## Question 616:

1

1. Q616_1_curve_above24.jpg
We can use the 1-sample t-test to test this assumption. First we need to identify the hypothesis.
1. The NULL hypothesis is that the average sleep cycle is 24 hours. Ho: μ = 24
2. The alternative hypothesis is that the average sleep cycle is not equal to 24 hours. Ho: μ 24
3. We will reject the NULL hypothesis if there is sufficient evidence (α = .05).
To find our test statistic t*, we use the following procedure:
1. Find the difference between the sample mean and test mean. The sample mean of the 8 values is 25 minus test mean of 24 = 1.
2. Find the standard error of the sample mean (SE) which is the sample standard deviation divided by the square root of the sample size. The sample standard deviation is 1.19522, making the SE = 1.19522/SQRT(8) = .422577
3. The test statistic t*, is the difference (Step1) divided by the SE (Step 2) = 1/.422577 = 2.36643.
4. We lookup this value from a t-table or using the Percentiles from the t-distribution calculator on 7 degrees of freedom (8-1) and we get the 2-sided p-value of 0.0499.
Since the p-value of 0.0499 is less than (just barely) our alpha of .05, we'd reject the null hypothesis and conclude that 24 hours is not the natural sleep cycle.

b. See the attached file for a visual sketch of the distribution.

c. The average sleep duration of 8 individuals was 25 hours. The probability a difference of 1 hour or more coming from a sample of 8 due to chance alone is less than 5%.