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 nequals800. 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. nothingless than or equalspiless than or equals nothing ​(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?

c. Repeat parts​ (a) and​ (b), assuming that the survey had a sample size of n= blank

Discuss the effect of sample size on the confidence interval estimate.

Answer 1

a.)

Given Sample size, n = 800

Propotion who made purchase, p = 27% = 0.27

Formula to calculate 95% confidence interval( = 0.05) for population proportion is given as:

CI = p Z_(0.05.2) *

Z_(0.05.2) = 1.96

CI = 0.27 1.96*

CI = 0.27 1.96 *0.015694

CI = 0.27 1.96* 0.0157

CI = 0.27 0.03077

CI = (0.2392, 0.3007)

CI = (23.92%, 30.07%)

b.) Since above Confidence interval lower limit is less than 25%, So it contains value less than 25% as well. We can't say that quarter of social media users have purchased an item promoted by celebrity

c.) Here new sample size is not provided, so i am assuming it to be 10000

So, n = 10000

Now solving a) again

CI = p Z_(0.05.2) *

Z_(0.05.2) = 1.96

CI = 0.27 1.96*

CI = 0.27 1.96 *0.00443

CI = 0.27 1.96* 0.0044

CI = 0.27 0.0087

CI = (0.2613, 0.2787)

CI = (26.13%, 27.87%)

Solution for part b) with new sample size, n = 10000 is:

Since above Confidence interval lower limit is more than 25%, So it always contains value more than 25%. We can say that quarter of social media users have purchased an item promoted by celebrity.

Impact of Sample Size on Confidence Interval

From above results, it's clear that as we increase Sample size, range of Confidence Interval decreases. Upper and lower limits value shrink after increase in sample size, and move close to actual proportion value of population.