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In the television show Star Trek, different characters wear uniforms of different colors based on their positions or jobs. There are three colors: blue, red, and gold. A random sample of episodes was chosen and watched to determine how many characters were killed in each episode and which color uniform they were wearing. A claim was made that the color of the uniform is independent of the number killed. Use the StatCrunch output below to test the claim at a [tex]$5\%$[/tex] alpha level and select the appropriate summary sentence.

Options:

Contingency table results:
Rows: Uniform Color Columns: None
\begin{tabular}{|l|r|r|r|}
\hline & Dead & Alive & Total \\
\hline Blue & 7 & 129 & 136 \\
\hline Gold & 9 & 46 & 55 \\
\hline Red & 24 & 215 & 239 \\
\hline Total & 40 & 390 & 430 \\
\hline
\end{tabular}

Chi-Square test:
\begin{tabular}{|c|r|c|r|}
\hline Statistic & DF & Value & P-value \\
\hline Chi-square & 2 & 6.1886148 & 0.0453 \\
\hline
\end{tabular}

A. There is sufficient evidence to warrant rejection of the claim that the color of the uniform is independent of the number killed.

B. There is not sufficient evidence to warrant rejection of the claim that the color of the uniform is independent of the number killed.



Answer :

GinnyAnswer
To test the claim that the color of the uniform is independent of the number killed, we can use a chi-square test for independence. Here's a step-by-step solution to the problem:

1. Construct the Contingency Table:
We are given data categorized by uniform color and whether the character was dead or alive.
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Uniform Color} & \text{Dead} & \text{Alive} & \text{Total} \\ \hline \text{Blue} & 7 & 129 & 136 \\ \hline \text{Gold} & 9 & 46 & 55 \\ \hline \text{Red} & 24 & 215 & 239 \\ \hline \text{Total} & 40 & 390 & 430 \\ \hline \end{array} \][/tex]

2. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The color of the uniform and the number killed are independent.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The color of the uniform and the number killed are not independent.

3. Calculate the Chi-Square Statistic:
The chi-square statistic given is 6.1886148.

4. Degrees of Freedom:
The degrees of freedom ([tex]\(df\)[/tex]) for a chi-square test of independence is calculated as:
[tex]\[ df = (r-1) \times (c-1) \][/tex]
Where [tex]\(r\)[/tex] is the number of rows and [tex]\(c\)[/tex] is the number of columns. In this case:
[tex]\[ df = (3-1) \times (2-1) = 2 \][/tex]

5. Determine the P-Value:
The p-value associated with the chi-square statistic (6.1886148) and 2 degrees of freedom is 0.0453.

6. Compare the P-Value to the Significance Level:
We are using a significance level ([tex]\(\alpha\)[/tex]) of 0.05.
- If the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
In this case, 0.0453 < 0.05.

7. Make a Decision:
Since the p-value (0.0453) is less than the significance level (0.05), we reject the null hypothesis.

8. Conclusion:
There is sufficient evidence to warrant rejection of the claim that the color of the uniform is independent of the number killed. In other words, the data suggest that the color of the uniform is associated with the number of characters killed.

Therefore, the appropriate summary sentence is:
"There is sufficient evidence to warrant rejection of the claim that the color of the uniform is independent of the number killed."

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