Question

PLEASE SHOW ALL WORK AND EXPLAIN

Four runners were randomly sampled and it was found they ran 11, 15, 11, 11 miles per week.

Assuming the population is normally distributed, if we wish to test the claim that the mean running

distance is more than 10 miles per week, what conclusion would you reach at the 5% level of

significance? Show all the steps of hypothesis testing. You must state your conclusion in terms of the

mean running distance of runners.

Answer 1

Sample observations = 11,15,11,11.

As, the parameters of the normal distribution are unknown we need to estimate them. Lets the samples are from .

Sample mean = = 12.

Sample size = n = 4

Sample variance = = 2.

Let us formulate the hypothesis :

The test statistic for this test is ,

which follow t distribution with 3 degrees of freedom.

Under ,

Now we calculate the p value , i.e , the probability of finding the observed values when the null hypothesis is true. Here, the p-value is

Clearly the p value is much higher than 0.05 (the significance level), hence we cannot reject the null hypothesis. Hence the above claim made in the problem is wrong.