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From the image provided, it seems to be a hypothesis testing problem based on a test of sobriety with the following details:

### Part (a) - Hypotheses:
The hypotheses for the given test are as follows:
- Null Hypothesis (\(H_0\)): The mean (\(\mu\)) is equal to 350, i.e., \(H_0: \mu = 350\).
- Alternative Hypothesis (\(H_1\)): The mean (\(\mu\)) is different from 350, i.e., \(H_1: \mu \neq 350\) (this is a two-tailed test).

### Part (b) - Calculation of Test Statistic:
The values provided are:
- Sample mean (\(\bar{x}\)) = 341.0
- Population mean (\(\mu_0\)) = 350
- Standard deviation (\(s\)) = 31
- Sample size (\(n\)) = 50
- Significance level (\(\alpha\)) = 0.05

The test statistic \(t\) is calculated using the formula:
\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]

Substitute the given values:
\[
t = \frac{341.0 - 350}{31 / \sqrt{50}} = \frac{-9.0}{31 / 7.071} = \frac{-9.0}{4.382} \approx -2.054
\]

### Step (c) - Decision:
At a significance level of 0.05, the critical \(t\)-value for a two-tailed test with \(df = n-1 = 49\) can be found from a \(t\)-distribution table or software. The approximate critical values are \(t = \pm 2.0096\).

- If \(|t| > t_{\text{critical}}\), we reject the null hypothesis.
- In this case, \(|t| = 2.054\) is greater than \(t_{\text{critical}} = 2.0096\), so we **reject the null hypothesis**.

### Conclusion:
There is enough evidence to suggest that the mean test score for sobriety is significantly different from 350 at the 0.05 significance level.

Would you like further explanation or have any additional questions?

### Related Questions:
1. How does the critical value change with the significance level (e.g., 0.01)?
2. Can you explain why this is a two-tailed test and not one-tailed?
3. What are the implications of rejecting the null hypothesis in this context?
4. How would the conclusion change if the sample size were smaller?
5. How do we interpret the test statistic in terms of practical significance?

### Tip:
When performing hypothesis testing, always verify if the assumptions of the test (e.g., normality of data, independence) are met to ensure valid results.

A test of sobriety involves measuring the subject’s motor skills. Fifty randomly selected sober subjects took the test and produced a mean score of 341.0 with a standard deviation of 31. At the 0.05 significance level, test the claim that the true mean score for all sober subjects is different than 350.

Learn how to test a claim about the mean sobriety score using hypothesis testing at a 0.05 significance level. This problem involves calculating the test statistic, identifying the critical value, and making a decision based on a two-tailed t-test.

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