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A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was 鈥淲hat is the main problem facing the country?鈥� Twenty per cent answered 鈥渃rime.鈥� 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鈥� 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.

i. State the confidence interval.

ii. Sketch the graph.

iii. 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 卤3%. In one to three complete sentences, explain what the 卤3% represents.

Step by step solution

01

part (a) explanation

$X$represents the number of American adults who feel that crime is a major problem and$P\text{'}$represents the proportion of American adults who feel that crime is a major problem.

02

part (b) explanation

In estimating proportions we use,

$p^{\mathrm{鈥�}}=0.2$

$n=1000$

$鈬�N\left(0.2,\sqrt{\frac{0.2脳0.8}{1000}}\right)$

03

part (c) explanation

(i) American adults who feel crime is a major problem,

$p脳n=0.2脳1000=200$

now, $x=200$and$n=1000$

Confidence level =$95\%$

Confidence Interval =$(0.17521,0.22479)$

(ii) The graph is shown,

(iii) Error Bound is

$EB=\left(\frac{\text{Upper limit-lower limit}}{2}\right)$

$EB=\left(\frac{0.22479鈭�0.17521}{2}\right)$

Error Bound$=0.002479$

04

part (d) explanation

By extending the sample size we lower the sampling error.

05

part (e) explanation

The maximum error bound is represented by$卤3\mathrm{\%}$.

This means that while surveying the maximum error that can occur is$3\%$.

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