To solve this question, we need to understand what a null hypothesis is in the context of hypothesis testing.
A null hypothesis ([tex]\(H_0\)[/tex]) is a statement that there is no effect or no difference, and it represents the current state of affairs or a baseline measure. In this case, the null hypothesis will reflect the status quo regarding the average number of hours of sleep college students get per night.
The recommended sleep duration for adults is 8 hours per night. The researcher is interested in testing whether college students get less than the recommended amount, but the null hypothesis typically assumes that there is no deviation from the recommended amount. Therefore, our null hypothesis should state that college students, on average, get 8 hours of sleep per night.
Let's look at the options provided:
A. [tex]\(H_0: B_x = 7.9\)[/tex] hours per night
- This is incorrect because the null hypothesis should not state the sample mean but rather the population mean, assuming no effect (in this case, assuming college students get the recommended 8 hours).
B. [tex]\(H_0: \mu_x < 8\)[/tex] hours per night
- This is a hypothesis that suggests students sleep less than the recommended 8 hours, but this would be our alternative hypothesis (not the null hypothesis).
C. [tex]\(H_0: H_x < 7.9\)[/tex] hours per night
- This is also suggesting that students actively sleep less than a certain amount, which again would be more aligned with the alternative hypothesis.
D. [tex]\(H_0: H_x = 8\)[/tex] hours per night
- This is the correct statement for the null hypothesis, indicating that there is no deviation and students get the recommended 8 hours, on average.
Therefore, the null hypothesis is:
[tex]\[H_0: H_{x} = 8 \text{ hours per night}\][/tex]
Thus, the correct answer is:
D. [tex]\(H_0: H_{x} = 8\)[/tex] hours per night
