What is the value of the sample test statistic? (Round your answer to two decimal places.)

Question

The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, with σ = 8. A random sample of 96 matchboxes shows the average number of matches per box to be 42.9. Using a 1% level of significance, can you say that the average number of matches per box is more than 40?

Answer 1

The standard error of the mean in this case is σ/sqrt(n) = 8/sqrt(96) = .816. The meaning of the standard error in this context is that if we repeatedly took samples of size 96 from a population with a standard deviation of 8, we would expect the averages of these samples (i.e. the average number of matches per box of the first 96 matchboxes selected , average number of matches per box of the second 96 matchboxes selected, average number of matches per box of the 3rd 96 matchboxes selected, etc.) to have a standard deviation of .816.

Our null hypothesis is that mu (the true average number of matches per box) = 40. Our alternative hypothesis is that mu > 40. This is a one-sided Z test. Thus, we reject if we get a z-value such that P(Z > z) < .01.

Our Z value in this case is thus (Xbar - mu)/standard error where Xbar denotes the sample average of matches per box, or the average number of matches per box in the 96 matchboxes that we actually sampled and examined. Thus Z = (42.9-40)/.816 = 3.56. P(Z > 3.56) is well less than .01 we reject the null hypothesis and we can say at a 1% level of significance that the average number of matches per box is greater than 40.