Exercises - Central Limit Theorem

  1. Compare the probability distribution for rolling a single 6-sided die to the probability distribution for the mean of two 6-sided dice (draw the histograms).

  2. 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.

    2. Why can the central limit theorem be applied?

  3. 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?

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

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

  4. 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.

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

    3. Why can the central limit theorem be used in part (b) even though the sample size does not exceed 30?

  5. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.

    1. If one pregnant woman is randomly selected, find the probability that her length of pregnancy is less than 260 days.

    2. If 25 pregnant women are put on a special diet just before they become pregnant, find the probability that their lengths of pregnancy have a mean that is less than 260 days (assuming that the diet has no effect).

    3. If the 25 women do have a mean of less than 260 days, does it appear that the diet has an effect on the length of pregnancy, and should the medical supervisors be concerned?

  6. Assume that a test has a mean score of 75 and a standard deviation of 10. Assume the distribution of scores is approximately normal.

    1. What is the probability that a person chosen at random will make 100 or above on the test?

    2. What score should be used to identify the top 2.5%?

    3. In a group of 100 people, how many would you expect to score below 60?

    4. What is the probability that the mean of a group of 100 will score below 70?