Question

According to sleep researchers, if you are between the ages of $12$and $18$years old, you need 9 hours of sleep to function well. 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 standardized test statistic for the hypothesis test ?

1. localid="1654221307488" $t=7.9-92.128$and$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$
2. $t=9-7.92.128$and$t=\frac{9-7.9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$
3. $t=7.9-92.128$and $t=\frac{7.9-9}{\sqrt{{\displaystyle \frac{2.1}{28}}}}$
   
4. localid="1654221494579" $t=7.9-92.127$and$t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{27}}}}$
5. localid="1654221497857" $t=9-7.92.127$and$t=\frac{9-7.9}{{\displaystyle \frac{2.1}{\sqrt{27}}}}$

Short answer

Option (a) is correct answer :$t=7.9-92.128,t=\frac{7.9-9}{{\displaystyle \frac{2.1}{\sqrt{28}}}}$

Step-by-step solution

Given Information

We are given following information:

No. of samples, $n=28$

Sample mean , $\overline{X}=7.9$

Sample standard deviation, $s=2.1$

Hypothesized value,${\mu }_0=9$

Explanation

For this situation, the test statistic formula is $t=\frac{\overline{X}-{\mu }_0}{{\displaystyle \frac{s}{\sqrt{n}}}}$

This assumes the data is normally distributed with unknown population standard deviation.

Calculating standardized test statistic for present case we get.;

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

Hence, option (a) is correct option.