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. 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 ?

1. $H_0:\mu =7.9$and $H_a:\mu <7.9$
   
2. $H_0:\mu =7.9$and $H_a:\mu \ne 7.9$
   
3. $H_0:\mu =9$and $H_a:\mu \ne 9$
   
4. $H_0:\mu =9$and width="69" height="24" role="math">Ha:μ<9
   
5. $H_0:\mu \le 9$and$H_a:\mu \ge 9$
   

We are given following information:

Mean of responses = $7.9$

Standard deviation =$2.1$

Remember that the null is identical to the hypothesized value while constructing the null and alternative.

The sample mean is $7.9$, which differs from the hypothesized value. So the hypothesized value is $9$.

$\Rightarrow$Null Hypothesis $H_0:\mu =9$

Whether the test is a left or right tail, or both, is determined by the alternative hypothesis.

We're curious if kids are receiving less sleep than we will go for the left tail test. It would be a right tail test if we were interested in pupils sleeping too much. Because the researchers are deliberately looking for too little sleep. So researchers will go for left tail test and alternative hypothesis will be less than null hypothesis.

$\Rightarrow$Alternative Hypothesis $H_a:\mu <9$

Hence, option (d) is correct.