Question

From a July 2019 survey of 889 randomly selected American households, it was discovered that 725 of them have an Amazon Prime membership.

a) Calculate the sample proportion of American households that have an Amazon Prime membership. Round this value to four decimal places.

b) Write one sentence each to check the three conditions of the Central Limit Theorem. Show your work for the mathematical check needed to show a large sample size was taken.

Answer 1

Solution:
Given:

Sample size = n = Number of American households  randomly selected = 889

x = Number of American households   have an Amazon Prime membership = 725

Part a) Calculate the sample proportion of American households that have an Amazon Prime membership.

Part b) Write one sentence each to check the three conditions of the Central Limit Theorem.

Central Limit Theorem for using Normal approximation to Proportion is:

For large sample size , sampling distribution of sample proportions is approximately Normal with mean of sample proportions is:

and standard deviation of sample proportions is:

if following conditions are satisfied:

and sample size should be no more than 10% of the population.

Thus

Number of successes =

Number of failures =

Since population of American households is very large , so that the sample size taken n= 889 is less than 10% of the population size.

Thus central limit theorem can be applied.