The range of​ P-values of the test is 

▼
 
between 0.025 and 0.05.

less than 0.01.

between 0.05 and 0.10.

between 0.01 and 0.025.

greater than 0.10.

Part 5

Based on the​ results, do the colors follow the same distribution as stated in the​ problem?

A.

Reject Upper H 0Reject H0. 

There is 

sufficientsufficient 

evidence that the distribution of colors is not the same as stated by the manufacturer.

B.

Reject Upper H 0Reject H0. 

There is 

not sufficientnot sufficient 

evidence that the distribution of colors is not the same as stated by the manufacturer.

C.

Do not reject Upper H 0Do not reject H0. 

There is 

not sufficientnot sufficient 

evidence that the distribution of colors is not the same as stated by the manufacturer.

D.

Do not reject Upper H 0Do not reject H0. 

There is 

sufficientsufficient 

evidence that the distribution of colors is not the same as stated by the manufacturer.




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.

Explore the concept of P-values and Chi-squared tests in hypothesis testing. Learn how to determine P-value ranges and interpret their significance. Includes a detailed explanation of rejecting or not rejecting the null hypothesis based on calculated statistics. Ideal for advanced undergraduate students and researchers.

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