SOLUTION: 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 o

Question 588398: 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.
I need: M= S2= t= t needed= =/- and Decision. I need this today. Thank you
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
Hypothesis:

Ho: mu = 24
Ha: mu =/= 24

-------------------------------------------------------

Critical T-value (for t-distribution with df = 8-1 = 7) is t = 2.365
Note: use alpha/2 = 0.05/2 = 0.025
You can find the Inverse T-Distribution Table here:
http://www.zweigmedia.com/RealWorld/finitetopic1/t_table.html

-------------------------------------------------------
Test Statistic:

For this, we need the mean xbar and the sample standard deviation s. We can find both with a calculator to get xbar = 25 and s = 1.1952286

t = (xbar - mu)/(s/sqrt(n))

t = (25 - 24)/(1.1952286/sqrt(8))

t = 2.3664319

Since the test statistic (2.3664319) is greater than the critical t-value (2.365), this means that the test statistic lies in the rejection region.

So this tells us to reject the null hypothesis.

So this allows us to conclude that the mean number of hours differs significantly from 24.

=======================================================
b)

The distribution involved is the t-distribution with 7 degrees of freedom and it's shown below

I've also included the rejection region, the critical values and the test statistic. You can see that the test statistic lies in the red rejection region.

=======================================================
c)

Essentially, the distribution shown in part B is the set of all possible cycles (we're assuming it is anyway). The values in the red regions are values that are very unlikely (at a significance level of 5%). So if you find any values in these regions, then it's unlikely that the assumed distribution is true.
RELATED QUESTIONS
Here is a question that I am working on for statistics, I already have the answer from a... (answered by richwmiller)

Evolutionary theories often emphasize that humans have adapted to their physical... (answered by stanbon)

*NEED HELP WITH THIS LAST ASSIGNMENT FOR ALGEBRA 2. I'M HAVING TROUBLE FIGURING IT OUT.... (answered by ewatrrr,Tipporules)

J. McWhorter of College of Health Science at the University of Nevada studied physical... (answered by stanbon)

I need help solving this problem. Can you please help me? A survey report states that... (answered by stanbon)

Humans have about 1,400,000 hairs on their head and lose an average of 75 hairs each day. (answered by Nate,bucky)

The weight distribution of a sample of adults with physical inabilities is approximately (answered by CPhill)

Researchers have noted a decline in cognitive functioning as people age. However the... (answered by Boreal)