Question

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.At a .05 level of significance, it can be concluded that the mean age is _____. a. significantly less than 25 b. not significantly different from 24 c. significantly less than 24 d. significantly different from 24

Answer 1

Option D) significantly different from 24

Explanation:

We are given the following in the question:

Population mean, μ = 24

Sample mean,
`\bar{x}` = 25

Sample size, n = 16

Alpha, α = 0.05

Population standard deviation, σ = 2

First, we design the null and the alternate hypothesis

`H_(0): \mu = 24\text{ years}\\H_A: \mu \\eq 24\text{ years}`

We use Two-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(25 - 24)/((2)/(√(16)) ) = 2`

Now,
`z_(critical) \text{ at 0.05 level of significance } = \pm 1.96`

Since,

The calculated z statistic does not lie in the acceptance region, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.

Thus, it can be concluded that the mean age is

Option D) significantly different from 24