You are conducting a multinomial Goodness of Fit hypothesis test for the claim that all 5 categories are equally likely to be selected. Complete the table. Report all answers correct to three decimal
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\begin{tabular}{|c|c|c|c|}
\hline Category & \begin{tabular}{c}
Observed \\
Frequency
\end{tabular} & \begin{tabular}{c}
Expected \\
Frequency
\end{tabular} & $\frac{(O-E)^{2}}{E}$ \\
\hline A & 11 & & \\
\hline B & 25 & \\
\hline C & 10 & \\
\hline D & 11 & \\
\hline E & 21 & \\
\hline
\end{tabular}
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What is the chi-square test-statistic for this data?
\[
\chi^{2}=
\]

For significance level alpha 0.005 , what is the chi-square Critical Value?
Critical Value $=$

What would be the conclusion of this hypothesis test?
Fail to reject the Null Hypothesis
Reject the Null Hypothesis

Step 1 ：Calculate the total observed frequency, which is the sum of the observed frequencies for all categories: \(11 (A) + 25 (B) + 10 (C) + 11 (D) + 21 (E) = 78\)

Step 2 ：Calculate the expected frequency for each category, which is the total observed frequency divided by the number of categories: \(78 / 5 = 15.6\)

Step 3 ：Fill in the expected frequencies in the table and calculate the chi-square statistic for each category: \(\frac{(O-E)^{2}}{E}\)

Step 4 ：Calculate the chi-square test statistic, which is the sum of the values in the last column: \(1.344 + 5.769 + 2.019 + 1.344 + 1.859 = 12.335\)

Step 5 ：Calculate the degrees of freedom for a goodness of fit test, which is the number of categories minus 1: \(5 - 1 = 4\)

Step 6 ：Find the chi-square critical value for a significance level of 0.005 and 4 degrees of freedom, which is approximately 14.860

Step 7 ：Since the chi-square test statistic (12.335) is less than the critical value (14.860), we fail to reject the null hypothesis. This means that the observed frequencies do not provide enough evidence to conclude that the categories are not equally likely to be selected. \(\boxed{\text{Fail to reject the null hypothesis}}\)