Question

According to sleep researchers, if you are between the ages of $12$and $18$years old, you need $9$hours of sleep to be fully functional. A simple random sample of $28$students was chosen from a large high school, and these students were asked how much sleep they got the previous night. The mean of the responses was $7.9$hours, with a standard deviation of $2.1$hours.

Which of the following is the test statistic for the hypothesis test?

(a)$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$

(b)$t=\frac{9-7.9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$

(c)$t=\frac{7.9-9}{\sqrt{{\displaystyle \frac{2.1}{28}}}}$

(d)$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{27}}}}$

(e)$t=\frac{9-7.9}{{\displaystyle \frac{2.1}{\sqrt{27}}}}$

Short answer

The test statistic for the hypothesis test is (a)$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$

Step-by-step solution

Given information

We are given that if you are between the ages of $12$and $18$years old.The mean of the responses was $7.9$hours, with a standard deviation of $2.1$ hours. We have to find the test statistic for the hypothesis test

Simplify

As we know,

$t=\frac{\overline{X}-{\mu }_0}{{\displaystyle \frac{s}{\sqrt{n}}}}$

Where,

${\mu }_0=9$(Hypothesized claim)

$\overline{X}=7.9$(Sample mean )

$n=28$(Sample size)

$s=2.1$(Sample standard deviation)

By putting the values in above formula

$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$