To determine whether James and his friends are significantly more skilled than the average gamer, we need to conduct a hypothesis test using the given data. We will perform the following steps:
1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The average time to reach level 10 Paladin for James and his friends is the same as the claimed average time, which is 3 hours.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The average time to reach level 10 Paladin for James and his friends is less than the claimed average time, implying they are more skilled.
2. Determine the level of significance and critical z-values:
- We are given several significance levels to consider: 5%, 2.5%, and 1%. The corresponding critical z-values for upper-tail tests are [tex]\(z = 1.65\)[/tex], [tex]\(z = 1.96\)[/tex], and [tex]\(z = 2.58\)[/tex], respectively.
3. Calculate the standard error of the mean (SEM):
- The standard error of the mean is calculated using the given standard deviation and the sample size ([tex]\(n = 5\)[/tex]):
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Given [tex]\(\sigma = 0.4\)[/tex]:
[tex]\[
\text{SEM} = \frac{0.4}{\sqrt{5}} \approx 0.17888543819998318
\][/tex]
4. Compute the z-score:
- The z-score indicates how many standard errors the sample mean (2.5 hours) is away from the claimed average time (3 hours):
[tex]\[
z = \frac{\text{Sample Mean} - \text{Claimed Mean}}{\text{SEM}}
\][/tex]
Substituting the values:
[tex]\[
z = \frac{2.5 - 3}{0.17888543819998318} \approx -2.7950849718747373
\][/tex]
Since we are performing a left-tail test (checking if the sample mean is less than the claimed mean), we take the absolute value of the z-score:
[tex]\[
|z| \approx 2.7950849718747373
\][/tex]
5. Compare the z-score with the critical z-values:
- The calculated z-score is compared against each of the critical values:
- For 5% significance level: [tex]\(z > 1.65\)[/tex]
- For 2.5% significance level: [tex]\(z > 1.96\)[/tex]
- For 1% significance level: [tex]\(z > 2.58\)[/tex]
6. Identify the most restrictive level:
- The z-score [tex]\(2.7950849718747373\)[/tex] is greater than all three critical values ([tex]\(1.65\)[/tex], [tex]\(1.96\)[/tex], and [tex]\(2.58\)[/tex]).
- Therefore, the most restrictive level that validates James’ claim is the 1% significance level.
Hence, the most restrictive level that would validate James’s claim that he and his friends are more skilled than the average gamer is [tex]\(1\%\)[/tex].
