(4 pts.) 4. The popular sport of mountain biking involves riding bikes off-road, often over very rough terrain. Trek is a leading manufacturer of mountain bikes, specially designed for durability and performance. A random sample of 15 Trek mountain bikes was obtained and a sample mean of 10.67 kg was found with a sample standard deviation of 2.272 kg.

(2 pts.) a) Assume the underlying distribution is normal. Find and interpret a 99% confidence interval for the true mean weight of Trek mountain bikes.

(1 pts.) b) Suppose Trek claims to have the lightest mountain bikes on the market, with a mean weight of 10 kg. Using your answer in part (a), is there any evidence to suggest this claim is false? Justify your answer.

(1 pt.) c) Determine the number of bikes that need to be sampled to ensure that the half-width of the interval in a) is at most 0.75 kg.