Question

In engineering and product design, it is important to consider the weights of people so that airplanes or elevators aren't overloaded. Based on data from the National Health Survey, we can assume the weight of adult males in the US has a mean weight of 197 pounds and standard deviation of 32 pounds. We randomly select 64 adult males. What is the probability that the average weight of these 64 adult males is over 205 pounds?

Answer 1

Answer:

There is a 2.28% probability that the average weight of these 64 adult males is over 205 pounds.

Step-by-step explanation:

To solve this question, we have 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 random variable X, with mean $\mu$ and standard deviation $\sigma$, a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s= \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 197, \sigma = 32, n = 64, s = \frac{32}{\sqrt{64}} = 4$

What is the probability that the average weight of these 64 adult males is over 205 pounds?

This is 1 subtracted by the pvalue of Z when X = 205.

So

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

$Z = \frac{205 - 197}{4}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

1 - 0.9772 = 0.0228

There is a 2.28% probability that the average weight of these 64 adult males is over 205 pounds.