You observe an Olympic Athlete in the long-jump. Suppose the distance of each jump is a random variable that follows a normal distribution with mean ? and variance ?2. He jumps 8 times and records the following distances.

Jump Distance (meters)

1       8.79

2      8.87

3      8.82

4      8.49

5       8.74

6      8.32

7      8.84

8      8.50

The athlete says that he is better than Carl Lewis, who jumped an average distance of 8.50 throughout his career. Test that hypothesis through the following steps:

a.) Re-state the hypothesis in terms of what it says about the population parameters.

b.) Calculate the t-statistic for the hypothesis test

c.) Find the critical value and rejection region for a 5% significance level

d.) Given your answers to b.) and c.), what do you say to the athlete's statement that he is better than Carl Lewis?

2. A dictator wins an election with 75% of the votes. The opposition polls 100 people, only 45 of whom say they voted for the dictator.

a.) Test the hypothesis that the dictator lied (i.e. that he was really supported by 75% of the population), by following steps a.)-d.) of question 1.

b.) Test the hypothesis that the dictator should have lost the election (i.e. that he was really supported by 50% of the population) by following steps a.)-d.) of question 1.