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Chapter 7: Problem 109

A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard deviation of 25 pounds. If 35 of these people board the elevator, what is the probability that their combined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.

Short Answer

Expert verified
The probability that 35 people will exceed the 6000 pound limit is 0.0064 or 0.64%

Step by step solution

01

Calculate Mean and Standard Deviation

Since the people constitute a random sample of the population, we can calculate the mean and standard deviation of the total weight of 35 people. In this case, the mean \(\mu_T = n \times \mu = 35 \times 160 = 5600\) and the standard deviation \(\sigma_T = \sqrt{n} \times \sigma = \sqrt{35} \times 25 = 147.73\) where \(n\) is the sample size, \(\mu\) is the population mean and \(\sigma\) is the population standard deviation.
02

Calculate Z-Score

The z-score of the legal carrying capacity can be found using the formula \[Z = \frac{X - \mu_T}{\sigma_T}\] where \(X = 6000\), the legal carrying capacity. After substitution, \[Z = \frac{6000 - 5600}{147.73} = 2.7\].
03

Find Probability

As we want the values that exceed 6000 pounds, we need the area to the right of 2.7. Standard z-tables give the area to the left. Hence the required probability \(P(X > 6000) = 1 - P(Z < 2.7) = 1 - 0.9936 = 0.0064\) after looking up the z-score on the table. This indicates there is a 0.64% chance that the combined weight of 35 people will exceed 6000 pounds.

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