Question

16. Confidence intervals, again 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?

a) For a given sample size, reducing the margin of error will mean lower confidence.
b) For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample.
c) For a fixed margin of error, smaller samples will mean lower confidence.
d) For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

Answer 1

The creation of confidence intervals is influenced by sample size, confidence level, and margin of error. Statements b), c), and d) are true, confirming that larger sample sizes bring about smaller margins of error and vice versa, and the square of the sample size change affects the margin of error directly. Statement a), which suggests reducing the margin of error results in lower confidence for a fixed sample size, is generally false.

Step-by-step explanation:

Regarding the creation of confidence intervals, several statements can be assessed for their truthfulness based on the relationship between sample size, level of confidence, and margin of error.

• a) For a given sample size, reducing the margin of error will mean lower confidence. This statement is generally false. Reducing the margin of error usually means increasing the confidence level or the sample size, not decreasing the confidence.
• b) For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample. This statement is true. Increasing the sample size will typically reduce the margin of error when the confidence level is held constant.
• c) For a fixed margin of error, smaller samples will mean lower confidence. This statement is true. Smaller sample sizes generally lead to a lower confidence level if you wish to maintain the same margin of error.
• d) For a given confidence level, a sample 9 times as large will make a margin of error one third as big. This statement is true. The margin of error is inversely proportional to the square root of the sample size, so increasing the sample size by a factor of 9 will indeed reduce the margin of error by a factor of 3 (since the square root of 9 is 3).

In summary, the effect of changing sample size and confidence level on the margin of error is crucial in determining the accuracy and reliability of a confidence interval.

Answer 2

Step-by-step explanation:

Confidence interval is one which has centre as mean. The width of the interval would be 2 times margin of error.

Margin of error = Critical value * Std error/sq rt of sample

Hence if margin of error is lower for same n, then it means lower confidence level. Option a is right

Similarly, if bigger the sample size, lower is margin of error . Hence option b is right

c) False. Smaller samples make margin of error big and hence confidence intervals bigger.

d) If n becomes 9n, then margin of error becomes 1/3 Hence option d is right.