SPA 3e_ Teachers Edition _ Ch 6

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C H A P T E R 6 • Sampling Distributions
Alternate Example
Thick hair?
Mean and standard deviation of the
sampling distribution of x
PROBLEM: Suppose that the true mean
number of hair follicles on a human head
is 100,000 with a standard deviation of
40,000 follicles. The mean number of hair
follicles on the heads of 20 randomly
selected humans will be computed.
(a) Calculate the mean and standard
deviation of the sampling distribution of x.
(b) Interpret the standard deviation
from part (a).
SOLUTION:
(a) m x = 100,000 follicles and
s x = 40,000 = 8944 follicles
"20
(b) In SRSs of size n 5 20, the sample
mean number of hair follicles will typically
vary by about 8944 follicles from the
population mean of 100,000 follicles.
Activity Overview
Time: 15–18 minutes
Materials: An Internet-connected device
for each student or group of students
Teaching Advice: This activity helps
students understand the shape of the
sampling distribution of x. Although
the applet doesn’t have high-resolution
graphics, it is an excellent visual display
of key concepts in this lesson.
If you don’t have enough devices,
students can work in groups or you
can demonstrate the applet to the
entire class. Showing the applet as a
demonstration also saves time, although
it doesn’t engage students as much.
Even if students are doing the activity
individually, it is helpful to show them
the layout of the applet and demonstrate
taking a few samples. Point out that
sample size is denoted with a capital N,
instead of the usual lowercase n.
This applet gives a visual of the
population distribution (the top/first
number line), the distribution of one
sample (the second number line), and
the sampling distribution (the third and
fourth number lines). Point out these three
distributions. Note that this activity doesn’t
make use of the fourth number line.
There are two mysterious values
reported by the applet: skew and kurtosis.
Neither is important for this course.
a
e XAMPLe
Seen any good movies lately?
Mean and standard deviation of the sampling distribution of x
PROBLEM: The number of movies viewed in the last year by students at a large high school has
a mean of 19.3 movies with a standard deviation of 15.8 movies. Suppose we take an SRS of 100
students from this school and calculate the mean number of movies viewed by the members of
the sample.
(a) Calculate the mean and standard deviation of the sampling distribution of x.
(b) Interpret the standard deviation from part (a).
SOLUTION:
(a) m x = 19.3 movies and s x = 15.8 = 1.58 movies
"100
(b) In SRSs of size n 5 100, the sample mean number of
movies will typically vary by about 1.58 movies from
the population mean of 19.3 movies.
AcT iviT y
Shape
Sampling from a normal population
Professor David Lane of Rice University has
developed a wonderful applet for investigating
the sampling distribution of x. In this activity, you’ll
use Professor Lane’s applet to explore the shape of
the sampling distribution when the population is
normally distributed.
1. go to http://onlinestatbook.com/stat_sim/
sampling_dist/ or search for “online statbook
sampling distributions applet” and go to the website.
When the BEgIN button appears on the left
side of the screen, click on it. You will then see a
yellow page entitled “Sampling Distributions” like
the one in the screen shot.
2. There are choices for the population distribution:
normal, uniform, skewed, and custom. The
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Skewness is a measure of the skewness of the
distribution; kurtosis measures how light or
heavy the tails of the distribution are relative to a
normal distribution.
Answers:
1. Students should launch the applet.
2. The black boxes represent individuals
being randomly selected from the
population. The blue square represents the
sample mean of the sample on the second
number line.
3.
• The simulated sampling distribution has
an approximately normal shape.
Recall that m x = m and s x = s "n .
FOR PRACTICE TRY EXERCISE 1.
The shape of the sampling distribution of the sample mean x depends on the shape of
the population distribution. In the following activity, you will explore what happens
when sampling from a normal population.
default is normal. Click the “Animated” button.
What happens? Click the button several more
times. What do the black boxes represent? What
is the blue square that drops down onto the
plot below?
• The mean and median of the sampling
distribution are 16, just like the population.
(It is possible that some students will get
values slightly different from 16.)
• The standard deviation of the sampling
distribution is smaller than the standard
deviation of the population.
4. The sampling distribution of x for n 5 20
has the same shape and center, but the
variability is even less than the sampling
distribution for n 5 5.
5. The shape of the sampling distribution is
normal when the population distribution
has a normal shape.
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C H A P T E R 6 • Sampling Distributions
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