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Find the critical values for a
99% confidence interval using the chi-square distribution with 27 degrees of freedom. Round the answers to three decimal places.
The critical values are
◻ and
◻.

Find the critical values for a $99 \%$ confidence interval using the chi-square distribution with $27$ degrees of freedom. Round the answers to three decimal places. $\newline$ The critical values are $\square$ and $\square$ .

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Interpret confidence intervals for population means

Full solution

Q. Find the critical values for a

99 \%

confidence interval using the chi-square distribution with

27

degrees of freedom. Round the answers to three decimal places.

\newline

The critical values are

\square

and

\square

.

Understand the problem: Understand the problem. $\newline$ We need to find the critical values for a $99$ \% confidence interval using the chi-square distribution with $27$ degrees of freedom. This means we are looking for the chi-square values that correspond to the lower and upper tails of the distribution, leaving $1\%$ of the distribution outside the confidence interval ( $0.5\%$ in each tail since it's a two-tailed test).
Determine the alpha level: Determine the alpha level. $\newline$ Since we want a $99\%$ confidence interval, the alpha level ( $\alpha$ ) is $1\%$ or $0.01$ . This alpha level is split between the two tails of the chi-square distribution, so each tail will have an alpha level of $0.005$ .
Find lower tail value: Find the critical chi-square value for the lower tail. $\newline$ Using a chi-square distribution table or calculator, we look up the value that corresponds to an area of $0.005$ to the left (lower tail) for $27$ degrees of freedom. This value is the lower critical value.
Find upper tail value: Find the critical chi-square value for the upper tail. $\newline$ Similarly, we look up the value that corresponds to an area of $0.005$ to the right (upper tail) for $27$ degrees of freedom. This value is the upper critical value.
Use chi-square calculator: Use a chi-square distribution calculator or table. $\newline$ Since the exact values are not provided in this format, we would typically use a chi-square distribution calculator or table to find the critical values. For $27$ degrees of freedom and an alpha level of $0.005$ in each tail, the critical values are approximately: $\newline$ Lower critical value ( $\chi^2_{\text{lower}}$ ): $13.121$ $\newline$ Upper critical value ( $\chi^2_{\text{upper}}$ ): $44.314$ $\newline$ These values should be rounded to three decimal places as per the problem statement.
Round critical values: Round the critical values to three decimal places. $\newline$ Rounding the values obtained in Step $5$ , we get: $\newline$ Lower critical value ( $\chi^2_{\text{lower}}$ ): $13.121 \rightarrow 13.121$ (already at three decimal places) $\newline$ Upper critical value ( $\chi^2_{\text{upper}}$ ): $44.314 \rightarrow 44.314$ (already at three decimal places)

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