Question

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?
upper-tail values
a
5%
2.5%
1%
critical z-values
1.65
1.96
2.58
1%
2.5%
5%
10%

Explanation:

Step1: Recall z - score formula for sample mean

The formula for the z - score of a sample mean \(\bar{x}\) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Here, \(\mu = 3\) hours, \(\sigma=0.4\) hours, \(\bar{x} = 2.5\) hours, and \(n = 5\) (James and his four friends).

Step2: Calculate the z - score

First, calculate the standard error \(\frac{\sigma}{\sqrt{n}}=\frac{0.4}{\sqrt{5}}\approx\frac{0.4}{2.236}\approx0.179\)

Then, calculate the z - score: \(z=\frac{2.5 - 3}{0.179}=\frac{- 0.5}{0.179}\approx - 2.79\) (we take the absolute value for one - tailed test, so \(|z|\approx2.79\))

Step3: Compare with critical z - values

The critical z - values for 5%, 2.5%, and 1% upper - tail are 1.65, 1.96, and 2.58 respectively.

Our calculated \(|z|\approx2.79\) is greater than the critical z - value for 1% (2.58), greater than 2.5% (1.96), and greater than 5% (1.65). The most restrictive level is the smallest significance level for which our test statistic is in the critical region. Since \(2.79>2.58\) (critical z for 1%), \(2.79 > 1.96\) (critical z for 2.5%), and \(2.79>1.65\) (critical z for 5%), the most restrictive level (smallest \(\alpha\)) that validates the claim is 1% because at 1% significance level, the test statistic is in the critical region.

Answer:

1%