Download 1. A New York Times poll on women`s issues interviewed

Confidence Intervals
Chapter 8 Review
Name: _________________
1. A New York Times poll on women’s issues interviewed 1025 women randomly selected from the United
States, excluding Alaska and Hawaii. The poll found that 47% of the women said they do not get enough
time for themselves.
(a) Construct and interpret a 90% confidence interval that estimates the proportion of women in the
United States who do not feel that they get enough time for themselves. Use the Four-step process.
State: We wish to estimate, with 90% confidence, the true proportion of women in the continental
48 states who feel that they don’t get enough time for themselves.
Plan: We will use a one-sample z-interval for a population proportion.
Conditions: Random: The problem states that women were randomly selected.
Normal:
Independent: It seems reasonable to assume that individual observations are independent, and the
population is certainly more than 10 times the sample size
Do: 90% confidence interval is
Conclude (i.e., interpret the interval): We are 90% confident that the interval 0.444 to 0.496
contains the true proportion of women in the U.S. (excluding Alaska and Hawaii) who feel that they
don’t have enough time for themselves.
(b) Explain, in the context of this problem, what “90% confidence” means.
If a large number of confidence intervals were constructed in this way, about 90% of them would
contain the true proportion of women in the U.S. who feel that they don’t have enough time for
themselves.
(c) Suppose this poll was conducted by telephone calls made from 9 am to 5 pm. Explain how using this
method might result in biased results, and speculate about the direction of bias.
Women who work outside the home are unlikely to be home between 9 am and 5 pm, so they would
be under-represented in the sample. If we assume that these women are busier than those who do
not work outside the home—or at least are less likely to be alone at home for some part of the day—
our sample will underestimate the proportion of women who feel they don’t have enough time for
themselves.
2. You want to conduct a poll at your school to estimate with 95% confidence the proportion of students in
your school who have outside jobs in the evenings and on weekends. You’d like your margin of error to
be less than ±5%.
(a) How large must you sample be to produce a 95% confidence interval with the desired margin of error
of ±5%?
Using the conservative value pˆ = 0.5, we want a sample size n such that
Solving as an equality produces n = 384.16, so n must be 385 to ensure a margin of error below
0.05.
(b) How big does your school have to be for this interval to be accurate? Explain.
To satisfy the independence condition when sampling a finite population, the number of students
must be at least 10x385 = 3,850. A big school!
3. Below are the graphs of a standard Normal distribution and a t-distribution with 3 degrees of freedom.
(a) Indicated which graph is which
and explain how you know.
Dotted graph = standard Normal
Solid graph = t-distribution with df = 3.
(b) On the same figure sketch a graph of a t-distribution with 1 degree of freedom.
t-distribution with df = 1 should be drawn so that the peak is lower than other graphs and so that it
is above the other graphs at both tails.
4. Find the critical t* value for each of the following confidence intervals:
(a) 95% confidence interval with 8 degrees
of freedom.
t* for 95% confidence and df = 8 is 2.306.
(b) 80% confidence interval when n = 20
t* for 80% confidence and df = (n – 1) = 19 is
1.328.
5. You want to estimate the mean fuel efficiency of Ford Focus automobiles with 99% confidence and a
margin of error of no more than 1 mile per gallon. Preliminary data suggests that σ = 2.4 miles per gallon
is a reasonable estimate of the standard deviation for all cars of this make and model. How large a sample
do you need?
We want a sample size n such that
Solving for n produces n greater than 38.044 , so n should be 39.
6. National Fuelsaver Corporation manufactures the Platinum Gasaver, a device they claim “may increase
gas mileage by 30%.” Here are the percent changes in gas mileage for 15 identical, randomly-selected
vehicles, as presented in one of the company’s advertisements:
(a) The sample mean is
error of the mean for these data.
and the sample standard deviation is s = 16.23. Calculate the standard
(b) Construct and interpret a 90% confidence interval to estimate the mean change (in percent) in gas
mileage. Does the data support the company’s claim? Use the four-step process.
State: We wish to estimate, with 90% confidence, the “true” mean percentage change in gas mileage
when the Platinum Gasaver is used.
Plan: We will use a one-sample t-interval for a population mean.
Conditions:
Random: The study used “15 identical cars.” We will have to assume that these cars represented an
SRS of cars of that make and model.
Normal: Since n = 15, we should examine the data. The dotplot below shows some left skew, but no
outliers. We should be safe using t-procedures.
Independent: It seems reasonable to assume that individual mileage changes are independent, and
the number of cars of this make and model is certainly more than 10 times the sample size.\
Do: Critical t for 90% confidence and 14 df is 1.761, so a 90% confidence interval is
Conclude (i.e., interpret the interval): We are 90% confident that the interval 28.339 to 30.521
contains the true mean percentage change in gas mileage when the Platinum Gasaver is used. Since
30% is in this interval, the data supports the company’s claim. (But then again, the company
provided the data…).