A Population Proportion · Exercise 42

A telephone poll of 1,000 adult Americans was reported in an issue of a national magazine. One of the questions asked, “What is the main problem facing the country?” Twenty percent responded "crime". We are interested in the population proportion of adult Americans who believe 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 percent confidence interval for the population proportion of adult Americans who believe 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 the group of researchers who conducted the poll is ±3 percent. In one to three complete sentences, explain what the ±3 percent represents.

(a) X = the number of adult Americans who believe that crime is the main problem; P′ = the proportion of adult Americans who believe that crime is the main problem.
(b) Because we are estimating a proportion, that P′ = 0.2 and n = 1,000, 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) Check student’s solution.
(c) EBM: 0.02
(d) One way to lower the sampling error is to increase the sample size.
(e) The stated ± 3 percent represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3 percent. Thus, they estimate the percentage of adult Americans who the percentage of adult Americans who that crime is the main problem to be between 18 percent and 22 percent.

Refer to. Another question in the poll asked, “[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 percent 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 the group of researchers who conducted the poll is ±3 percent. In one to three complete sentences, explain what the ±3 percent represents.

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

The error bound is approximately _____.

(a) 1.581
(b) 0.791
(c) 0.059
(d) 0.030

OpenStax, Statistics · §8.3 A Population Proportion · Exercise 42