Show your answers by following the 5 steps of testing hypothesis.

Question

Research suggests that most people need at least 7 hours of sleep each night in order to function well the next day. Some research suggests that lack of sleep can adversely affect health, memory, learning. creativity, productivity, emotional stability, and even life expectancy. Suppose adults need 7 hours of sleep per night. You as a researcher obtained a random sample of 33 adults, and the sleeping time for each was recorded. The sample mean was 6.36 hours and assume 

the standard deviation is 0.67 hours. Is there any evidence to suggest that the mean sleeping time for adults is less than 7 hours? Use significance level = 0.05 or α=0.05. Show the steps in testing hypothesis.

Answer 1

Step One: Null and Alternative Hypothesis.

H₀: μ ≥ 7.0 ("research suggest the mean is at least 7.0 hours". "At least" is greater than or equal to which includes an equality. Equality always goes in the null hypothesis.

H₁: μ < 7.0. (" is there evidence to suggest the mean is less than 7.0?". " Less than" is a strict inequality and therefore goes in the alternative hypothesis.

Step Two. State the significance level of the test. If you are using the critical value approach they also want the critical value of the test statistic.

For the p-value approach, just state that the significance level of the test is given as α = .05.

To find the critical value of the test statistic, first note that this is a lower tailed test (the alternative is "less than") . Next, determine if the critical value is a Z or a t. Since we are told to assume the standard deviation is 0.67, I believe that they are saying σ = 0.67. Because the population standard deviation is known (σ = 0.67), use a Z score. The Z score for the lower 5% of the probability distribution of X-bar is -1.645. Therefore the critical value of our test statistic is: Z_critical = -1.645.

Step Three. State your decision rule.

For the p-value approach to hypothesis testing, our decision rule is "reject the null if the p-value corresponding to our X-bar value is less than α = .05"

For the critical value approach to hypothesis testing, we will reject the null hypothesis if the Z score that we calculate for our sample mean, X-bar, is less than -1.645.

Step Four. Calculate the Z-score for X-bar = 6.36 and find the corresponding p-value.

Z _calc = (xbar -μ)/(σ/sqrt(n)) = (6.36-7.0)/(.67/sqrt(33))= -5.49. The p-value for a Z-score of -5.49 is practically zero.

Step Five.

Using the p-value approach, have p-value ≈ 0 < α = 0.05 Reject the null hypothesis, there is enough evidence to conclude that the mean sleep time is less than 7.0 hours.

Using the critical value approach, Z _calc = -5.49 < Z critical =-1.645 . So, we can reject the null hypothesis. There is enough evidence to conclude that the mean sleep time is less than 7.0 hours.