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 5% 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.

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To test the theory that humans follow a 24-hour cycle of sleeping and waking even without exposure to sunlight, we can use hypothesis testing. Hypothesis testing involves comparing the observed data to a hypothesis or assumption and determining if the difference between them is statistically significant. In this case, our null hypothesis (H0) is that the average cycle length is 24 hours, and the alternative hypothesis (Ha) is that the average cycle length is different from 24 hours.

(a) Steps of hypothesis testing:
1. Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The average cycle length is 24 hours.
- Alternative hypothesis (Ha): The average cycle length is different from 24 hours.

2. Choose a significance level (α) – in this case, 5%. This determines the probability of rejecting the null hypothesis when it is true.

3. Collect data – the given data consists of the cycle lengths of the eight volunteers after a month in the room.

4. Calculate the test statistic – in this case, we will use a t-test since the sample size is small (less than 30) and the population standard deviation is unknown.

5. Determine the critical region – this is the region of values that would lead us to reject the null hypothesis. Since the alternative hypothesis is two-sided (average cycle length is different), we will calculate the critical t-values for a two-tailed test at the 5% significance level, with the degrees of freedom as n-1 (n=8 in this case).

6. Compare the test statistic to the critical region – if the test statistic falls within the critical region, we reject the null hypothesis; otherwise, we fail to reject it.

(b) Sketching the distributions involved:
To sketch the distributions, we would plot a t-distribution with degrees of freedom (n-1) and calculate the critical t-values for the two-tailed test. We will compare the observed t-value from the data to this distribution.

(c) Explanation for someone unfamiliar with statistics:
To determine if the average cycle length under the given conditions differs significantly from 24 hours, we use hypothesis testing. We set up a null hypothesis (H0) that assumes the average cycle length is 24 hours and an alternative hypothesis (Ha) that suggests it is different from 24 hours. Then, we collect data on the cycle lengths of the eight volunteers.

Next, we calculate a test statistic (t-value) to see how far the observed data deviates from the null hypothesis. Since our sample size is small and the population standard deviation is unknown, we use a t-test. The resulting t-value is then compared to critical t-values obtained from a t-distribution.

By comparing the test statistic to the critical region (values that would lead us to reject the null hypothesis), we can determine if the average cycle length differs significantly from 24 hours. If the test statistic falls within the critical region, we reject the null hypothesis; otherwise, we fail to reject it.