Question

Suppose we are interested in inferring about the true mean height of highschool basketball players in Ontario. We are interested in determining if this average height has increased over the years. Twenty years ago the true average height was said to be 67 inches. A sample of 100 highschool basketball players in Ontario is taken and their average height is found to be

x=68.1 inches with a standard deviation of s=1 inch.
(a) Conduct an appropriate test at 5% level of significance to determine whether there is evidence that the true mean height of Ontario basketball players has increased over the years using the rejection region approach. Show all your steps including stating α, hypotheses, determining rejection region and a carefully worded conclusion.
(b) Suppose instead, at 5% level of significance, we had been interested in determining if the true mean height has simply changed (compared to 20 years ago). Furthermore suppose a sample, again of size 100 , was collected and the same sample mean and standard deviation were found. Use an appropriate confidence interval to make a conclusion regarding this hypothesis. 4

Answer 1

To determine whether there is evidence that the true mean height of Ontario basketball players has increased over the years, we conduct a hypothesis test. The null hypothesis is that the true mean height is not greater than 67 inches, and the alternative hypothesis is that the true mean height is greater than 67 inches. Using the rejection region approach at a 5% level of significance, the test statistic falls in the rejection region, leading to the rejection of the null hypothesis. This provides evidence to suggest that the true mean height of Ontario basketball players has increased over the years.

Step-by-step explanation:

To conduct the hypothesis test, we need to state the null and alternative hypotheses:

1. Null hypothesis (H0): The true mean height is not greater than 67 inches (μ ≤ 67)
2. Alternative hypothesis (Ha): The true mean height is greater than 67 inches (μ > 67)

Next, we determine the rejection region based on the level of significance α = 0.05. Since we are conducting a one-tailed test in the positive direction, the rejection region is to the right of the critical value.

Using the z-distribution, compute the test statistic:

z = (x - μ) / (s / √n) = (68.1 - 67) / (1 / √100) = 1.1 / (1 / 10) = 11

Since the test statistic falls in the rejection region (z > 1.645), we reject the null hypothesis. There is evidence to suggest that the true mean height of Ontario basketball players has increased over the years.