Question

Several factors are involved in the creation of a confidence interval. Among them are the sample​ size, the level of​ confidence, and the margin of error. Which statements are​ true?

Select all true statements below.

A.

For a certain confidence​ level, you can get a smaller margin of error by selecting a bigger sample.

B.

For a given sample​ size, reducing the margin of error will mean lower confidence.

C.

For a given confidence​ level, a sample 9 times as large will make a margin of error one third as big.

D.

For a fixed margin of​ error, smaller samples will mean lower confidence.

E.

None of these statements are true.

Answer 1

A.:   For a certain confidence​ level, you can get a smaller margin of error by selecting a bigger sample.

Answer: True

Justification: The margin of error is the range of value below and above the sample statistic in confidence interval.

When the sample size increases, we get more information about the population proportion. As a result our estimate becomes more accurate. The more accurate estimates implies narrower confidence interval. The margin of the error is dependent on confidence interval. We get smaller margin of error for narrower confidence interval.

Thus, you can get a smaller margin of error by selecting a bigger sample.

B. For a given sample​ size, reducing the margin of error will mean lower confidence.

Answer: True

Justification: As the margin of the error reduces, confidence interval becomes narrower and we get less possible values for population proportion. That means we become less confident ( lower confidence level).

Thus, reducing the margin of error will mean lower confidence.

C. For a given confidence​ level, a sample 9 times as large will make a margin of error one third as big.

Answer: True

Justification:

where   

p = proportion of successes. q = proportion of failures and n = sample size

When n becomes 9n

Thus, a sample 9 times as large will make a margin of error one third as big.

D. For a fixed margin of​ error, smaller samples will mean lower confidence.

Answer: True

Justification: As the sample becomes smaller, we get less information about the population proportion and as a result our estimate will be less accurate.

The less accurate estimate implies that we are less confident that the population proportion has certain margin of error. Thus, confidence level decreases.

Thus, smaller samples will mean lower confidence.