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 create a sketch of the distributions involved, you can use a bell curve or normal distribution graph.
In this case, the null hypothesis states that the average cycle length is 24 hours, while the alternative hypothesis suggests that the average cycle length is not equal to 24 hours.
To sketch the distributions, you would plot the x-axis representing the cycle lengths and the y-axis representing the frequency or probability. Since the standard deviation is given as 1.95229, you can use this information to plot the normal distribution curve.
Start by drawing a bell curve with a peak at 24 hours to represent the null hypothesis. The curve would be symmetrical around the mean of 24 hours. Label the x-axis with the possible cycle lengths.
Next, you can plot another bell curve with a peak at the observed mean of 25 hours. This curve represents the alternative hypothesis. Again, label the x-axis with the possible cycle lengths.
Remember that the null hypothesis assumes that the average cycle length is 24 hours, so the peak of the curve for the null hypothesis would be a bit higher than the curve for the alternative hypothesis (since the average cycle length observed in the study was slightly higher than 24 hours).
Additionally, you would need to shade or mark the region where the observed t-value of 2.36649 falls on the distribution. This would help demonstrate the probability of observing the mean cycle length of 25 hours if the true population mean was 24 hours.
Note that I am an AI and cannot directly generate an image or sketch. I recommend using software like Excel, Google Sheets, or statistical software packages that can generate normal distribution graphs based on the mean and standard deviation values provided.