Question

Which of the following statements is true regarding the confidence interval of a population mean?

a. The higher the confidence level required, the smaller the confidence interval reported.
b. The lower the confidence level required, the larger the confidence interval reported.
c. There is no relationship between the confidence level and the confidence interval reported.
d. The higher the confidence level required, the larger the confidence interval reported.

Answer 1

Answer to Question #53551 – Math – Statistics and Probability

Question

Which of the following statements is true regarding the confidence interval of a population mean?

a. The higher the confidence level required, the smaller the confidence interval reported.

b. The lower the confidence level required, the larger the confidence interval reported.

c. There is no relationship between the confidence level and the confidence interval reported.

d. The higher the confidence level required, the larger the confidence interval reported.

Solution

If $C$ percent confidence level is given and the poll of survey is repeated over and over again, the results would match the results from the actual population $C$ percent of time.

If the population standard deviation is known or there are enough observations ($n \geq 30$), then the formula for the confidence interval is

$$\bar{x} \pm Z \frac{s}{\sqrt{n}},$$

where

$\bar{x}$ is the mean, $Z$ is the $Z - value$, $s$ is the standard deviation, $n$ is the number of observations.

For a confidence level $90\%$ the corresponding $Z = 1.645$. For a larger confidence level $95\%$ the corresponding $Z = 1.960$ will be larger, hence the confidence interval will be larger.

Thus, the correct option is d) 'the higher the confidence level required, the larger the confidence interval reported'.

Answer: d.

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