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Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value
t_(alpha//2), (b) find the critical value
z_(alpha//2), or (c) state that neither the normal distribution nor the
t distribution applies.
The confidence level is
99%,sigma=3262 thousand dollars, and the histogram of 59 player salaries (in thousands of dollars) of football players on a team is as shown.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Clear all

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value $t_{\alpha / 2}$ , (b) find the critical value $z_{\alpha / 2}$ , or (c) state that neither the normal distribution nor the $t$ distribution applies. $\newline$ The confidence level is $99 \%, \sigma=3262$ thousand dollars, and the histogram of $59$ player salaries (in thousands of dollars) of football players on a team is as shown. $\newline$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. $\newline$ Clear all

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Find values of normal variables

Full solution

Q. Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value

t_{\alpha / 2}

, (b) find the critical value

z_{\alpha / 2}

, or (c) state that neither the normal distribution nor the

t

distribution applies.

\newline

The confidence level is

99 \%, \sigma=3262

thousand dollars, and the histogram of

59

player salaries (in thousands of dollars) of football players on a team is as shown.

\newline

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\newline

Clear all

Determine Critical Value: To determine the critical value for constructing a confidence interval, we need to know whether the population standard deviation is known and whether the sample size is large enough to use the normal distribution. Since the population standard deviation is given $\sigma = 3262$ thousand dollars), and the sample size is $59$ , which is greater than $30$ , we can use the $z$ -distribution for constructing the confidence interval.
Confidence Level: The confidence level is $99\%$ , which means that the area in the tails ( $\alpha$ ) is $1\%$ or $0.01$ . Since the confidence interval is two-tailed, we need to find the critical value $z_{\alpha/2}$ that corresponds to the upper $0.5\%$ of the $z$ -distribution.
Find $z_{\alpha/2}$ : To find the critical value $z_{\alpha/2}$ , we can use a standard normal distribution table or a calculator with inverse normal distribution functions. We look up or calculate the z-score that corresponds to the cumulative area of $0.995$ (since $1 - 0.005 = 0.995$ ).
Calculate z-score: Using a standard normal distribution table or calculator, we find that the z-score that corresponds to a cumulative area of $0.995$ is approximately $2.576$ .
Final Critical Value: Therefore, the critical value $z_{\alpha/2}$ for a $99\%$ confidence interval is $2.576$ .

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