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

Rejection region:

z < -z/2 or zz/2

Here significance level is 0.05,

So, z0.025 = 1.96 (Using Statistical Ratio Calculator

From for calculating z with 0.05 significance)

I am assuming the sample is selected independently and randomly from population. Population size is sufficiently large in sample.

Now,

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.

Calculating t-test statistics:

z = 25-24/(1.195/√8) = 1* √8/1.195= 2.828/1.195 = 2.366

Calculating p-value:

Degree of freedom = DF = 8-1 = 7

Absolute value of calculated t-test statistics = |z| = |2.366| = 2.366

P(|z7| < 2.366) = 0.049899

(Using with t=0.8695 and DF = 24)

The p-value of 4.9% is very slightly less than significant level of 5%. Hence, we reject the null hypothesis and the average cycle length is not equal to 24hours.

(b) Sketch the distributions involved.

(c) Explain your answer to someone who has never taken a course in statistics.

Statistical result shows that 24 hours is not the natural cycle in present case which is found to be slightly higher than expected mean. As there is only slight difference between statistical significance which can be practically ignored for validating the hypothesis.

Null Hypothesis: H0: (1 - 2) =0 vs. Ha: (1 - 2)0

Rejection region can be defined as

z < -z

Here significance level is 0.05,

So, z0.05= -1.645 [critical value](Using Statistical Ratio Calculator

From for calculating z with 0.05 significance)

Now,

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.

Calculating t-test statistics:

z = 24.4-26/(9.2/√25) = -1.6/(9.2/5) = -1.6*5/9.2 = -8/9.2 = -0.8695

Calculating p-value:

Degree of freedom = DF = 25-1 = 24

Absolute value of calculated t-test statistics = |z| = |-0.8695| = 0.8695

P(|z24| 0.8695) = 0.1965

(Using t=0.8695 and DF = 24)

The p-value of 19.65% is greater than significant level of 5%. Hence, we can say that null hypothesis is true and the mean age of the skating population in one city is 26 years.