SPA 3e_ Teachers Edition _ Ch 6

Answers continued
X
18. (c) p^ =
n , so X = p^ n; (0.33)(1500) =
495 and (0.37)(1500) = 555
495 − 525
(d) z = = −1.62;
18.47
555 − 525
z = = 1.62
18.47
P(495 # X # 555)= P(−1.62 ≤ Z # 1.62)
= 0.9474 − 0.0526 = 0.8948
Using technology: Applet/normalcdf(lower:
495, upper:555, mean:525, SD:18.47)
5 0.8957. This answer matches the answer
from the example that uses the sampling
distribution of p^ , except for rounding.
19. (a) Shape: Both distributions are
skewed to the right. Center: Drivers
generally take longer to leave when
someone is waiting for the space. Spread:
There is more variability for the drivers
with someone waiting. Outliers: There
were no outliers for those with someone
waiting, but there were two high outliers
for those with no one waiting.
(b) Not necessarily; the researchers
merely observed what was happening,
and they did not randomly assign the
treatments of either having a person
waiting or not to the drivers of the cars
leaving the lot.
20. (a)
Relative frequency
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Key
Other
Hazel
Green
Brown
Blue
432
C H A P T E R 6 • Sampling Distributions
(a) Write a few sentences comparing these distributions.
(b) Can we conclude that a waiting car causes drivers to
leave their spaces more slowly? Why or why not?
20. Those baby blues (2.1, 4.4) The two-way table
summarizes information about eye color and gender
in a random sample of 200 high school students.
Gender
Male Female Total
Blue 21 29 50
Brown 35 40 75
Eye color green 14 21 35
Hazel 12 23 35
Other 2 3 5
Total 84 116 200
(a) Is there an association between eye color and gender
in this group of students? Support your answer
with an appropriate graphical summary of the data.
(b) Select a student at random. Are the events “Student
is male” and “Student has blue eyes” independent?
Justify your answer.
Lesson 6.5
The Sampling Distribution
of a Sample Mean
L e A r n i n g T A r g e T S
d
d
Find the mean and standard deviation of the sampling distribution of a sample
mean x and interpret the standard deviation.
Use a normal distribution to calculate probabilities involving x when sampling
from a normal population.
When sample data are categorical, we often use the count or proportion of successes
in the sample to make an inference about a population. When sample data are quantitative,
we often use the sample mean x to estimate the mean m of a population. When
we select random samples from a population, the value of x will vary from sample
to sample. To understand how much x varies from m and what values of x are likely
to happen by chance, we want to understand the sampling distribution of the sample
mean x.
Male
Gender
Female
Yes, because the percentages for a
given eye color are not the same for
each gender. In other words, knowing
a person’s gender helps us predict eye
color. Males are more likely than females
to have brown eyes, while females are
more likely than males to have hazel or
green eyes.
(b) P(blue eyes | male) 5 21/84 5 0.25;
P(blue eyes | female) 5 29/116 5 0.25.
Because the probabilities are equal,
the events “male” and “blue eyes” are
independent. Knowing that a student
is male does not change the probability
that he has blue eyes.
Starnes_3e_CH06_398-449_Final.indd 432
Learning Target Key
The problems in the test bank are
keyed to the learning targets using
these numbers:
d 6.5.1
d 6.5.2
BELL RINGER
Thinking back to the “A penny for your
thoughts?” activity for samples of
5 pennies, did every sample produce the
same sample mean year? What is the
name for this phenomenon?
Teaching Tip
Be picky with students about using the correct
symbols! The sample mean is denoted x and
the population mean is denoted m.
Common Error
This lesson and the next are about means, not
proportions. Make sure students don’t use the
symbols p and p^ when working with means.
18/08/16 5:02 PMStarnes_3e_CH0
432
C H A P T E R 6 • Sampling Distributions
Starnes_3e_ATE_CH06_398-449_v3.indd 432
11/01/17 3:56 PM