To help determine if James and his friends are more skilled on average than other gamers, we need to perform a hypothesis test for the mean. Specifically, we want to see if the difference in their average time (2.5 hours) is statistically significant compared to the population mean (3 hours) given the population's standard deviation (0.4 hours).
First, let's outline the steps for the hypothesis test:
1. State the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\):
- \(H_0: \mu = 3\) (the average time to reach level 10 Paladin is 3 hours)
- \(H_a: \mu < 3\) (the average time to reach level 10 Paladin is less than 3 hours, indicating greater skill)
2. Calculate the sample mean and standard deviation:
- Sample mean (\(\bar{x}\)): 2.5 hours
- Population mean (\(\mu\)): 3 hours
- Population standard deviation (\(\sigma\)): 0.4 hours
- Sample size (\(n\)): 5 (James and his four friends)
3. Compute the standard error of the mean (SEM):
- Standard error = \(\frac{\sigma}{\sqrt{n}} = \frac{0.4}{\sqrt{5}} \approx 0.17888543819998318\)
4. Calculate the z-score:
- \(z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{2.5 - 3}{0.17888543819998318} \approx -2.7950849718747373\)
5. Determine the significance level to validate James’s claim:
- Compare the computed z-score against the critical z-values for the given significance levels.
- For a 5% significance level, the critical z-value is 1.65.
- For a 2.5% significance level, the critical z-value is 1.96.
- For a 1% significance level, the critical z-value is 2.58.
Since we observe a negative z-score of -2.7950849718747373, we look at the magnitude of this z-score. This implies that we are more than \(2.58\) standard deviations away from the mean, but in the negative direction. For a result to be statistically significant and validate James' claim at:
- 5% significance level, z-score would need to exceed \( -1.65 \).
- 2.5% significance level, z-score would need to exceed \( -1.96 \).
- 1% significance level, z-score would need to exceed \( -2.58 \).
Since -2.7950849718747373 is less than all these thresholds, the most restrictive significance level that validates his claim is \(1 \%\).
So, we conclude:
James and his friends can statistically claim that they are more skilled than the average gamer at the [tex]\(1\%\)[/tex] significance level.
