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Chapter 8: Q.3.2 (page 494)

1. In the company鈥檚 prior-year survey, 80% of customers surveyed said they were 鈥渟atisfied鈥� or 鈥渧ery satisfied.鈥� Using this value as a guess for p藛, find the sample size needed for a margin of error of 3% at a 95% confidence level.

What if the company president demands 99% confidence instead? Determine how this would affect your answer to Question 1.

Short Answer

Expert verified

The required sample size is 1180.

From the calculations, we know that the increase in confidence level is leading to an increase in the sample size.

Step by step solution

01

Given Information

Given that

Population proportion(p^)=80%=0.80

Margin of error(E)=3%=0.03

Confidence level=95%

02

Explanation

From the standard normal table, thez-score at 99%the confidence level is 2.579

The sample size is calculated as:

n=(p^)(1-p^)zE2

=0.80(1-0.80)2.5760.032

=1179.537

鈮�1180

Thus, the required sample size is 1180.

From the calculations, we know that the increase in confidence level is leading to an increase in the sample size.

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Most popular questions from this chapter

In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met.

1. An AP Statistics class at a large high school conducts a survey. They ask the first 100 students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say "Yes."

2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt.

School vouchers A national opinion poll found that 44% of all American adults agree that parents should be given vouchers that are good for education at any public or private school of their choice. The result was based on a small sample.

(a) How large an SRS is required to obtain a margin of error of 0.03(that is, 卤3%) in a 99% con铿乨ence interval? Answer this question using the previous poll鈥檚 result as the guessed value for p鈭�.

(b) Answer the question in part (a) again, but this time use the conservative guess p鈭�=0.5. By how much do the two sample sizes differ?

Rating dorm food Latoya wants to estimate what proportion of the seniors at her high school like the cafeteria food. She interviews an SRS of 50of the 175 seniors living in the dormitory. She finds that 14 think the cafeteria food is good.

A radio talk show host with a large audience is interested in the proportion pof adults in his listening area who think the drinking age should be lowered to eighteen. To find this out, he poses the following question to his listeners: "Do you think that the drinking age should be reduced to eighteen in light of the fact that eighteen-year-olds are eligible for military service?" He asks listeners to phone in and vote "Yes" if they agree the drinking age should be lowered and "No" if not. Of the 100people who phoned in, 70answered "Yes." Which of the following conditions for inference about a proportion using a confidence interval are violated?

I. The data are a random sample from the population of interest.

II. nis so large that both np铃�andn1-p铃�are at least 10.

III. The population is at least 10times as large as the sample.

(a) I only

(c) III only

(c) I, II, and III

(b) II only

(d) I and II only

62. Blink When two lights close together blink alternately, we 鈥渟ee鈥� one light moving back and forth if the time between blinks is short. What is the
longest interval of time between blinks that preserves the illusion of motion? Ask subjects to turn a knob that slows the blinking until they 鈥渟ee鈥� two lights rather than one light moving. A report gives the results in the form 鈥渕ean plus or minus the standard error of the mean.鈥� Data for 12 subjects are summarized as 251卤45 (in milliseconds).
(a) Find the sample standard deviation sx for these measurements. Show your work.
(b) Explain why the interval 251卤45 is not a confidence interval.
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