To determine the significance level that validates James' claim using hypothesis testing, we need to follow several systematic steps:
1. State the Hypotheses:
- Null Hypothesis \((H_0)\): The average time to reach level 10 Paladin for James and his friends is the same as for the average gamer, which is 3 hours.
- Alternative Hypothesis \((H_1)\): The average time to reach level 10 Paladin for James and his friends is less than the average gamer, indicating they are more skilled.
2. Given Data:
- Population mean \((\mu)\) = 3 hours
- Population standard deviation \((\sigma)\) = 0.4 hours
- Sample mean \((\bar{x})\) = 2.5 hours
- Number of friends (sample size, \(n\)) = 4 (including James)
3. Calculate the Standard Error (SE):
The Standard Error of the sample mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
SE = \frac{0.4}{\sqrt{4}} = \frac{0.4}{2} = 0.2
\][/tex]
4. Calculate the z-value:
The z-value is calculated using the formula:
[tex]\[
z = \frac{\mu - \bar{x}}{SE}
\][/tex]
Substituting the given values:
[tex]\[
z = \frac{3 - 2.5}{0.2} = \frac{0.5}{0.2} = 2.5
\][/tex]
5. Compare the z-value with Critical z-values:
We need to compare our calculated z-value with the critical z-values at different significance levels:
- For \(5\%\) significance level, the critical z-value is \(1.65\)
- For \(2.5\%\) significance level, the critical z-value is \(1.96\)
- For \(1\%\) significance level, the critical z-value is \(2.58\)
6. Determine the Most Restrictive Significance Level:
The calculated z-value is \(2.5\). We compare this with our critical z-values:
- \(2.5 > 1.65\) (for 5% significance level)
- \(2.5 > 1.96\) (for 2.5% significance level)
- \(2.5 < 2.58\) (for 1% significance level)
Therefore, James' claim that he and his friends are more skilled than the average gamer can be validated at the \(2.5\%\) significance level, which is the most restrictive level in this context.
Conclusion:
The most restrictive significance level that validates James' claim is [tex]\(2.5\%\)[/tex].
