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 (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 would be the same as the population mean, which is given as 155 pounds.
(b) The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:
Standard Error = (Standard Deviation of Population) / √(Sample Size)
Standard Error = 29 / √(16) = 29 / 4 = 7.25 pounds
(c) In order to exceed the weight limit of 2,500 pounds, the total weight of the sample of 16 people needs to be greater than 2,500 pounds. The average weight would be the total weight divided by the sample size:
Average Weight = Total Weight / Sample Size
Total Weight = Average Weight * Sample Size
Total Weight = X * 16
To exceed the weight limit of 2,500 pounds:
X * 16 > 2500
X > 2500 / 16
X > 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 calculate the probability that a random sample of 16 people will exceed the weight limit, we need to use the concept of the sampling distribution of the sample mean.
First, we need to calculate the z-score for the weight limit of 156.25 pounds using the formula:
Z = (X - μ) / σ
Z = (156.25 - 155) / (29 / √16)
Z = 1.25 / (29 / 4)
Z = 0.1724
Next, we need to find the probability that the z-score is greater than 0.1724. Using a standard normal distribution table or calculator, we find that the probability is 0.4325.
Therefore, the probability that a random sample of 16 people will exceed the weight limit is approximately 0.4325.
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