Question

 

A researcher is interested to learn if there is a linear relationship between the hours in a week spent exercising and a person’s life satisfaction. The researchers collected the following data from a random sample, which included the number of hours spent exercising in a week and a ranking of life satisfaction from 1 to 10 ( 1 being the lowest and 10 the highest).

Participant	Hours of Exercise	Life Satisfaction
1	3	1
2	14	2
3	14	4
4	14	4
5	3	10
6	5	5
7	10	3
8	11	4
9	8	8
10	7	4
11	6	9
12	11	5
13	6	4
14	11	10
15	8	4
16	15	7
17	8	4
18	8	5
19	10	4
20	5	4

 

 

 

 

Find the mean hours of exercise per week by the participants.

Find the variance of the hours of exercise per week by the participants.

Determine if there is a linear relationship between the hours of exercise per week and the life satisfaction by using the correlation coefficient.

Describe the amount of variation in the life satisfaction ranking that is due to the relationship between the hours of exercise per week and the life satisfaction.

Develop a model of the linear relationship using the regression line formula.

Insomnia has become an epidemic in the United States. Much research has been done in the development of new pharmaceuticals to aide those who suffer from insomnia. Alternatives to the pharmaceuticals are being sought by sufferers. A new relaxation technique has been tested to see if it is effective in treating the disorder. Sixty insomnia sufferers between the ages of 18 to 40 with no underlying health conditions volunteered to participate in a clinical trial. They were randomly assigned to either receive the relaxation treatment or a proven pharmaceutical treatment. Thirty were assigned to each group. The amount of time it took each of them to fall asleep was measured and recorded. The data is shown below. Use the appropriate t-test to determine if the relaxation treatment is more effective than the pharmaceutical treatment at a level of significance of 0.05.

Relaxation	Pharmaceutical
98	20
117	35
51	130
28	83
65	157
107	138
88	49
90	142
105	157
73	39
44	46
53	194
20	94
50	95
92	161
112	154
71	75
96	57
86	34
92	118
75	41
41	145
102	148
24	117
96	177
108	119
102	186
35	22
46	61
74	75

A researcher is interested to learn if there is a relationship between the level of interaction a women in her 20s has with her mother and her life satisfaction ranking. Below is a list of women who fit into each of four level of interaction. Conduct a One-Way ANOVA on the data to determine if a relationship exists.

No Interaction	Low Interaction	Moderate Interaction	High Interaction
2	3	3	9
4	3	10	10
4	5	2	8
4	1	1	5
7	2	2	8
8	2	3	4
1	7	10	9
1	8	8	4
8	6	4	1
4	5	3	8

Is there a relationship between handedness and gender? A researcher collected the following data in hopes of discovering if handedness and gender are independent (Ambidextrous individuals were excluded from the study). Use the Chi-Square test for independence to explore this at a level of significance of 0.05.

	Left-Handed	Right-Handed
Men	13	22
Women	27	18

A researcher is interested in studying the effect that the amount of fat in the diet and amount of exercise has on the mental acuity of middle-aged women. The researcher used three different treatment levels for the diet and two levels for the exercise. The results of the acuity test for the subjects in the different treatment levels are shown below.

	Diet
Exercise	<30% fat	30% - 60% fat	>60% fat
<60 minutes	4	3	2
	4	1	2
	2	2	2
	4	2	2
	3	3	1
60 minutes	6	8	5
or more	5	8	7
	4	7	5
	4	8	5
	5	6	6

Perform a two-way analysis of variance and explain the results. (Show all work to receive full credit)

Find the effect size for each factor and the interaction and explain the results. (Show all work to receive full credit)

 

Answer 1

Problem 1: Descriptive Statistics: Hours of Exercise

Variable Mean Variance   
Hours of Exercise 8.850 13.397

Pearson correlation of Hours of Exercise and Life Satisfaction = -0.103

(-0.103)²x100=0.0106x100=1.06% the amount of variation in the life satisfaction ranking that is due to the relationship between the hours of exercise per week and the life satisfaction.

The regression equation is
Life Satisfaction = 5.67 - 0.070 Hours of Exercise

Problem 2:

Two-Sample T-Test and CI: Relaxation, Pharmaceutical

Two-sample T for Relaxation vs Pharmaceutical (assume equal variances)

N Mean StDev SE Mean
Relaxation 30 74.7 28.9 5.3
Pharmaceutical 30 102.3 53.6 9.8

Difference = mu (Relaxation) - mu (Pharmaceutical)
Estimate for difference: -27.6
95% upper bound for difference: -9.0
T-Test of difference = 0 (vs <): T-Value = -2.48 P-Value = 0.008 DF = 58
Both use Pooled StDev = 43.0730

Two-sample T for Relaxation vs Pharmaceutical (Assume variances are not equal)

N Mean StDev SE Mean
Relaxation 30 74.7 28.9 5.3
Pharmaceutical 30 102.3 53.6 9.8

Difference = mu (Relaxation) - mu (Pharmaceutical)
Estimate for difference: -27.6
95% upper bound for difference: -8.9
T-Test of difference = 0 (vs <): T-Value = -2.48 P-Value = 0.008 DF = 44

From the above test we observed that p-value<0.05 hence  the relaxation treatment is more effective than the pharmaceutical treatment at a level of significance of 0.05.

Problem 3:

Source DF SS MS F P
Interaction 3 38.27 12.76 1.57 0.213
Error 36 292.50 8.13
Total 39 330.77

Since p-value=0.213>0.05 so the population means of 4 interactions are same and there are no relationship exist.

Problem 4:

Observed frequencies: O11=13, O12=22, O21=27, O22=18

n=13+22+27+18=80, O10=13+22=35, O20=27+18=45, O01=13+27=40, O02=22+18=40

Expected frequecies: E11=(35x40)/80=17.5, E12=(35x40)/80=17.5, E21=(45x40)/80=22.5,

E22=(45x40)/80=22.5

Hence there is a relationship between handedness and gender.