(b) What is the chi-square test-statistic for this data?
Test Statistic:
χ
2
=

(d) The p-value is...

greater than
α

less than (or equal to)
α

(e) The p-value leads to a decision to...

fail to reject the null
reject the null
(f) What is the final conclusion?

There is sufficient evidence to conclude sport preference is dependent on age.
There is not sufficient evidence to conclude sport preference is dependent on age.

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0.1547]

To solve this problem, we'll go through the steps required for a Chi-square test of independence. Here's what each step involves:

### (b) Chi-square Test Statistic Calculation
1. **Observed Frequencies (O)**: These are given in the contingency table of your data.
2. **Expected Frequencies (E)**: Calculated as:
   \[
   E = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}
   \]
3. **Chi-square Test Statistic (\(\chi^2\))**:
   \[
   \chi^2 = \sum \frac{(O - E)^2}{E}
   \]
   This is calculated by summing over all the cells of the contingency table.

To proceed, you'd need to have the observed data to calculate the expected frequencies and then compute the chi-square statistic.

### (d) P-value Comparison
- Once you calculate \(\chi^2\), you'll find the corresponding p-value from the chi-square distribution with \((\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)\) degrees of freedom.
- Compare this p-value with the significance level \(\alpha\):
  - If **p-value > \(\alpha\)**, the p-value is greater than \(\alpha\).
  - If **p-value \(\leq \alpha\)**, the p-value is less than or equal to \(\alpha\).

### (e) Decision Based on P-value
- **If p-value > \(\alpha\)**: Fail to reject the null hypothesis.
- **If p-value \(\leq \alpha\)**: Reject the null hypothesis.

### (f) Final Conclusion
- **Fail to reject the null hypothesis**: There is not sufficient evidence to conclude that sport preference is dependent on age.
- **Reject the null hypothesis**: There is sufficient evidence to conclude that sport preference is dependent on age.

### Example Walkthrough
1. **Calculate the test statistic \(\chi^2\)**: This requires the observed data, which you haven’t provided. 
2. **Find the p-value**: Use the \(\chi^2\) statistic and degrees of freedom to find the p-value.
3. **Make a decision**: Based on whether the p-value is greater or less than \(\alpha\).
4. **Draw a conclusion**: Based on whether the null hypothesis is rejected or not.

Do you have the observed frequency data available? This is necessary to calculate the chi-square test statistic and proceed with the p-value and final conclusion.

Would you like to proceed with details on calculating the expected frequencies or need any clarifications?

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### Related Questions:
1. How do you calculate expected frequencies in a chi-square test?
2. What is the null hypothesis in a chi-square test of independence?
3. How do degrees of freedom affect the chi-square test statistic?
4. How is the chi-square test used to determine independence between two variables?
5. What does a large chi-square statistic indicate about the data?
6. How do you interpret a p-value in the context of hypothesis testing?
7. What are the limitations of the chi-square test?
8. How does sample size affect the chi-square test result?

### Tip:
Always ensure your data meets the assumptions of the chi-square test: expected frequency in each cell should be at least 5 for the test to be valid.

Understand how to calculate the chi-square test statistic and interpret the p-value in hypothesis testing scenarios. This detailed guide covers steps to determine if there is sufficient evidence to conclude sport preference is dependent on age.

Math Problem Statement

(b) What is the chi-square test-statistic for this data? Test Statistic: χ 2

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