Question

Are you likely to purchase an item promoted by a celebrity on a social media​ site? According to a​ survey,

27​%

of social media users have made such a purchase. Complete parts​ (a) through​ (d) below.

a. Suppose that the survey had a sample size of

nequals=900.

Construct a

90 %90%

confidence interval estimate for the population proportion of social media users that have purchased an item promoted by a celebrity on a social media site.

____less than or equals≤piπless than or equals≤____

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

b. Based on​ (a), can you claim that more than a quarter of all social media users have purchased an item promoted by a celebrity on a social media​ site? A. ​Yes, with 95 % confidence because the sample proportion is greater than a​ quarter, and it falls within the limits of the confidence interval estimate. B. ​Yes, with 95 % confidence because all the values contained in the confidence interval are greater than 0.25. C. ​No, because the confidence interval contains proportion values that are less than 0.25. Your answer is correct.D. ​No, because a 95 % confidence interval is not indicative of a guarantee. A​ 99.9% confidence interval would be needed to make such a claim. c. Repeat parts​ (a) and​ (b), assuming that the survey had a sample size of nequals10 comma 000. Construct a 95 % confidence interval estimate for the population proportion of social media users that have purchased an item promoted by a celebrity on a social media site. 0.2613less than or equalspiless than or equals 0.2787 ​(Type an integer or a decimal. Round to four decimal places as​ needed.)

Based on the confidence interval created using a sample size of 10 comma 000​, can you claim that more than a quarter of all social media users have purchased an item promoted by a celebrity on a social media​ site? A. ​Yes, with 95 % confidence because the sample proportion is greater than a​ quarter, and it falls within the limits of the confidence interval estimate. B. ​Yes, with 95 % confidence because all the values contained in the confidence interval are greater than 0.25. Your answer is correct.C. ​No, because a 95 % confidence interval is not indicative of a guarantee. A​ 99.9% confidence interval would be needed to make such a claim. D. ​No, because the confidence interval contains proportion values that are less than 0.25.

d. Discuss the effect of sample size on confidence interval estimation. Choose the correct answer below. A. A larger sample size has no effect on a confidence​ interval, holding everything else constant. Your answer is not correct.B. A larger sample size results in a wider confidence​ interval, holding everything else constant. C. A larger sample size results in a narrower confidence​ interval, holding everything else constant. This is the correct answer.D. It is impossible to compare confidence interval estimates with difference sample sizes because the sample size is part of the formula for finding confidence interval estimate limits.

Answer 1

a)

Given,

p = 0.27

n = 900

Standard Error (SE) = sqrt(p*(1-p)/n) = 0.015

Alpha = 0.1

ZCritical = 1.64

90% CI = p +/- ZCritical * SE = 0.27 +/- 1.64*0.015 = {0.246,0.294}

b)

Option C.

No, because the confidence interval contains proportion values that are less than 0.25. Your answer is correct.

c)

Given,

p = 0.27

n = 10000

Standard Error (SE) = sqrt(p*(1-p)/n) = 0.004

Alpha = 0.05

ZCritical = 1.96

99% CI = p +/- ZCritical * SE = 0.27 +/- 2.58*0.004 = {0.261,0.279}

Option B

Yes, with 95 % confidence because all the values contained in the confidence interval are greater than 0.25. Your answer is correct.

d)

Option C

A larger sample size results in a narrower confidence​ interval, holding everything else constant. This is the correct answer.