To determine the P-value range and decide whether to reject the null hypothesis ($H_0$), we need to compare our calculated chi-squared statistic to the critical values from the chi-squared distribution table. Given that we have 6 categories (colors), the degrees of freedom (df) is:

$df = \text{number of categories} - 1 = 6 - 1 = 5$

Using the chi-squared distribution table for 5 degrees of freedom, we compare our test statistic (17.471) to the critical values to find the P-value range.

Chi-squared critical values for 5 degrees of freedom are approximately:

• $\chi^2_{0.10} = 9.236$
• $\chi^2_{0.05} = 11.070$
• $\chi^2_{0.025} = 12.833$
• $\chi^2_{0.01} = 15.086$
• $\chi^2_{0.001} = 20.515$

Since our test statistic (17.471) is greater than 15.086 but less than 20.515, the P-value is:

$\text{P-value} < 0.01$

Thus, the range of P-values is:

• less than 0.01.

Part 5: Conclusion

Based on the calculated chi-squared statistic and the corresponding P-value range, we need to determine whether to reject $H_0$ (the null hypothesis that the colors follow the same distribution as stated by the manufacturer).

If the P-value is less than the significance level (usually 0.05), we reject $H_0$.

Since our P-value is less than 0.01, which is less than 0.05, we reject $H_0$.

Therefore, the correct conclusion is:

A. Reject $H_0$. There is sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.

Would you like more details or have any questions?

Here are 8 relative questions that expand on this information:

1. What is the significance level ($\alpha$) and how is it chosen in hypothesis testing?
2. How do you interpret a P-value in the context of hypothesis testing?
3. What are the consequences of Type I and Type II errors in hypothesis testing?
4. Can you explain the steps involved in conducting a chi-squared test for goodness of fit?
5. How do you handle multiple comparisons in chi-squared tests?
6. What are the alternative methods to chi-squared tests for goodness of fit?
7. How do you calculate the expected counts in a chi-squared test for different types of data?
8. What are the limitations of the chi-squared test, and how can they be addressed?

Tip: When reporting the results of a chi-squared test, always include the degrees of freedom and the P-value to provide context for the test statistic.