A sign in the elevator of a college library indicates a limit of 16 persons. In addition, there is a weight limit
of 2,500 pounds. Assume that the average weight of students, faculty, and staff at this college is
155 pounds, that the standard deviation is 29 pounds, and that the distribution of weights of individuals on
campus is approximately normal. A random sample of 16 persons from the campus will be selected.
(a) What is the mean of the sampling distribution of (in pounds)?
(b) What is the standard deviation of the sampling distribution (in pounds)?
(c) What average weights (in pounds) for a sample of 16 people will result in the total weight exceeding
the weight limit of 2,500 pounds?
The average weight of 16 persons needs to be greater than _______ lbs to exceed the weight limit of the elevator
(d) What is the probability that a random sample of 16 people will exceed the weight limit? (Round your
answer to four decimal places.)
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(a) The mean of the sampling distribution of the average weight of 16 persons is equal to the mean weight of the population, which is given to be 155 pounds.
(b) The standard deviation of the sampling distribution is equal to the standard deviation of the population divided by the square root of the sample size. Therefore, the standard deviation of the sampling distribution is given by:
Standard deviation = 29 pounds / √16 = 7.25 pounds
(c) To find the average weight that will result in the total weight exceeding the weight limit of 2,500 pounds, we need to divide the weight limit by the sample size.
Weight limit for each person = 2,500 pounds / 16 = 156.25 pounds
Therefore, the average weight of 16 persons needs to be greater than 156.25 pounds to exceed the weight limit of the elevator.
(d) To find the probability that a random sample of 16 people will exceed the weight limit, we can use the z-score formula. First, we need to find the z-score for the weight limit.
z = (x - μ) / σ
Where x is the weight limit, μ is the mean weight of the population, and σ is the standard deviation of the population.
z = (156.25 - 155) / 7.25 = 0.1724
Using a standard normal distribution table or a calculator, we can find the probability corresponding to this z-score.
The probability that a random sample of 16 people will exceed the weight limit is approximately 0.4325 (rounded to four decimal places).
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