To determine whether James and his friends are indeed more skilled than the average gamer, we need to test whether their average time to reach level 10 differs significantly from the claimed average of 3 hours. We can do this using a one-sample z-test for the mean.
Here's the detailed step-by-step solution:
1. State the hypotheses:
- Null hypothesis ([tex]\( H_0 \)[/tex]): James and his friends' average time is equal to the claimed average time, i.e., [tex]\( \mu = 3 \)[/tex] hours.
- Alternative hypothesis ([tex]\( H_1 \)[/tex]): James and his friends' average time is less than the claimed average time, i.e., [tex]\( \mu < 3 \)[/tex] hours.
2. Given values:
- Claimed population mean ([tex]\( \mu \)[/tex]): 3 hours
- Population standard deviation ([tex]\( \sigma \)[/tex]): 0.4 hours
- Sample mean ([tex]\( \bar{x} \)[/tex]): 2.5 hours
- Sample size ([tex]\( n \)[/tex]): 5 (James and his four friends)
3. Calculate the standard error ([tex]\( SE \)[/tex]):
[tex]\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.4}{\sqrt{5}} \approx 0.179
\][/tex]
4. Calculate the z-score:
[tex]\[
z = \frac{\bar{x} - \mu}{SE} = \frac{2.5 - 3}{0.179} \approx -2.795
\][/tex]
5. Determine the critical z-values for the given significance levels:
In this context, the critical z-values for the left-tailed test at different significance levels ([tex]\( \alpha \)[/tex]) are given by:
- [tex]\( 5\% \)[/tex] significance level: [tex]\( z_{0.05} = -1.65 \)[/tex]
- [tex]\( 2.5\% \)[/tex] significance level: [tex]\( z_{0.025} = -1.96 \)[/tex]
- [tex]\( 1\% \)[/tex] significance level: [tex]\( z_{0.01} = -2.58 \)[/tex]
6. Compare the calculated z-score to the critical values:
- The calculated z-score is [tex]\( -2.795 \)[/tex].
- This z-score is less than all the critical values: -1.65, -1.96, and -2.58.
7. Determine the most restrictive level the claim is validated at:
- Since [tex]\( -2.795 \)[/tex] is less than the critical value at the [tex]\( 1\% \)[/tex] significance level (which is -2.58), the null hypothesis can be rejected at this most stringent level.
Therefore, the most restrictive level that would validate James' claim is [tex]\( 1\% \)[/tex]. This result implies that there is strong evidence to support the claim that James and his friends are more skilled than the average gamer, at the 1% significance level.
