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.If we are interested in whether students at this high school are getting too little sleep, which of the following represents the appropriate null and alternative hypotheses?

(a) $H_0:\mu =7.9and{H}_a:\mu <7.9$

(b) $H_0:\mu =7.9and{H}_a:\mu \ne 7.9$

(c) $H_0:\mu =9and{H}_a:\mu \ne 9$

(d)$H_0:\mu =9and{H}_a:\mu <9$

(e)$H_0:\mu \le 9and{H}_a:\mu \ge 9$

Short answer

The appropriate null and alternative hypotheses are (d) $H_0:\mu =9,H_a:\mu <9$

Step-by-step solution

Given information

We are given that if you are between the ages of $12$and $18$years old, you need hours of sleep to be fully functional.The mean of the responses was $7.9$ hours, with a standard deviation of $2.1$hours. We have to find appropriate null and alternative hypothesis.

Simplify

For Null hypothesis ,

To construct the null and alternative hypothesis. the sample mean is$7.9$this is not hypothesized value. The hypothesized value is $9$.

$H_0:\mu =9$

For alternative hypothesis,

The alternative determines wether the test is left tail or right tail or both left and right.

since,the researcher is specifically looking for too little sleep, we are intrested wether or not the students are getting less sleep . We are intrested in students getting too much sleep , it would be a right tail test.

Therefore,$H_a:\mu <9$