10.3 Estimating a Population Proportion (pp.663-683) On time:
Finished:
1. In statistics, what is meant by a sample proportion:𝑝̂ ?
2. Give the mean and standard deviation for the sampling distribution of 𝑝̂ ?
3. How does the standard deviation differ to standard error for the sampling distribution of 𝑝̂ ?
4. How do you calculate the standard error of 𝑝̂ ?
5. What conditions must be met in order to use z procedures for inference about a proportion?
6. Describe how to construct a level C confidence interval for a population proportion.
7. What formula is used to determine the sample size necessary for a given margin of error?
For the following exercises (8-10) (a) describe the population of interest and explain in words what
the parameter of p is. (b) Give the numerical value of the statistic 𝑝̂ that estimates p. (c) Determine
whether each of the conditions is met for calculating a confidence interval for the population
proportion p.
8. Tonya wants to estimate what proportion of the students in her dormitory like the dorm food. She
interviews an SRS of 50 of the 175 students living in the dormitory. She finds that 14 think that the
dorm food is good.
9. Glenn wonders what proportion of the students at his school think that tuition is too high. He
interviews an SRS of 50 of the 2400 students at his college. Thirty-eight percent of those
interviewed think tuition is too high.
10. In the National AIDS Behavioral Surveys sample of 2673 adult heterosexuals, 0.2% had both
received a blood transfusion and had a sexual partner from a group at high risk of AIDS. We want to
estimate the proportion p in the population who share these two risk factors.
11. The National Survey of Student Engagement found that 87% of students reported that their peers
at least “sometimes” copy information from the Internet in their reports without citing the source.
Assume that the sample size is 430,000.
(a) Find the margin of error for 95% confidence.
(b) Here are some items from the report that summarize the survey. More than 430,000 students
from 730 four-year colleges and universities participated. The average response rate was 43% and
ranged from 15% to 89%. Institutions pay a participant fee of between $3000 and $7500 based on
the size of their undergraduate enrollment. Are these issues part of the margin of error you found in
part (a)? What impact might these issues have on the survey results?
12. In the Harvard School of Public Health survey, 2105 of the 10,904 respondents were classified
as abstainers (nondrinkers).
(a) Define the population and parameter of interest.
(b) Confirm that the conditions for constructing a confidence interval for the true population
proportion p are satisfied.
(c) Calculate a 99% confidence interval for p.
(d) Interpret the interval in the context of this problem.
13. Have efforts to promote equality for women gone far enough in the United States? A poll on the
issue by the cable network MSNBC contracted 1019 adults. A newspaper article about the poll said,
“Results have a margin of sampling error of plus or minus 3 percentage points.”
(a) Overall, 54% of the sample (550 of 1019) answered “Yes.” Construct and interpret a 95%
confidence interval for the proportion in the adult population who would say “Yes” if asked. Is the
report’s claim about the margin of error roughly correct? (Assume that the sample is an SRS.)
(b) The news article said that 65% of men, but only 43% of women, think that efforts to promote
equality have gone far enough. Explain why we do not have enough information to give confidence
intervals for men and women separately.
(c) Would a 95% confidence interval for women alone have a margin of error less than 0.03, about
equal to 0.03, or greater than 0.03? Why? You see that the news article’s statement about the
margin of error for poll results is a bit misleading.
14. You are planning a survey of students at a large university to determine what proportion favor an
increase in student fees to support an expansion of the student newspaper. Using records provided by
the registrar, you can select a random sample of students. You will ask each student in the sample
whether he or she is in favor of the proposed increase. Your budget will allow a sample of 100
students.
(a) For a sample of 100, construct a table of the margins of error for 95% confidence intervals when
𝑝̂ takes the values 0.l, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b) A former editor of the student newspaper offers to provide funds for a sample of size 500.
Repeat the margin of error calculations in (a) for the larger sample size. Then write a short thankyou note to the former editor describing how the larger sample size will improve the results of the
survey.
15. A national poll found that 44% of all American adults agree that parents should be given
vouchers good for education at any public or private school of their choice. The result was based on
a small sample. How large an SRS is required to obtain a margin of error of 0.03 (that is ±3%) in a
95% confidence interval?
(a) Answer this question using the previous poll’s results as the guessed value 𝑝∗ .
(b) Do the problem again using the conservative guess 𝑝∗ = 0.5. By how much do the two sample
sizes differ?
16. In the United States approximately 900 people die in bicycle accidents each year. One study
examined the records of 1711 bicyclists aged 15 or older who were fatally injured in bicycle
accidents were tested for alcohol. Of these, 542 tested positive for alcohol (blood alcohol
concentration of 0.01% or higher).
(a) Find a 99% confidence interval for p. Follow the Inference Toolbox.
(b) Can you conclude from your statistical analysis of this study that alcohol causes fatal bicycle
accidents? Explain.
17. PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The
ability to taste PTC is inherited. About 75% of Italians can taste PTC, for example. You want to
estimate the proportion of Americans with at least one Italian Grandparent who can taste PTC.
Starting with the 75% estimate for Italians, how large a sample must you test in order to estimate the
proportion of PTC tasters within ±0.04 with 95% confidence?