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Central Limit Theorem: Probabilities and Means of Normally Distributed Data, Assignments of Probability and Statistics

Walters State Community CollegeProbability and Statistics

Problems related to the central limit theorem, focusing on finding probabilities and means of normally distributed data. The problems involve calculating probabilities for individual data points and means of groups, given mean and standard deviation values. The central limit theorem states that the distribution of the means of large random samples from a population approaches a normal distribution, regardless of the shape of the population distribution.

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

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THE CENTRAL LIMIT THEOREM
1. For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 and a
standard deviation of 13.1
a. If a woman between the ages of 18 and 24 is randomly selected, find the probability that her
systolic blood pressure is above 120.
b. If 30 women in that age bracket are randomly selected, find the probability that their mean
systolic blood pressure is greater than 120.
c. If a group of 30 twenty-four year old women had a mean systolic blood pressure greater than 120,
what might we conclude.
2. The average number of pounds of meat a person consumes a year is 218.4 pounds. Assume that the
standard deviation is 25 pounds and the distribution is approximately normal.
a. If a person is selected at random, find the probability that a person consumes less than 224
pounds per year.
b. If a sample of 40 individuals is selected, find the probability that the mean of the sample will be
less than 224 pounds per year.
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THE CENTRAL LIMIT THEOREM

  1. For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 and a standard deviation of 13.

a. If a woman between the ages of 18 and 24 is randomly selected, find the probability that her systolic blood pressure is above 120.

b. If 30 women in that age bracket are randomly selected, find the probability that their mean systolic blood pressure is greater than 120.

c. If a group of 30 twenty-four year old women had a mean systolic blood pressure greater than 120, what might we conclude.

  1. The average number of pounds of meat a person consumes a year is 218.4 pounds. Assume that the standard deviation is 25 pounds and the distribution is approximately normal.

a. If a person is selected at random, find the probability that a person consumes less than 224 pounds per year.

b. If a sample of 40 individuals is selected, find the probability that the mean of the sample will be less than 224 pounds per year.

  1. The average amount of a purchase at a newsstand is $2.12. The standard deviation is $0.45. If 25 customers purchase items at the stand, find the probability that the mean of the purchases is greater than $2.30. Assume that the distribution is normal.
  2. The average time it takes a group of adults to complete a certain achievement test is 46.2 minutes. The standard deviation is 8 minutes. Assume that the distribution is normal.

a. Find the probability that a randomly selected adult will complete the test in less than 43 minutes.

b. Find the probability that if 50 randomly selected adults take the test, the mean time it takes the group to complete the test will be less than 43 minutes.

c. Does it seem reasonable that an adult would finish the test in less than 43 minutes?

d. Does it seem reasonable that the mean of 50 adults could be less than 43 minutes.