March 18, 2013
14.) Evolutionary theory often emphasizes that human have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking-even though 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 were no light from outside and no clocks or other indications of time. They could turn the lights on and off is they wish. After a month in the room, each individual tended to develop a study cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23, 24, 25, 26, and 25.
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Finding the variance of the distribution of mean:
S^2/M = S^2/N = 1.43/8 = 0.179
Finding the standard deviation of the distribution of means:
S/M = √S^2/M = √0.179 = 0.423
3. Define the region cutoff sample score:
Sample mean = 25
SD = 1.195229
4. Determine the sample score on the comparison distribution:
t = (M - µ) / S/M = (25 – 24) / 0.423 =1/0.423 = 2.36
t = 2.36
5. Select the appropriate theory:
Since the t – score is greater than the 0.5 level of significance we reject the null theory and conclude that natural cycle is not 24 hours.
(b) Sketch of distributions
(c) Explain yourself to someone who has never had a course in statistics. The natural cycle is less than 24 hours, so the critical value is the difference between means and the hypothesized value which makes it statistically significant. There was a small difference with t score being greater than the t value.
18.) Twenty students randomly assigned to an experimental group received an instructional program; 30 in a control group do not. After six months, both groups are tested on their knowledge. The experimental group as a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35(with an estimated population standard deviation of 5). Using the .05 level; what should the experimenter conclude? (a) Use the steps of hypothesis testing. (b) Sketch the distributions involved, and (c) explain your answer to someone who is familiar with the t-test for a single sample but not with the t-test for independent means.
(a) Steps of hypothesis testing
1. Restate the question as a research hypothesis and a null hypothesis about the populations.
There are two populations
Population 1: Students who are engaged in an instrumental group
Population 2: Students who engage in other neutral instrumental group
Null Hypothesis: µ1 = µ2
Alternative Hypothesis: µ1 ≠ µ2
2. Define the features of the compared distribution. The compared distribution is a distribution to find the difference between the two means.
Find the estimated population variances based on each sample
S^2 1 = 3; S^2 2 = 5
Find the pooled sum of the population variance:
df/1 = N/1- 1 = 20 -1 = 19; df/2 -1 = N/1 -1 = 30 -1 = 29
df/ total = df /1 + df/2 = 19 + 29 = 48
S^2 / pooled = df/1/ df total ( S^2/1)
19/48(3) + 29/48(5)
1.187 + 3.020
S^2 pooled= 4.207
S^2 / M1 = 4.207/ 20 = 0.21035 and S^2/ M2 = 4.207/ 30 = 0.1402
S^2 difference = 0.21035 + 0.1402 = .35055
S difference = √S^2 difference = √.35055 = .5921
t- distribution w/ df total = 48
3. Define the region cut off score
The t cutoff score for .05 level one tailed, w/ df = 48
4. Determine the sample score on the comparison distribution
t= (M1-M2)/ S difference = (38 – 35) /.5921 = 5.067
5. Select the appropriate theory
We reject the null theory because t statistics is greater than t...