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¡ªeven 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.
So fay I have been ablet o answer a, but I have no idea about how to make a sketch. Help please!!
a) Null Hypoithesis ¦Ì=24
Alternative Hypothesis ¦Ì ¡Ù24
mean=25
Standard deviation=1.95229
t= ((25-24))/(1.19520/¡Ì8)=2.36649
To sketch the distributions involved in this question, you can create a visual representation of the data using a histogram or a bar chart. Here's how you can do it:
1. Create a horizontal axis to represent the different cycle lengths (in hours) and a vertical axis to represent the frequency or number of individuals in each cycle length.
2. Start with a bar or rectangle representing the cycle length of 23, and make its height proportional to the frequency of individuals with a cycle length of 23.
3. Repeat this step for each cycle length (24, 25, 26, and so on), creating bars or rectangles of appropriate heights relative to the frequencies.
4. Once you have plotted all the bars, you should have a visual representation of the distribution of cycle lengths.
In this case, you should expect most of the bars to have heights close to the average cycle length of 25, as that is the mean of the data. The shape of the distribution will help you understand how the data is spread around the mean.
To interpret the sketch, you can explain that the histogram or bar chart shows the distribution of cycle lengths among the eight volunteers. Each bar represents a specific cycle length, and the height of the bar shows how many individuals had that particular cycle length. The shape of the distribution can indicate whether the average cycle length significantly differs from 24 hours.
To determine the conclusion about the theory that 24 hours is the natural cycle, you need to compare the calculated t-value of 2.36649 with the critical t-value at the 0.05 level of significance. If the calculated t-value exceeds the critical t-value, you would reject the null hypothesis (that the average cycle length is 24 hours) and conclude that the average cycle length under these conditions significantly differs from 24 hours. On the other hand, if the calculated t-value does not exceed the critical t-value, you would fail to reject the null hypothesis and conclude that there is not enough evidence to support a significant difference from 24 hours.
Just remember that the sketch is a visual representation that can help understand the distribution, but the final conclusion should be based on the hypothesis testing process itself, comparing the calculated test statistic (t-value) with the critical value.