According to the Bureau of Labor Statistics, the average age of American workers is $41.9$years. The manager of a large technology company believes that the company’s employees tend to be younger, on average. So she takes a random sample of 12 employees and records their ages.

Here are the data:

27 38 32 24 30 47 42 38 27 43 37 33

a. State appropriate hypotheses for testing the manager’s belief. Be sure to define the parameter of interest.

b. State the conditions for performing a test of the hypotheses in (a), and determine whether each condition is met.

c. The P-value of the test is$0.003$. Interpret this value. What conclusion would you make?

$\text{Claim is that mean is less than}41.6\text{(younger)}$

The null hypothesis statement is that the population value is equal to the given in the claim:

$H_0:\mu =41.6$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is that the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

$H_1:\mu <41.6$

$\mu$is the mean age of the employees at the company.

Conditions for performing a test of the hypotheses

The three conditions are: Random, independent $(10\%$condition),

Normal/Large sample.

Random: Satisfied, because the sample is a random sample.

Independent: Satisfied, because the sample of $12$employees is less than$10\%$of the population of all employees of $12$employees is less than $10\%$of the population of all employees

Normal/ Large sample: Satisfied, because the pattern in the normal quintile plot is roughlyLinear, which indicates that the distribution is approximately Normal

Since all conditions are satisfied, Therefore we can perform a hypothesis test for the population mean

$P=0.003=0.3\%$

$\text{Claim: mean less than}41.9.$

The claim is either the null hypothesis or the alternative hypothesis. The null hypotheses statement is that the population mean is equal to the value given in the claim. If the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of null hypothesis.

$H_0:\mu =41.9$

$H_1:\mu <41.9$

The $P$-value is the probability of getting the value of the test statistic or a value more extreme, once the null hypothesis is true.

There is a $0.3\%$ probability of getting the sample results or more extreme, when the mean age of the company's employees is$41.9$ years.