From the image, we are solving a hypothesis testing problem with a Z-test. Here's a step-by-step breakdown:

Given Information:

1. Claim: The mean player rating $\mu$ is higher than 14.3.
2. Sample Size ($n$): 82 players.
3. Sample Mean ($\bar{x}$): 15.6.
4. Population Standard Deviation ($\sigma$): 4.
5. Significance Level ($\alpha$): 0.05.
6. Test Type: One-tailed test (as we are testing for "higher than").

Part (a): State the Null and Alternative Hypotheses

• Null Hypothesis ($H_0$): $\mu = 14.3$.
• Alternative Hypothesis ($H_a$): $\mu > 14.3$.

Part (b): Perform the Z-test and Find the p-value

The formula for the Z-test statistic is: $Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Step 1: Calculate the Z-statistic

$Z = \frac{15.6 - 14.3}{\frac{4}{\sqrt{82}}}$

1. Calculate the denominator: $\frac{4}{\sqrt{82}} = \frac{4}{9.0554} \approx 0.4417$

2. Calculate the Z-statistic: $Z = \frac{15.6 - 14.3}{0.4417} = \frac{1.3}{0.4417} \approx 2.943$

Step 2: Find the p-value

For a Z-statistic of 2.943 in a one-tailed test, use a Z-table or a statistical tool to find the p-value.

• From a Z-table:
  • The cumulative probability for $Z = 2.943$ is approximately 0.9984.
  • Since it's a one-tailed test, the p-value is: $p = 1 - 0.9984 = 0.0016$

Step 3: Compare the p-value to $\alpha$

• $p = 0.0016$
• $\alpha = 0.05$

Since $p < \alpha$, we reject the null hypothesis. There is enough evidence to support the claim that the mean player rating is higher than 14.3.

Would you like further details or explanations?

Related Questions

1. How do you calculate a p-value using software instead of a Z-table?
2. What is the effect of sample size on the Z-statistic and p-value?
3. How does a two-tailed test differ from a one-tailed test in hypothesis testing?
4. Can you explain how the level of significance $\alpha$ influences hypothesis test outcomes?
5. How is the decision rule $p < \alpha$ applied in real-world scenarios?

Tip

Always double-check your hypothesis test setup, especially the direction of the test (one-tailed vs. two-tailed), to avoid misinterpreting results.