Answer:
The standard error of the mean is 1.3.
87.64% probability that the sample mean age of the employees will be within 2 years of the population mean age
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
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 random variable X, with mean
and standard deviation
, a large sample size can be approximated to a normal distribution with mean
and standard deviation, which is also called standard error ![s = \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
In this problem, we have that:
![\sigma = 8.2, n = 40](https://tex.z-dn.net/?f=%5Csigma%20%3D%208.2%2C%20n%20%3D%2040)
Computer the standard error of the mean
![s = \frac{8.2}{\sqrt{40}} = 1.3](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B8.2%7D%7B%5Csqrt%7B40%7D%7D%20%3D%201.3)
The standard error of the mean is 1.3.
What is the probability that the sample mean age of the employees will be within 2 years of the population mean age
This is the pvalue of Z when
subtracted by the pvalue of Z when
. So
![X = \mu + 2](https://tex.z-dn.net/?f=X%20%3D%20%5Cmu%20%2B%202)
![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{\mu + 2 - \mu}{1.3}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B%5Cmu%20%2B%202%20-%20%5Cmu%7D%7B1.3%7D)
![Z = 1.54](https://tex.z-dn.net/?f=Z%20%3D%201.54)
has a pvalue of 0.9382
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![X = \mu - 2](https://tex.z-dn.net/?f=X%20%3D%20%5Cmu%20-%202)
![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{\mu - 2 - \mu}{1.3}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B%5Cmu%20-%202%20-%20%5Cmu%7D%7B1.3%7D)
![Z = -1.54](https://tex.z-dn.net/?f=Z%20%3D%20-1.54)
has a pvalue of 0.0618
0.9382 - 0.0618 = 0.8764
87.64% probability that the sample mean age of the employees will be within 2 years of the population mean age