Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation, which is also called standard error [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\sigma = 8.2, n = 40[/tex]
Computer the standard error of the mean
[tex]s = \frac{8.2}{\sqrt{40}} = 1.3[/tex]
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 [tex]X = \mu + 2[/tex] subtracted by the pvalue of Z when [tex]X = \mu - 2[/tex]. So
[tex]X = \mu + 2[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{\mu + 2 - \mu}{1.3}[/tex]
[tex]Z = 1.54[/tex]
[tex]Z = 1.54[/tex] has a pvalue of 0.9382
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[tex]X = \mu - 2[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{\mu - 2 - \mu}{1.3}[/tex]
[tex]Z = -1.54[/tex]
[tex]Z = -1.54[/tex] 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
