Suppose a local manufacturing company claims their production line has a variance of at most 9. A quality control engineer decides to test this claim by sampling 23 parts. She finds that the standard deviation of the sample is 3.75.

Is this enough evidence at the 2.5% level of significance, to continue assuming the manufacturing companies claim is valid?Determine the p-value (round to 4 decimal places as needed) 

To test the company's claim that the variance of their production line is at most 9, we need to perform a hypothesis test for the population variance. We will use a chi-square (\(\chi^2\)) test for variance with the following hypotheses:

- Null Hypothesis (\(H_0\)): \(\sigma^2 \leq 9\) (The variance is at most 9)
- Alternative Hypothesis (\(H_1\)): \(\sigma^2 > 9\) (The variance is greater than 9)

### Step 1: Calculate the Test Statistic

The test statistic for a chi-square test for variance is given by:

\[
\chi^2 = \frac{(n - 1) \cdot s^2}{\sigma^2_0}
\]

where:
- \(n = 23\) is the sample size,
- \(s = 3.75\) is the sample standard deviation,
- \(\sigma^2_0 = 9\) is the hypothesized population variance.

First, let's calculate \(s^2\) (the sample variance):

\[
s^2 = (3.75)^2 = 14.0625
\]

Now, compute the test statistic:

\[
\chi^2 = \frac{(23 - 1) \cdot 14.0625}{9}
\]

### Step 2: Calculate the Test Statistic

Let's compute this value:

\[
\chi^2 = \frac{22 \cdot 14.0625}{9}
\]The calculated chi-square test statistic is:

\[
\chi^2 = 34.375
\]

### Step 3: Determine the P-Value

The degrees of freedom (df) for the chi-square test is \(n - 1 = 23 - 1 = 22\). We need to find the p-value corresponding to \(\chi^2 = 34.375\) with 22 degrees of freedom for a one-tailed test.

The p-value is the area under the chi-square distribution curve to the right of the test statistic.

Let's compute the p-value.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the p-value using a statistical calculator or software like Python, R, or any online chi-square distribution calculator with the following steps:

1. Use the calculated chi-square value: \(\chi^2 = 34.375\).
2. Use the degrees of freedom: \(df = 22\).
3. Look up the cumulative distribution function (CDF) value for the chi-square distribution with 22 degrees of freedom at \(\chi^2 = 34.375\).
4. Compute the p-value as \(1 - \text{CDF value}\).

Would you like to know more about these steps, or do you have any questions?

### Related Questions:
1. What is the chi-square distribution, and how is it used in statistics?
2. How do you interpret p-values in hypothesis testing?
3. How do you calculate critical values for chi-square tests?
4. What are some common applications of variance tests in quality control?
5. How do you determine the required sample size for variance tests?

**Tip:** Always check assumptions like normality of data when using chi-square tests for variance.

Suppose a local manufacturing company claims their production line has a variance of at most 9. A quality control engineer decides to test this claim by sampling 23 parts. She finds that the standard deviation of the sample is 3.75. Is this enough evidence at the 2.5% level of significance, to continue assuming the manufacturing companies claim is valid? Determine the p-value (round to 4 decimal places as needed).

Learn how to perform a chi-square test for variance to evaluate a manufacturing company's claim that their production variance is at most 9. The problem uses hypothesis testing at the 2.5% level of significance, applying the chi-square distribution with 22 degrees of freedom. Includes calculation of the chi-square statistic and determination of the p-value.

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