Question

Assume the general population gets an average of 7 hours of sleep per night. You randomly select 45 college students and survey them on their sleep habits. From this sample, the mean number of hours of sleep is found to be 6.87 hours with a standard deviation of 0.55 hours. You claim that college students get less sleep than the general population. That is, you claim the mean number of hours of sleep for all college students is less than 7 hours. Test this claim at the 0.10 significance level.

(a) What type of test is this?

i. This is a right-tailed test.
ii. This is a left-tailed test.
iii. This is a two-tailed test.
(b) What is the test statistic? Round your answer to 2 decimal places.

tx= _________
(c) What is the critical value of t? Use the answer found in the t-table or round to 3 decimal places.

tα = _________
(d) What is the conclusion regarding the null hypothesis?

i. reject H0
ii. fail to reject H0
(e) Choose the appropriate concluding statement.

i. The data supports the claim that college students get less sleep than the general population.
ii. There is not enough data to support the claim that college students get less sleep than the general population.
iii. We reject the claim that college students get less sleep than the general population.
iv. We have proven that college students get less sleep than the general population.

Answer 1

a) ii. This is a left-tailed test.

b) -1.59

c) -1.301

d) i. reject null hypothesis

e) Option i) The data supports the claim that college students get less sleep than the general population.

Explanation:

We are given the following in the question:

Population mean, μ = 7 hours

Sample mean,
`\bar{x}` = 6.87 hours

Sample size, n = 45

Alpha, α = 0.10

Sample standard deviation, s = 0.55 hours

First, we design the null and the alternate hypothesis

`H_(0): \mu = 7\text{ hours}\\H_A: \mu < 7\text{ hours}`

a) We use one-tailed(left) t test to perform this hypothesis.

b) Formula:

`t_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`t_(stat) = \displaystyle(6.87 - 7)/((0.55)/(√(45)) ) =-1.59`

c) Now,

`t_(critical) \text{ at 0.10 level of significance, 44 degree of freedom } = -1.301`

Since,

`t_(stat) < t_(critical)`

d) We fail to accept the null hypothesis and reject it.

We accept the alternate hypothesis and conclude that mean number of hours of sleep for all college students is less than 7 hours.

e) Option i) The data supports the claim that college students get less sleep than the general population.