A random sample of 16 students selected from the student body of a large university had an average age of 25 years. We want to determine if the average age of all the students at the university is significantly different from 24. Assume the distribution of the population of ages is normal with a standard deviation of 2 years. Refer to Exhibit 9-4. At a .05 level of significance, it can be concluded that the mean age is .

At a 0.05 level of significance, it can be concluded that the mean age is significantly different from 24.

Step-by-step explanation:

For hypothesis testing, we first clearly state our null and alternative hypothesis.

The null hypothesis is that the mean age of students isn't significantly different from 24.

And the alternative hypothesis is that the mean age of students is significantly different from 24.

Mathematically, the null hypothesis is

H₀: μ₀ = 24

The alternative hypothesis is

Hₐ: μ₀ ≠ 0.49

To do this test, we will use the z-distribution because we have information about the population standard deviation.

The population of ages is normal with a standard deviation of 2 years.

So, we compute the z-test statistic

z = (x - μ)/σₓ

x = 25

μ = μ₀ = 24

σₓ = standard error of the mean = (σ/√n)

σ = 2

where n = Sample size = 16

σₓ = (2/√16) = 0.50

z = (25 - 24) ÷ 0.50

z = 2.00

checking the tables for the p-value of this z-statistic

p-value (for z = 2.00, at 0.05 significance level, with a two tailed condition) = 0.0455

The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

So, for this question, significance level = 5% = 0.05

p-value = 0.0455

0.0455 < 0.05

Hence,

p-value < significance level

This means that we reject the null hypothesis & accept the alternative hypothesis and we conclude that there is significant evidence to show that the mean age of students is significantly different from 24.

Hope this Helps!!!