Question

The elevator in a car rental building has a placard stating that the maximum capacity "4000 lb or 27 passengers." Because 4000/27 = 148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 "Body Data" in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb. a.) Find the probability that 1 randomly selected adult male has a weight greater than 148 lb. b.) Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.

Answer 1

Answer:

a) 85.31% probability that 1 randomly selected adult male has a weight greater than 148 lb.

b) 100% probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that

$\mu = 189, \sigma = 39$

a.) Find the probability that 1 randomly selected adult male has a weight greater than 148 lb.

This is 1 subtracted by the pvalue of Z when X = 148.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{148 - 189}{39}$

$Z = -1.05$

$Z = -1.05$ has a pvalue of 0.1469

1 - 0.1469 = 0.8531

85.31% probability that 1 randomly selected adult male has a weight greater than 148 lb.

b.) Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.

Now $n = 27, s = \frac{39}{\sqrt{27}} = 7.51$

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{148 - 189}{7.51}$

$Z = -5.46$

$Z = -4.56$ has a pvalue of 0

1 - 0 = 1

100% probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.