AP Statistics

The Central Limit Theorem

1. A survey found that the American family generates an average of 17.2 pounds of glass garbage each year. Assume the standard deviation of the distribution is 2.5 pounds.
   1. Find the probability that the mean of a sample of 55 families will be between 17 and 18 pounds.

1. Why can the central limit theorem be applied?

1. The average teacher's salary in New Jersey is $52,174. Suppose that the distribution is normal with standard deviation $7500.
   1. What is the probability that a randomly selected teacher makes less than $50,000 per year?

1. If we sample 100 teachers' salaries, what is the probability that the sample mean is less than $50,000 per year?

1. Why is the probability in part (a) higher than the probability in part (b)?

1. Assume SAT scores are normally distributed with mean 1518 and standard deviation 325.
   1. If one SAT score is randomly selected, find the probability that it is between 1440 and 1480.

1. If 16 SAT scores are randomly selected, find the probability that they have a mean between 1440 and 1480.

1. Why can a normal distribution be used in part (b) even though the sample size does not exceed 30?

1. Carbon monoxide (CO) emissions for a certain kind of car are normally distributed with mean 2.9 gm/mi and standard deviation 0.4 gm/mi. A company has 80 of these cars in its fleet, acquired from various (i.e., random) sources.
   1. What's the probability that a randomly selected car from the fleet has CO emissions in excess of 3.1 gm/mi?

1. What's the probability that the average CO emissions for all 80 cars is in excess of 3.1?

1. There is only a 1% chance that the fleet's mean CO level is greater than what value?

1. Although most of us buy milk by the quart or gallon, farmers measure daily production in pounds. Ayrshire cows average 47 pounds of milk a day, with a standard deviation of 6 pounds. For Jersey cows, the mean daily production is 43 pounds, with a standard deviation of 5 pounds. Assume that Normal models describe milk production for these breeds.
   1. If we select an Ayrshire at random, what's the probability that she averages more than 50 pounds of milk a day?

1. A farmer has 20 Jerseys. What's the probability that the average production for this small herd exceeds 45 pounds of milk a day?

1. A farmer has 20 Ayrshires. There's a 99% chance each day that this small herd produces at least how many pounds of milk?