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. Statistics for 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
NOTE: SOME OF THE SYMBOLS I HAVE TYPED OUT HAVE BEEN ALTERED AS I PASTED IT INTO THIS WINDOW- SO I WILL ALSO ATTACH A FILE OF MY WORK FOR MORE CLARITY.
The first thing I do is write out what I know:
n = 8
Then calculate the mean: 25+27+25+23+24+25+26+25 = 200 / 8 = 25
Knowing that we are using 5% confidence interval we find the corresponding points on the t-table that make up the cut-off (or critical region) that corresponds to 5%.
We know this is a two-tailed test because the question is open-ended " ... does the average cycle... differ... ?" [if it were 1-tailed then the language would indicate it with something along the lines of "... is the cycle longer than the 24hr ...
The solution includes a description of all of the steps of hypothesis testing, presents the necessary equations used for t-tests and explains the notion of significance.