Answer:
0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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.
The heights of adults in a certain town have a mean of 65.42 inches with a standard deviation of 2.32 inches.
This means that ![\mu = 65.42, \sigma = 2.32](https://tex.z-dn.net/?f=%5Cmu%20%3D%2065.42%2C%20%5Csigma%20%3D%202.32)
Sample of 144:
This means that ![n = 144, s = \frac{2.32}{\sqrt{144}} = 0.1933](https://tex.z-dn.net/?f=n%20%3D%20144%2C%20s%20%3D%20%5Cfrac%7B2.32%7D%7B%5Csqrt%7B144%7D%7D%20%3D%200.1933)
Find the probability that a sample of this size would have a mean of 64.82 inches or less.
This is the pvalue of Z when X = 64.82. 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{64.82 - 65.42}{0.1933}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B64.82%20-%2065.42%7D%7B0.1933%7D)
![Z = -3.1](https://tex.z-dn.net/?f=Z%20%3D%20-3.1)
has a pvalue of 0.001
0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.