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
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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
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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.
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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.
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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.
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