Question

Check the requirements for confidence intervals using the following information:

Suppose we want to estimate the proportion of American teenagers who would rather own movies physically vs those who would rather own movies digitally. A survey of 45 American teenagers finds that 12 of them would rather own movies physically. With this information, you plan to create a 95% Confidence Interval for the true proportion of Americans who would rather own movies physically.

Conditions:

i. The method of sampling                            [ Select ]                       ["can be safely assumed to be Random.", "is not Random.", "is stated to be Random."]      

ii. The sample size is                            [ Select ]                       ["not large enough, since 12 ≤ 30.", "large enough, since 45 ≥ 30.", "not large enough, since 2.3467 ≤ 10.", "not large enough, since 8.8 ≤ 10."]      

iii. The sample is                            [ Select ]                       ["more than 5% of all American teenagers, therefore the sample can not be assumed to be independent.", "less than 5% of all American teenagers, therefore the sample can be assumed to be independent.", "more than 5% of all American teenagers, therefore the sample can be assumed to be independent..", "less than 5% of all American teenagers, therefore the sample can not be assumed to be independent."]      

Because of our answers from the three conditions above, we                            [ Select ]                       ["have not met the requirements to construct a 95% Confidence Interval.", "have met the requirements to construct a 95% Confidence Interval."]      

Answer 1

Based on the given data, it is known that:

Sample size (n) = 45 Let X = American teenager would rather own movies physically than digitally (Occurance of this event is considered success)

By definition of proportion, based on the sample data given,

Our objective is to ensure whether all the conditions for constructing a 95% CI for this proportion is met:

Conditions:

i. The method of sampling: Since, the only information given is that the data has been collected from a survey, but there is no clear evidence that the respondents were chosen in a random or non-random manner. Hence, a distinct conclusion on the type of sampling is not possible. Hence, it can only be assumed that the data is random. Ans. Can be safely assumed to be Random.

ii. The sample is said to be large enough (by Central limit theorem), if by a thumb of rule, . As mentioned at the start, hee, n = 45.Hence, the correct option would be:

The sample size is "large enough, since 45 ≥ 30."

iii. For satisfying the independence assumption, it must be ensured that the sample size is not more than 5% of the population.The correct option would be:

The sample is  "less than 5% of all American teenagers, therefore the sample can be assumed to be independent."

Because of our answers from the three conditions above, we "have met the requirements to construct a 95% Confidence Interval."