Evolutionary theories often emphasize that humans have adapted to their physical

environment. One such theory hypothesizes that people should spontaneously
follow a 24-hour cycle of sleeping and waking—even if they are not exposed to
the usual pattern of sunlight. To test this notion, eight paid volunteers were
placed (individually) in a room in which there was no light from the outside and
no clocks or other indications of time. They could turn the lights on and off as
they wished. After a month in the room, each individual tended to develop a
steady cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23,
24, 25, 26, and 25.
Using the .05 level of significance, what should we conclude about the
theory that 24 hours is the natural cycle? (That is, does the average cycle length
under these conditions differ significantly from 24 hours?) (a) Use the steps of
hypothesis testing. (b) Sketch the distributions involved. (c) Explain your answer
to someone who has never taken a course in statistics.

To test the theory that humans have a natural 24-hour cycle of sleeping and waking even without exposure to sunlight, the scientists conducted an experiment with eight volunteers. They placed each individual in a room with no external light or clocks for a month and allowed them to control the lights. The individuals developed their own sleeping and waking patterns during this time.

To determine if the average cycle length differs significantly from 24 hours, we will use hypothesis testing. Here are the steps:

Step 1: State the null hypothesis (H0) and the alternate hypothesis (Ha).
The null hypothesis is that the average cycle length under these conditions is equal to 24 hours: H0: μ = 24. The alternate hypothesis is that the average cycle length under these conditions is not equal to 24 hours: Ha: μ ≠ 24.

Step 2: Choose the level of significance (α) you want to test at.
In this case, the level of significance is given as 0.05 (5%).

Step 3: Collect the data and calculate the test statistic.
The data provided gives the cycle lengths for the eight individuals: 25, 27, 25, 23, 24, 25, 26, and 25. To calculate the test statistic, we will use a one-sample t-test since we are comparing the sample mean to a known value (24). We will use the t-distribution because the population standard deviation is unknown.

Step 4: Determine the critical value(s) and rejection region.
Since the test is two-tailed (we are testing for not equal), we will divide the level of significance (α) by 2 to get α/2 = 0.025. We will then identify the critical values from the t-distribution with degrees of freedom (n-1) = 7 and α/2 = 0.025. Looking up the critical values in the t-distribution table or using software, the values are approximately ±2.365.

Step 5: Compare the test statistic to the critical value(s) and make a decision.
Calculate the test statistic using the sample mean, sample standard deviation, and sample size. With the given data, the sample mean is 25.125, the sample standard deviation is 1.032, and the sample size is 8. The formula for the t-test statistic is: t = (x̄ - μ) / (s / sqrt(n)), where x̄ is the sample mean, μ is the population mean (in this case, 24), s is the sample standard deviation, and n is the sample size.

In this case, t = (25.125 - 24) / (1.032 / sqrt(8)) ≈ 2.451.

Since the absolute value of the test statistic (2.451) is greater than the critical value of 2.365, we reject the null hypothesis. This means that the average cycle length under these conditions is significantly different from 24 hours.

(b) To visualize the distributions involved, we would create two overlapping bell-shaped curves representing the null hypothesis and alternative hypothesis. The null hypothesis curve would be centered at 24, while the alternative hypothesis curve would be centered at the sample mean of 25.125. The critical values (±2.365) will represent the cutoff points where we reject or fail to reject the null hypothesis.

(c) If you have never taken a course in statistics, here's the explanation: The experiment tested whether humans, without exposure to sunlight or any time cues, still follow a 24-hour sleep-wake cycle. They observed eight individuals for a month and recorded their cycle lengths. Using statistical analysis, they compared the observed average cycle length with the expected 24-hour cycle. The analysis involved testing a null hypothesis that the average cycle length is 24 hours. By calculating a test statistic (2.451) and comparing it to the critical value (±2.365), they found that the observed average cycle length significantly differs from 24 hours. Therefore, the experiment provides evidence against the theory that humans naturally follow a 24-hour cycle regardless of external cues.