 154 91 166<br />
2 165 12 152 22 174 32 167 42 165 52 171 62 172 72 150 82 181 92 165<br />
3 166 13 161 23 174 33 158 43 166 53 168 63 164 73 150 83 155 93 170<br />
4 168 14 144 24 164 34 154 44 177 54 173 64 178 74 158 84 165 94 175<br />
5 180 15 174 25 152 35 167 45 148 55 175 65 153 75 162 85 180 95 175<br />
6 157 16 172 26 155 36 140 46 147 56 160 66 152 76 166 86 168 96 158<br />
7 153 17 165 27 160 37 143 47 166 57 167 67 167 77 163 87 158 97 160<br />
8 150 18 157 28 172 38 167 48 184 58 172 68 165 78 159 88 158 98 177<br />
9 179 19 174 29 156 39 178 49 165 59 179 69 174 79 148 89 175 99 166<br />
10 157 20 159 30 163 40 165 50 162 60 153 70 145 80 170 90 176 100 180<br />
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Figure 4: Table of 10 sets of 5 samples filled in by each pair of students<br />
Heights<br />
in<br />
Sample<br />
1<br />
Heights<br />
in<br />
Sample<br />
2<br />
Heights<br />
in<br />
Sample<br />
3<br />
Heights<br />
in<br />
Sample<br />
4<br />
heights<br />
in<br />
Sample<br />
5<br />
heights<br />
in<br />
Sample<br />
6<br />
heights<br />
in<br />
Sample<br />
7<br />
heights<br />
in<br />
Sample<br />
8<br />
heights<br />
in<br />
Sample<br />
9<br />
heights<br />
in<br />
Sample<br />
10<br />
Mean<br />
of each<br />
sample<br />
Using the 100 sample means collected from the full class the frequency table in Figure 5 can be filled<br />
in .The intervals used in the table are for one unit. For example, the interval labelled 165 includes all<br />
sample means x in the interval165 x<br />
166 .<br />
Figure 5 – Master table for Sample means of size n =5<br />
Class/cm Tally Frequency Class/cm Tally Frequency<br />
145 163<br />
146 164<br />
147 165<br />
148 166<br />
149 167<br />
150 168<br />
151 169<br />
152 170<br />
153 171<br />
154 172<br />
155 173<br />
156 174<br />
157 175<br />
158 176<br />
159 177<br />
160 178<br />
161 179<br />
162 180<br />
Sketch the histogram for the distribution of the 100 sample means of size n = 5 on figure 5.<br />
Figure 2 is shown first so that the distribution the heights can be compared with the distribution of<br />
the sample means of the heights.<br />
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Distribution of heights for the population of 100 fifteen year olds.<br />
Figure 5 Distribution of the sample means (sample size n = 5)<br />
10<br />
f<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
130 135 140 145 150 155 160 165 170 175 180 185 190 195 200<br />
x<br />
1) Compare the overall shape of the distribution of the sample means for n =5, with the shape of<br />
the distribution of the population heights.<br />
__________________________________________________________________________________<br />
________________________________________________________<br />
2) Calculate <br />
x<br />
(i) Compare the values of<br />
x<br />
<br />
x = ______________________<br />
and <br />
<br />
x = _______________ = ____________<br />
(ii) What do you notice about the mean of the sample means x<br />
and the mean of the<br />
population _______________________________________________________<br />
3) Calculate the standard deviation of the sample means.<br />
(i) Standard deviation of the sample means = _____________<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 49 of 66
(ii) How does this compare with the standard deviation for the population distribution?<br />
____________________________________________________________________<br />
Samples of size 5 are quite small but it would be very laborious to generate 100 random samples of a<br />
bigger size, for example 100 samples of size 20 each. However, the process would be the same as in<br />
the above exercise so we can use computer software to reduce the time factor as described in the<br />
following section.<br />
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Using computer generated samples to investigate a distribution of sample means<br />
A computer program e.g. Autograph or Excel can be used to take any number of samples of different<br />
sizes from the data for a population. One such program, Autograph, was used to generate 100<br />
different random samples of 20 students from the original population of 100 students and calculate<br />
the mean of each sample. The sample means are shown below taken from the Statistics box in<br />
Autograph.<br />
See “Using Autograph to investigate the distribution of Sample Means”<br />
Figure 6 Frequency Table for the above results (100 samples of size 20)<br />
Sample mean x for n = 20 Frequency Cumulative frequency<br />
160 3 3<br />
161 4 7<br />
162 8 15<br />
163 21 36<br />
164 14 50<br />
165 18 68<br />
166 11 79<br />
167 13 92<br />
168 4 96<br />
169 4 100<br />
Sketch the histogram for the distribution of the 100 sample means of size n = 20 on figure 6.<br />
Figure 7 Distribution of the sample means (sample size n = 20)<br />
10<br />
f<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
130 135 140 145 150 155 160 165 170 175 180 185 190 195 200<br />
x<br />
1) Compare the overall shape of the distribution of the sample means for n =20, with the shape<br />
of the distribution for n = 5 and with the distribution of the population heights.<br />
_____________________________________________________________________<br />
_____________________________________________________________________<br />
(ii) Compare the values of<br />
<br />
x and <br />
<br />
x = _______________ = _______<br />
What do you notice about the mean of the sample means x<br />
and the mean of the<br />
population ? __________________________________________<br />
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2) Calculate the standard deviation of the sample means.<br />
(i) Standard deviation of the sample means = _____________<br />
(ii) What is the approximate ratio of the standard deviation for the distribution<br />
of sample heights of size n = 20 heights compared with the standard<br />
deviation for the distribution of sample means of size n = 5?<br />
________________________________________<br />
3) What would you expect if 100 different samples of 30 heights were taken? What would you<br />
predict about<br />
(i) The shape of the histogram representing the distribution of the sample<br />
means _____________________________________________________<br />
(ii) The mean of the sample means _________________________<br />
(iii) The standard deviation of the sample means ____________ _______<br />
4) What effect is the size of the sample having on the mean of the sample means in relation to<br />
the mean of the population?<br />
__________________________________________________<br />
5) What effect is the size of the sample having on the standard deviation of the sample means?<br />
_____________________________________________________<br />
The standard deviation of the sample means can be estimated by the standard error of the<br />
mean <br />
x where<br />
x <br />
n<br />
6) Using the formula for standard error of the mean, calculate standard error of the mean for<br />
samples of size n = 40 and n = 50.<br />
for n 40 ______________ for n 50 ______________<br />
x<br />
x<br />
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We have seen that sample size affects the standard deviation (estimated by the standard<br />
error) of the distribution of sample means.<br />
Does the variation in standard deviation also depend on the number of samples taken?<br />
Graphs 1 to 6 show the distribution of sample means from a population using different sample<br />
sizes and different numbers of samples<br />
Graph 1: Histogram of the raw data from the population<br />
10<br />
f<br />
9<br />
8<br />
Graph 1: Graph of the raw data<br />
7<br />
6<br />
5<br />
N = 100<br />
Mean 164.72<br />
4<br />
3<br />
Standard deviation 10.21<br />
2<br />
1<br />
0<br />
135 145 155 165 175 185 195<br />
x<br />
Draft 01 © <strong>Project</strong> <strong>Maths</strong> Development Team Page 53 of 66
Graphs 2 – 4 vary the number of samples while keeping the sample size constant.<br />
(Note the change in scale on the y –axis as more samples are taken)<br />
Graph 2: Histogram of 100 sample means from samples of size 30<br />
30<br />
f<br />
Graph 2: Graph of sample means<br />
100 samples<br />
20<br />
n = 30<br />
Mean 164.7<br />
10<br />
Standard deviation 1.96<br />
0<br />
135 145 155 165 175 185 195<br />
x<br />
Graph 3: Histogram of 500 sample means from samples of size 30<br />
200<br />
f<br />
180<br />
160<br />
140<br />
Graph 3: Graph of sample means<br />
500 samples<br />
120<br />
100<br />
n = 30<br />
80<br />
60<br />
Mean 164.77<br />
40<br />
20<br />
Standard deviation 1.93<br />
x<br />
0<br />
135 145 155 165 175 185 195<br />
Graph 4 : Histogram of 1000 sample means from samples of size 30<br />
300<br />
f<br />
280<br />
260<br />
240<br />
Graph 4: Graph of sample means<br />
220<br />
200<br />
180<br />
1000 samples<br />
160<br />
140<br />
120<br />
100<br />
80<br />
60<br />
40<br />
20<br />
0<br />
135 145 155 165 175 185 195<br />
x<br />
n = 30<br />
Mean 164.8<br />
Standard deviation 1.92<br />
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Compare graph 5 with graph 3 – same number of samples but different sample sizes<br />
Graph 5 : Histogram of 500 sample means from samples of size 50<br />
200<br />
f<br />
180<br />
160<br />
140<br />
Graph 5: Graph of sample means<br />
500 samples<br />
120<br />
100<br />
80<br />
60<br />
40<br />
n = 50<br />
Mean 164.7<br />
Standard deviation 1.478<br />
20<br />
x<br />
0<br />
135 145 155 165 175 185 195<br />
Compare graph 6 with graph 4 – same number of samples but different sample sizes<br />
Graph 6 : Histogram of 500 sample means from samples of size 50<br />
300<br />
f<br />
280<br />
260<br />
Graph 6<br />
240<br />
220<br />
200<br />
180<br />
160<br />
140<br />
120<br />
1000 samples<br />
n = 50<br />
Mean 164.7<br />
100<br />
80<br />
60<br />
Standard deviation 1.47<br />
40<br />
20<br />
0<br />
135 145 155 165 175 185 195<br />
x<br />
The other five graphs show distributions of sample means from the population.<br />
Comment on the<br />
(i) shapes of all the graphs<br />
(ii) The means of the distributions of the sample means compared to the mean<br />
of the population<br />
(iii) The standard deviations of the sample means<br />
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Answers to questions on sample means<br />
N<br />
<br />
N<br />
<br />
2<br />
Xi<br />
( Xi<br />
)<br />
i1 i1<br />
1) = , where N = size of the population<br />
N<br />
N<br />
2) Use the calculator or Excel to find mean and standard deviation of the data.<br />
Questions and activities on the histogram in Figure 2<br />
1) Draw the frequency polygon.<br />
10<br />
f<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
x<br />
0<br />
130 135 140 145 150 155 160 165 170 175 180 185 190 195 200<br />
2) Are the heights normally distributed? No – but we are only looking at 100 heights<br />
3) The distribution is unimodal but is not symmetric about the mean<br />
4)<br />
5) In which interval will the median lie? 165 x<br />
166<br />
6) What is the mode of the distribution? 165<br />
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Questions on the distribution of sample means n = 5<br />
Below is one possible set of results for 100 samples of size 5.<br />
Sample mean x for n = 5 Frequency Cumulative frequency<br />
154 1 1<br />
155 0 1<br />
156 1 2<br />
157 4 6<br />
158 2 8<br />
159 4 12<br />
160 5 17<br />
161 9 26<br />
162 4 30<br />
163 9 39<br />
164 12 51<br />
165 8 59<br />
166 3 62<br />
167 8 70<br />
168 10 80<br />
169 5 85<br />
170 4 89<br />
171 7 96<br />
172 3 99<br />
173 0 99<br />
174 1 100<br />
Histogram showing the distribution of the above 100 samples of size 5<br />
20<br />
f<br />
15<br />
10<br />
5<br />
0<br />
130 135 140 145 150 155 160 165 170 175 180 185 190 195 200<br />
x<br />
1) The overall shape for the distribution of sample means is more clustered than the distribution of<br />
the heights for the population. Hence the standard deviation is less. The distribution is also<br />
closer to a normal distribution.<br />
2) The mean of the sample means x<br />
165.2 is very close to the mean of the population 164.7<br />
3) Standard deviation of the sample means <br />
x = 4.275, much smaller than , the population<br />
standard deviation<br />
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Answers to questions on the distribution of sample means<br />
Distribution of sample means of size n = 20<br />
1) This distribution of sample means of sample size 20 is approximately normal, with more<br />
clustering than the distribution for the population or for the distribution of sample means of<br />
size 5.<br />
2) x<br />
165 . This is approximately equal to the mean of the population 164.7<br />
3)<br />
(i)Standard deviation of the sample means x<br />
2.123<br />
(ii) The approximate ratio of the standard deviation for the distribution of sample heights of<br />
size n = 20 ( = 4.275) heights compared with the standard deviation for the distribution of<br />
x<br />
sample means of size n = 5( <br />
x<br />
2.123 ) is about 1:2. As n increased by 4, <br />
x decreased by<br />
about 2.<br />
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7) What would you expect if 100 different samples of 30 heights were taken? What would you<br />
predict about<br />
i. The histogram would show tighter clustering about the mean than for<br />
sample sizes of 5 or sample sizes of 20.<br />
ii. The mean of the sample means will be approximately the same as the<br />
population mean<br />
iii. The standard deviation of the sample means will be less than for the<br />
standard deviation for a distribution of sample means of size 20.<br />
8) The mean of the sample means is approximately the same as the population mean and does<br />
not depend on the size of the sample provided the sample is large enough (see Central Limit<br />
Theorem).<br />
9)<br />
10.72<br />
<br />
x<br />
1.69<br />
n 40<br />
Comments on the graphs 1-6<br />
(i) The first graph is not normally distributed but the graphs of the sample means are all roughly<br />
normal ( n 30 for all). The first graph of the population shows far more variation and has a<br />
higher standard deviation than the graphs of the distributions of the sample means.<br />
(ii) All the means of the distributions of the sample means are approximately equal to the<br />
population mean.<br />
(iii) Graphs 2, 3 and 4 have approximately the same standard deviation and they all have the same<br />
sample sizes.<br />
Graphs 5 and 6 are very similar. They are more clustered around the mean than graphs 2, 3 and<br />
4, have larger sample sizes and smaller standard deviations.<br />
Hence the standard deviation seems to depend on the size of the sample and not on the<br />
number of samples taken. Also as the sample size increases the standard deviation decreases.<br />
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Appendix 8<br />
Using Autograph to investigate the distribution of Sample Means<br />
Example: The data below shows the heights of a hundred 15-year-old students.<br />
1. Open an Autograph statistical page by clicking on the icon shown in Diagram 1.<br />
Statistic Icon<br />
Diagram 1<br />
2. From the Data dropdown menu select Enter Raw Data. Enter the data manually or paste from<br />
an Excel sheet.<br />
3. Group the data by pointing to Raw Data and right clicking. This will give you the option of<br />
grouping the data. Use class width = 1.<br />
4. Select the Histogram icon and the following will appear.<br />
5. Make sure that Frequency Density unit is 1. Click OK<br />
Change the axis by using Axis dropdown menu.<br />
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6. Select the Histogram by clicking on it. Once it is selected Right Click and select Sample<br />
Means from the menu. The following box will should appear.<br />
7. Click on Single Sample and see what happens to the<br />
Histogram. A sample of 5 will be selected which are shown<br />
by 5 arrows pointing at the axis. The mean of this sample<br />
of 5 is shown by a small square on the axis. See diagram below.<br />
Info on Population and<br />
Sample.<br />
Data point<br />
8. Click on Clear Samples and the sample and the mean will disappear. Type in 20 for Sample<br />
Size and 30 for the number of samples and click on Sample. The following will appear.<br />
Info on Population and<br />
Sample. What can be<br />
deduced?<br />
Distribution of Sample Means<br />
9. If we now take a Sample Size of 30 (n = 30) and 100 samples. What can be deduced? Refer to the<br />
Central Limit Theorem.<br />
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Appendix 9<br />
Student activity on confidence Intervals for the mean<br />
If we look at the distribution of 100 samples of 30 heights again, we see that it is approximately<br />
normal and would be perfectly normal if we took all possible samples of 30 heights.<br />
For the 100 samples of 30 heights, the population mean is 164.7(approximately equal to the<br />
mean of the sample means) and the standard error is 1.86.<br />
10.2<br />
x 1.86<br />
n 30<br />
30<br />
f<br />
20<br />
10<br />
0<br />
135 145 155 165 175 185 195<br />
x<br />
Listed below are the 100 samples of size 30 which gave rise to the above distribution<br />
164.1 168.3 163.7 165.1 164.1 163.1 163.7 163.7 164.3 165.3<br />
164.1 163.3 167.4 162.4 163.3 163.5 164.4 164.0 168.7 164.5<br />
165.2 165.7 165.0 164.1 166.2 163.9 161.2 161.4 162.6 163.9<br />
162.3 163.8 163.4 163.0 168.6 163.1 163.6 163.2 162.9 162.0<br />
164.3 164.1 162.1 163.9 166.1 164.0 166.8 162.1 162.8 165.2<br />
164.8 165.5 166.9 168.1 163.4 167.0 166.1 161.6 166.2 165.5<br />
165.8 164.4 166.3 162.9 160.1 160.4 165.8 165.6 165.5 164.1<br />
165.3 163.2 165.7 164.8 162.4 164.2 164.3 169.9 167.3 165.6<br />
166.8 163.7 166.2 166.2 165.0 169.6 167.9 167.7 166.4 164.9<br />
163.8 164.1 162.0 164.8 167.5 168.0 168.3 162.3 164.4 163.7<br />
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Answer the following questions , applying the characteristics of normal distributions:<br />
1) What percentage of samples should fall within one standard error of the population mean?<br />
_________<br />
2) What percentage of samples should fall within approximately two (1.96 to be more exact)<br />
standard errors of the of the population mean?<br />
_________<br />
3) What heights represent the interval within approximately two (1.96) standard errors of the<br />
population mean?<br />
___________________________________<br />
4) Using these values and the above 100 sample means, verify that the expected percentage of<br />
sample means falls within 1.96 standard errors of the population mean.<br />
5) What percentage of samples should fall within three standard errors of the of the population<br />
mean? _________<br />
6) For each of the following sample means, taken from the above table, calculate an interval within<br />
1.96 standard errors of the mean. Does this interval contain the population mean?<br />
(i) x1 160.4<br />
(ii) x1 165.5<br />
7) We often wish to estimate a population mean based on the mean of one sample. Establish the<br />
95% confidence interval for the population mean using a single random sample of 30 heights<br />
with x 162.0 .<br />
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Answers to questions on confidence intervals<br />
Q1 68%<br />
Q2 95%<br />
Q3. x 164.7 1.96(1.86) 168.34 and x 164.7 1.96(1.86) 161.05<br />
i.e. 161.05 x<br />
168.34<br />
Q4. 96% of the sample fall within 1.96 standard errors of the population mean which is reasonable<br />
as the distribution is not perfectly normal.<br />
Q5. 99.7%<br />
Q6. (i) 160.4+1.96(1.86) = 164.05 160.4 – 1.96(1.86) = 156.75<br />
The interval between 156.75 and 164.05 does not contain the population mean which is<br />
164.7.<br />
We expect 95% of samples to be within 1.96 x of the population mean and clearly not all<br />
samples have a mean within this interval.<br />
(ii) 165.5 +1.96(1.86) = 169.15 165.5-1.96(1.86) = 161.85<br />
The interval between 161.85 and 169.15 does contain the population mean.<br />
Q7. 162.0+1.96(1.86) =165.6 162.0-1.96(1.86) = 158.4<br />
We can say with 95% confidence that the interval from 158.4 to 165.6 contains the<br />
population mean.<br />
We expect that 95% of all sample means will be within 1.96 standard errors of the population<br />
mean.<br />
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Appendix 10<br />
Files from Student’s Disc JC and LC<br />
Student Disc Leaving Certificate Strand 1<br />
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Student Disc Junior Certificate Strand 1<br />
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