https://file.aiquickdraw.com/mathbot/file/1728683072994ybzegpuc.jpg

Let's break down the steps to compute the test statistic based on the image provided.

### Problem Summary:
- Sample size \( n = 12 \)
- Sample standard deviation \( s = 52 \)
- Null hypothesis \( H_0: \sigma = 69 \)
- Alternative hypothesis \( H_1: \sigma < 69 \) (this is a **left-tailed** test)
- Significance level \( \alpha = 0.05 \)
- We are testing the population standard deviation (\( \sigma \))

This is a hypothesis test for the population standard deviation. The test statistic for this kind of problem follows a chi-square (\( \chi^2 \)) distribution and is calculated using the formula:

\[
\chi^2 = \frac{(n - 1) \cdot s^2}{\sigma_0^2}
\]

Where:
- \( n = 12 \) is the sample size.
- \( s = 52 \) is the sample standard deviation.
- \( \sigma_0 = 69 \) is the hypothesized population standard deviation.
- \( \sigma_0^2 = (69)^2 = 4761 \)

### Step-by-Step Solution:

1. **Compute \( s^2 \) (the sample variance):**
   \[
   s^2 = 52^2 = 2704
   \]

2. **Compute the test statistic \( \chi^2 \):**
   \[
   \chi^2 = \frac{(12 - 1) \cdot 2704}{4761} = \frac{11 \cdot 2704}{4761} = \frac{29744}{4761} \approx 6.245
   \]

So, the test statistic is approximately \( \chi^2 \approx 6.245 \).

Would you like me to explain how to interpret this test statistic or proceed to find the p-value?

### Related Questions:
1. What is the rejection region for this left-tailed chi-square test?
2. How do you find the p-value for a chi-square test?
3. What assumptions must be met for this chi-square test to be valid?
4. How does the sample size affect the outcome of this test?
5. How do you interpret the significance level \( \alpha = 0.05 \) in this context?

### Tip:
When dealing with chi-square tests for standard deviations, always check if the sample size is large enough and the population is normally distributed for accurate results.

Compute the test statistic for a left-tailed hypothesis test with a sample size of 12, a sample standard deviation of 52, and hypotheses H0: σ = 69 vs. H1: σ < 69 at a 0.05 significance level.

Learn how to perform a left-tailed hypothesis test for a population standard deviation with a sample of size 12 and a standard deviation of 52. This example uses a chi-square distribution to test the hypotheses H0: σ = 69 vs. H1: σ < 69 at a 0.05 significance level.

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