Let’s break this problem down step-by-step:

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$:

• The null hypothesis $H_0$: The mean player rating $\mu$ is equal to or less than 78. Mathematically:
  $H_0: \mu \leq 78$

• The alternative hypothesis $H_1$: The mean player rating $\mu$ is greater than 78. Mathematically:
  $H_1: \mu > 78$

(b) Perform a Z-test and find the p-value:

Formula for Z-test:

$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Given values:

• Sample mean ($\bar{x}$) = 80.8
• Hypothesized mean ($\mu$) = 78
• Population standard deviation ($\sigma$) = 13.8
• Sample size ($n$) = 36

Step 1: Compute the Z-test statistic:

$Z = \frac{80.8 - 78}{13.8 / \sqrt{36}} = \frac{2.8}{13.8 / 6} = \frac{2.8}{2.3} \approx 1.217$

Step 2: Find the p-value:

The p-value is the area under the standard normal curve to the right of $Z = 1.217$.

Using a Z-table or calculator: $\text{p-value} \approx 0.1115$

(c) Conclusion at the 0.05 level of significance:

• The p-value (0.1115) is greater than the significance level (0.05).
• Therefore, we fail to reject the null hypothesis $H_0$.
• This means there is not enough evidence to support the claim that the mean player rating is higher than 78.

Correct answer:

"Since the p-value is greater than the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to support the claim that the mean player rating is higher than 78."

Summary of Results:

1. Null Hypothesis $H_0$: $\mu \leq 78$
2. Alternative Hypothesis $H_1$: $\mu > 78$
3. Test Statistic $Z$: 1.217
4. P-value: 0.1115
5. Conclusion: Fail to reject $H_0$.

Would you like a detailed explanation of any step? Let me know!

Here are 5 related questions to consider:

1. What happens to the Z-test result if the sample size increases significantly?
2. How does a one-tailed test differ from a two-tailed test?
3. How can the significance level ($\alpha$) affect the conclusion of a hypothesis test?
4. What is the relationship between confidence intervals and hypothesis testing?
5. How do you calculate the critical Z-value for a one-tailed test at a specific significance level?

Tip: Always check whether your test is one-tailed or two-tailed when interpreting p-values!