A Population Proportion · Exercise 40

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem.

(a) Define the random variables X and P′ in words.
(b) Which distribution should you use for this problem? Explain your choice.
(c) Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
(a) State the confidence interval.
(b) Sketch the graph.
(c) Calculate the error bound.
(d) Suppose we want to lower the sampling error. What is one way to accomplish that?
(e) The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.

(a) X = the number of adult Americans who feel that crime is the main problem; $p^{{}^{\prime }}$ = the proportion of adult Americans who feel that crime is the main problem
(b) Since we are estimating a proportion, given $p^{{}^{\prime }}$ = 0.2 and n = 1000, the distribution we should use is $N\left(0.2,\sqrt{\frac{\left(0.2\right)\left(0.8\right)}{1000}}\right)$.
(c) (a) CI: (0.18, 0.22)
(b) Answers may vary.
(c) EBM: 0.02
(d) One way to lower the sampling error is to increase the sample size.
(e) The stated “± 3%” represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3%. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%.

Refer to. Another question in the poll was “[How much are] you worried about the quality of education in our schools?” Sixty-three percent responded “a lot”. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

(a) Define the random variables X and P′ in words.
(b) Which distribution should you use for this problem? Explain your choice.
(c) Construct a 95% confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.
(a) State the confidence interval.
(b) Sketch the graph.
(c) Calculate the error bound.
(d) The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.

According to a Field Poll, 79% of Georgia adults (actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top issues facing the state. We wish to construct a 90% confidence interval for the true proportion of Georgia adults who feel that education and the schools is one of the top issues facing Georgia.

A point estimate for the true population proportion is:

(a) 0.90
(b) 1.27
(c) 0.79
(d) 400

OpenStax, Introductory Statistics 2e · §8.3 A Population Proportion · Exercise 40