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.