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)b 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 no and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycle at the end of the study was as follows: 25, 27, 25, 23, 24, 25, 26, and 25.
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 the 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. Solution:
(a) Use the steps of hypothesis testing.
Size of sample, n = 8
Degree of freedom = n-1 = 8-1 = 7
Sum of sample = i=1∑n=8xi = (25+27+25+23+24+25+26+25) = 200 | Time (in Hours)| Sum| Mean(xm)| (xi- xm)2| Standard Deviation| 1| 25| 200| Mean = Sum/n= 200/8= 25| 0| σ = √( i=1∑n=8(xi-xm)2/(n-1))= √(10/7)= √1.4285= 1.195| 2| 27| | | 4| |
3| 25| | | 0| |
4| 23| | | 4| |
5| 24| | | 1| |
6| 25| | | 0| |
7| 26| | | 1| |
8| 25| | | 0| |
Here X = 25
σ = 1.195
We can define two-tailed statistics for above observations as follows: Null Hypothesis: H0: =24 vs. Ha: 24
z < -z/2 or z > z/2
Here significance level is 0.05,
So, z0.025 = 1.96 (Using Statistical Ratio Calculator
z = (X - μ) / σx
Where X is a normal random variable, μ is the mean, and σ is the standard deviation.
Where n is the sample size.