One feature of the Mega Millions Lottery is that a "Mega" number is drawn to determine the winning lottery ticket. The Mega number is a value between 1 and 15 . If this lottery is fair, these values should occur with the same frequency. In other words, a " 3 " should come up just as often as a "7" or any other of the 15 values. Similarly, if we group the values into the categories "1-3", "4-6", "7-9", "10-12", and "13-15", each category should occur with the same frequency.
86 Mega Million Lottery's were observed to have the mega numbers appearing in those different categories as summarized in the table below.
Using a significance level of .01 , test the claim that the mega numbers appear with the same frequency.
Complete the table:
\begin{tabular}{|c|l|l|}
\hline Category & \begin{tabular}{l}
Observed \\
Frequency
\end{tabular} & \begin{tabular}{c}
Expected \\
Frequency
\end{tabular} \\
\hline $1-3$ & 23 & \\
\hline $4-6$ & 21 & \\
\hline $7-9$ & 6 & \\
\hline $10-12$ & 21 & \\
\hline $13-15$ & 15 & \\
\hline
\end{tabular}
Report all answers accurate to three decimal places. But retain unrounded numbers for future calculations.
What is the chi-square test-statistic for this data? (Report answer accurate to three decimal places, and remember to use the unrounded Pearson residuals in your calculations.)
\[
\chi^{2}=
\]
Answer
Finally, add up all the values to get the chi-square test statistic: \(\boxed{\chi^{2} = 11.212}\).
Step 1 ：First, calculate the expected frequency for each category. Since the lottery is fair, each category should occur with the same frequency. There are 5 categories and 86 observations, so the expected frequency for each category is \(\frac{86}{5} = 17.2\).
Step 2 ：Complete the table with the observed and expected frequencies for each category.
Step 3 ：Next, calculate the chi-square test statistic. The formula for the chi-square test statistic is \(\chi^{2} = \sum \frac{(O_i - E_i)^2}{E_i}\), where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency.
Step 4 ：Calculate the chi-square test statistic: \(\chi^{2} = \frac{(23 - 17.2)^2}{17.2} + \frac{(21 - 17.2)^2}{17.2} + \frac{(6 - 17.2)^2}{17.2} + \frac{(21 - 17.2)^2}{17.2} + \frac{(15 - 17.2)^2}{17.2}\).
Step 5 ：Simplify the chi-square test statistic: \(\chi^{2} = \frac{33.64}{17.2} + \frac{14.44}{17.2} + \frac{125.44}{17.2} + \frac{14.44}{17.2} + \frac{4.84}{17.2}\).
Step 6 ：Further simplify the chi-square test statistic: \(\chi^{2} = 1.956 + 0.839 + 7.297 + 0.839 + 0.281\).
Step 7 ：Finally, add up all the values to get the chi-square test statistic: \(\boxed{\chi^{2} = 11.212}\).
