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 variance2. 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

a. Calculate an unbiased estimate for the value of ?

b. Calculate an unbiased estimate for the value of ?2?

c. Construct an interval that has a 95% probability of containing ?.

The athlete says that he is better than Carl Lewis, who jumped an average distance of 8.50 throughout his career. What would you say to that? Suppose both of your guesses from parts a.) and b.) are exactly correct. What is the probability that the athlete's next jump is over the world record of 8.95 meters?