SOLUTION: 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

Question 1071968: 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?
1%
2.5%
5%
10%
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
sample mean is 2.5 hours.
sample size is 4.
assumed population mean is 3 hours.
assumed population standard deviation is .4 hours.

z-score = (sample mean minus assumed population mean) / sample error.

sample error is equal to population standard deviation / square root of sample size.

sample error = .4/sqrt(4) = .2

sample z-score = (2.5 - 3.0) / .2 = -2.5

z-score of -2.5 has an alpha of .006 * 100 = .6%.

most restrictive level is 1%.

following two pictures show this relationship.

first is to determine the critical z-score at 1% alpha.

second is to determine the alpha of the z-score of the sample.

$$$

if the z-score of the sample is less than the critical z-score, than the test results are significant.

if the alpha of the sample is less than the critical alpha of 1%, than the test results are significant.

these two criteria go together.
either one is enough to determine that the test is statistically significant.

since the test alpha is less than 1%, the results of the test are significant.

the test would be significant for all of the alphas shown, but 1% is the most restrictive.

the z-score calculator i used can be found at:

http://davidmlane.com/hyperstat/z_table.html

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