A popular video game claims that the average time needed to reach level 10 Paladin is 3 hours with a standard deviation 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$a$ & $5 \%$ & $2.5 \%$ & $1 \%$ \\
\hline \begin{tabular}{c}
Critical
$z$-values
\end{tabular} & 1.65 & 1.96 & 2.58 \\
\hline
\end{tabular}
A. 1%
B. 2.5%
C. 5%
D. 10%
Answer (2)
Calculate the z-score: z = 5 0.4 2.5 − 3 ≈ − 2.795 .
Compare the absolute z-score value (2.795) to the critical z-values for different significance levels.
Determine the most restrictive significance level where the absolute z-score is greater than the critical z-value.
The most restrictive level that validates James' claim is 1% .
Explanation
Understand the problem and provided data We are given a problem where James and his friends believe they are more skilled at a video game than the average player. To validate this claim, we need to determine the most restrictive significance level that supports their belief. The problem provides the average time to reach level 10, the standard deviation, the sample size (James and his friends), and their average time. We also have a table of upper-tail z-values for different significance levels.
Calculate the z-score First, we need to calculate the z-score to determine how many standard deviations James and his friends' average time is from the population mean. The formula for the z-score is: z = n σ x ˉ − μ where: x ˉ = sample mean (2.5 hours) μ = population mean (3 hours) σ = population standard deviation (0.4 hours) n = sample size (5)
Calculate the z-score value Plugging in the values, we get: z = 5 0.4 2.5 − 3 = 5 0.4 − 0.5 z = 0.4 − 0.5 × 5 z ≈ − 2.795
Compare the z-score with critical z-values Since James believes they are more skilled , we are looking for a lower-tail test. This means we want to find the significance level for which the calculated z-score is less than the negative of the critical z-value. We need to compare our calculated z-score (-2.795) with the negative of the critical z-values from the table.
The table provides upper-tail values, so we will compare the absolute value of our z-score with the critical z-values.
Critical z-values from the table:
5% significance level: 1.65
2.5% significance level: 1.96
1% significance level: 2.58
Determine the most restrictive significance level Now, we compare the absolute value of our z-score (|-2.795| = 2.795) with the critical z-values:
For a 5% significance level, the critical z-value is 1.65. Since 2.795 > 1.65, the result is significant at the 5% level.
For a 2.5% significance level, the critical z-value is 1.96. Since 2.795 > 1.96, the result is significant at the 2.5% level.
For a 1% significance level, the critical z-value is 2.58. Since 2.795 > 2.58, the result is significant at the 1% level.
Since James wants the most restrictive level, we choose the smallest significance level for which the result is still significant. In this case, it is the 1% level.
Final Answer Therefore, the most restrictive level that would validate James' claim is 1%.
Examples
In hypothesis testing, determining the significance level is crucial for validating claims. For instance, a company claiming a new drug is effective needs to choose a significance level to minimize the chance of falsely claiming effectiveness. Similarly, in quality control, manufacturers use significance levels to ensure product standards are met, balancing the risk of rejecting good products versus accepting faulty ones. Choosing a more restrictive level, like 1%, reduces the chance of a false positive but increases the risk of missing a real effect or issue.
James's average time of 2.5 hours is statistically significant when compared to the average of 3 hours. The most restrictive significance level that validates his claim is 1%. This is determined by comparing calculated z-scores against critical values at various levels of significance.
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