Answer:
92.65% probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.
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
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For sums the theorem can also be used, with mean
and standard deviation ![s = \sqrt{n}*\sigma](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bn%7D%2A%5Csigma)
In this problem, we have that:
![n = 47, \mu = 47*167 = 7849, s = \sqrt{47}*35 = 240](https://tex.z-dn.net/?f=n%20%3D%2047%2C%20%5Cmu%20%3D%2047%2A167%20%3D%207849%2C%20s%20%3D%20%5Csqrt%7B47%7D%2A35%20%3D%20240)
Use this information to estimate the probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.
This is 1 subtracted by the pvalue of Z when X = 7500. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{7500 - 7849}{240}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B7500%20-%207849%7D%7B240%7D)
![Z = -1.45](https://tex.z-dn.net/?f=Z%20%3D%20-1.45)
has a pvalue of 0.0735
1 - 0.0735 = 0.9265
92.65% probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.