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A popular video game claims that the average time needed to reach level 10 Paladin is 3 hours with a standard deviation of 0.4 hours. James thinks that he and his four friends are more skilled than the average gamer because it took them an average of only 2.5 hours. Which of the following is the most restrictive level that would validate his claim?

\begin{tabular}{|c|c|c|c|}
\hline \multicolumn{4}{|c|}{Upper-Tail Values} \\
\hline [tex]$a$[/tex] & [tex]$5 \%$[/tex] & [tex]$2.5 \%$[/tex] & [tex]$1 \%$[/tex] \\
\hline \begin{tabular}{c}
Critical \\
[tex]$z$[/tex]-values
\end{tabular} & 1.65 & 1.96 & 2.58 \\
\hline \hline
\end{tabular}

A. [tex]$1 \%$[/tex]

B. [tex]$2.5 \%$[/tex]

C. [tex]$5 \%$[/tex]

D. [tex]$10 \%$[/tex]

Sagot :

GinnyAnsweridn
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].
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