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.
M=
S^=
Sm=
t=
t needed=
To test the hypothesis that the average cycle length under the given conditions differs significantly from 24 hours, we can follow the steps of hypothesis testing:
Step 1: State the null and alternative hypotheses.
- Null hypothesis (H0): The average cycle length is equal to 24 hours.
- Alternative hypothesis (HA): The average cycle length is not equal to 24 hours.
Step 2: Set the significance level.
The given significance level is .05 (or 5%).
Step 3: Collect and analyze the data.
The cycle lengths at the end of the study were: 25, 27, 25, 23, 24, 25, 26, and 25.
To calculate the statistics needed for hypothesis testing, we need to find the sample mean (M), the sample standard deviation (S^), the standard error of the mean (Sm), the value of t, and the critical value of t needed.
M = (Σx) / n = (25+27+25+23+24+25+26+25) / 8 = 200 / 8 = 25 (average)
S^ = √[(Σ(x-M)^2)/(n-1)] = √[(0^2+2^2+0^2-2^2-1^2+0^2+1^2)/7] = √[8/7] ≈ 0.378 (sample standard deviation)
Sm = S^ / √n = 0.378 / √8 ≈ 0.134 (standard error of the mean)
Step 4: Calculate the test statistic (t) using the formula:
t = (M - μ) / Sm, where μ is the assumed population mean (24 in this case).
t = (25 - 24) / 0.134 ≈ 7.46
Step 5: Determine the critical value.
Since we have a two-tailed test, we need to find the critical value of t at the 0.025 level (since the significance level is 0.05 / 2).
Using a t-distribution table with (n-1) degrees of freedom (8-1 = 7), the critical value of t needed is approximately ±2.365.
Step 6: Make a decision.
If the absolute value of the calculated t statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
|t| = |7.46| > 2.365
Therefore, we reject the null hypothesis.
In simple terms, the results of the study provide strong evidence that the average cycle length under these conditions is significantly different from 24 hours. The sleep-wake cycle, in the absence of sunlight and external cues, tends to deviate from the traditional 24-hour cycle.