SPA 3e_ Teachers Edition _ Ch 6

L E S S O N 6.2 • Sampling Distributions: Center and Variability 415
4. Is the IQR unbiased? To investigate if the sample
IQR is an unbiased estimator of the population
IQR, 1000 SRSs of size n 5 10 were selected from
the population described. The sample IQR for each
of these samples was recorded on the dotplot. The
mean of the simulated sampling distribution is indicated
by an orange line segment. Does the sample
IQR appear to be an unbiased estimator of the
population IQR? Explain your reasoning.
dd
0
d
d
d
d
d
d
20
d
d
d
d
d
dd
d
d
d d
d
40 60
Sample IQR
d d d
d
d
d d dd d d
80
100
5. More about medians Refer to Exercise 1.
(a) What would happen to the sampling distribution of
pg 412 the sample median if the sample size were n 5 50
instead? Justify.
(b) What is the practical consequence of this change in
sample size?
6. More about minimums Refer to Exercise 2.
(a) What would happen to the sampling distribution of
the sample minimum if the sample size were n 5 50
instead? Justify.
(b) What is the practical consequence of this change in
sample size?
7. More about ranges Refer to Exercise 3.
(a) What would happen to the sampling distribution
of the sample range if the sample size were n 5 5
instead? Justify.
(b) What is the practical consequence of this change in
sample size?
8. More about IQRs Refer to Exercise 4.
(a) What would happen to the sampling distribution
of the sample IQR if the sample size were n 5 5
instead? Justify.
(b) What is the practical consequence of this change in
sample size?
Applying the Concepts
9. Cholesterol in teens A study of the health of teenagers
plans to measure the blood cholesterol levels of
an SRS of 13- to 16-year-olds. The researchers will
report the mean x from their sample as an estimate
of the mean cholesterol level m in this population.
Explain to someone who knows little about statistics
what it means to say that x is an unbiased
estimator of m.
10. Predict the election A polling organization plans to
ask a random sample of likely voters who they will
vote for in an upcoming election. The researchers
will report the sample proportion p^ that favors the
incumbent as an estimate of the population proportion
p that favors the incumbent. Explain to
someone who knows little about statistics what it
means to say that p^ is an unbiased estimator of p.
11. Sampling more teens Refer to Exercise 9. The sample
mean x is an unbiased estimator of the population
mean m no matter what size SRS the study chooses.
Explain to someone who knows nothing about statistics
why a large random sample will give more reliable
results than a small random sample.
12. Sampling more voters Refer to Exercise 10. The
sample proportion p^ is an unbiased estimator of
the population proportion p no matter what size
random sample the polling organization chooses.
Explain to someone who knows nothing about statistics
why a large random sample will give more
trustworthy results than a small random sample.
13. Housing prices In a residential neighborhood, the
median value of a house is $200,000. For which of
the following sample sizes, n 5 10 or n 5 100, is
the sample median most likely to be greater than
$250,000? Explain.
14. Houses with basements In a particular city, 74%
of houses have basements. For which of the following
sample sizes, n 5 10 or n 5 100, is the sample
proportion of houses with a basement more likely
to be greater than 0.70? Explain.
15. Bias and variability The histograms show sampling
distributions for four different statistics intended to
estimate the same parameter.
(i)
(ii)
(iii)
(iv)
Population parameter
Population parameter
Population parameter
Population parameter
10. If we chose many random samples
and calculated the sample proportion
p^ for each sample, the distribution of p^
would be centered at the value of p. In
other words, when we use p^ to estimate
p, we will not consistently underestimate
p or consistently overestimate p.
11. A larger random sample will provide
more information about the population
and, therefore, more precise results.
The variability of the distribution of x
decreases as the sample size increases.
12. A larger random sample will provide
more information about the population
and, therefore, more precise results.
The variability of the distribution of p^
decreases as the sample size increases.
13. n 5 10; the sampling distribution
of the sample median will be more
variable with n 5 10 than with n 5 100.
Because the distribution is more variable,
it is more likely to get a sample median
($250,000) that is far away from the true
median ($200,000).
14. n 5 100; the sampling distribution
of the sample proportion will be less
variable with n 5 100 than with n 5 10.
Because the distribution is less variable,
it is less likely to get a sample proportion
that is far away (less than 0.70) from
the true proportion (0.74). This makes it
more likely for the sample proportion to
be above 0.70 with n 5 100.
Teaching Tip
Don’t skip Exercise 15! It’s a wonderful
way to assess the two learning targets
from this Lesson. Consider having a
short class discussion on it after students
have had a chance to try the exercise for
themselves.
Lesson 6.2
18/08/16 5:00 PMStarnes_3e_CH06_398-449_Final.indd 415
4. Yes; the mean of the sampling distribution is
very close to 27.64, the value of the population
IQR. Population IQR 5 Q 3 2 Q 1 5 39.93 2 12.29
5 27.64.
5. (a) It will be less variable because the
sample size is larger.
(b) The estimated median amount spent will
typically be closer to the true median amount
spent. In other words, the estimate will be
more precise.
6. (a) It will be less variable because the
sample size is larger.
(b) The estimated minimum amount spent
will typically be closer to the true minimum
amount spent. In other words, the estimate
will be more precise.
18/08/16 5:00 PM
7. (a) It will be more variable because the
sample size is smaller.
(b) The estimated range amount spent
will typically be farther from the true range
amount spent. In other words, the estimate
will be less precise.
8. (a) It will be more variable because the
sample size is smaller.
(b) The estimated IQR amount spent will typically
be farther from the true IQR amount spent.
In other words, the estimate will be less precise.
9. If we chose many SRSs and calculated the
sample mean x for each sample, the distribution
of x would be centered at the value of m. In
other words, when we use x to estimate m,
we will not consistently underestimate m or
consistently overestimate m.
L E S S O N 6.2 • Sampling Distributions: Center and Variability 415
Starnes_3e_ATE_CH06_398-449_v3.indd 415
11/01/17 3:54 PM