Data Driven Statistics
You must include parts I and II from Individual
Project #4 as they will be graded again. Then, add the following
responses to your document:
Part III: Regression and Correlation
Based on what you have learned from your research on regression
analysis and correlation, answer the following questions about the Body
Fat Versus Weight data set:
When performing a regression analysis, it is important to first
identify your independent/predictor variable versus your
dependent/response variable, or simply put, your x versus y variables. How do you decide which variable is your predictor variable and which is your response variable?
Based
on the Body Fat Versus Weight data set, which variable is the predictor
variable? Which variable is the response variable? Explain.
Using Excel, construct a scatter plot of your data.
Using
the graph and intuition, determine whether there is a positive
correlation, a negative correlation, or no correlation. How did you come
to this conclusion?
Calculate the correlation coefficient, r,
and verify your conclusion with your scatter plot. What does the
correlation coefficient determine? Discuss the strength of the
correlation coefficient calculated, and if the correlation is positive
or negative.
Add a regression line to your scatter plot, and obtain the regression equation.
Does the line appear to be a good fit for the data? Why or why not?
Regression equations help you make predictions. Using your
regression equation, discuss what the slope means, and determine the
predicted value of body fat x = 0. Interpret the meaning of this
equation.
Part IV: Putting it Together
Your analysis is now complete, and you are ready to report your
findings to your boss. In one paragraph, summarize your results by
explaining your findings from the statistical measures, hypothesis test,
and regression analysis of body fat and weight for the 252 men
attending Silver’s Gym
INFO That MAY NEEDED
Part I
The mean, median, range, and
standard deviation for the Body Fat and Weight data are computed in the
following Table
Measure
BODYFAT
WEIGHT
Mean
18.94
178.92
Median
19.00
176.50
Range
45.10
244.65
Standard Deviation
7.75
29.39
Mean and median
are two commonly used measures of central tendency. Mean is the sum of observations divided by
the total number of observations. Median is the midpoint of the values after
all observations have been ordered from the smallest to the largest. The
average body fat and weight of men who attend the gym are 18.94 and 178.92
respectively. The median values of body fat and weight are 19.0 and 176.5
respectively. The mean and median values of body fat and weight are close.
Range and Standard deviation are measures of dispersion. They show the
variability in the data. Range is the difference between the largest and
smallest values in a data set. Standard deviation is the square root of the
arithmetic mean of the squared deviations from the mean.
The purpose of mean and median is to pinpoint the center of a
set of observations. Mean and median are useful measure for comparing two or
more populations.
The mean is not representative of data with extreme values.
The median is a useful measure when we encounter data with an extreme value.
In this data set, the mean may
be more useful than the median, because of the relatively large sample size
which will not be as heavily
impacted by statistical outliers.
A direct comparison of two sets
of data based only on two measures of location such as the mean and the median
can be misleading since an average does not tell us anything about the spread
of the data. In such situation we use the measures of dispersion range or
standard deviation.
Part II
Let μ denote the mean
body fat in men attending Silver’s Gym.
The null and
alternative hypotheses are
Ho:
μ = 20
Ha: μ ≠ 20
Since the alternative
hypothesis contains the not-equal-to symbol, the test is two-tailed test.
The significance level of
the test is given to be α = 0.05.
Since the sample size (n = 252 > 30) is large, we can use
the z–test for testing the hypothesis.
The critical values at α =
0.05 are –z0 = -1.96 and z0 = 1.96. Therefore, the
rejection regions are to the left of –z0
= -1.96 and to the right of z0
= 1.96.
The standardized test
statistic is
z = (xbar –
μ)/(σ/√n)
Since the sample size n > 30, we can use σ ≈ s, the sample
standard deviation.
For the body fat data, n = 252, xbar = 18.94 and s = 7.75.
Therefore, z = (18.94 – 20)/(7.75/√252) = –
2.17
The decision rule is:
Reject Ho if z 1.96.
Since z < -1.96 it is in the rejection region and we reject the null
hypothesis Ho.
Thus, there is sufficient
evidence to reject the claim that the mean body fat in
men attending Silver’s Gym is 20%.
Supplement:
IDNO
BODYFAT
WEIGHT
1
12.6
154.25
Description:
The dataset provided here
is based on a sample of 252 men.
2
6.9
173.25
Their body fat and
weights were recorded.
3
24.6
154.00
4
10.9
184.75
Statement:
The doctors office claims
that the mean body fat in men is 20%.
5
27.8
184.25
6
20.6
210.25
Source:
http://www2.stetson.edu/~jrasp/data.htm
7
19.0
181.00
8
12.8
176.00
9
5.1
191.00
10
12.0
198.25
11
7.5
186.25
12
8.5
216.00
13
20.5
180.50
14
20.8
205.25
15
21.7
187.75
16
20.5
162.75
17
28.1
195.75
18
22.4
209.25
19
16.1
183.75
20
16.5
211.75
21
19.0
179.00
22
15.3
200.50
23
15.7
140.25
24
17.6
148.75
25
14.2
151.25
26
4.6
159.25
27
8.5
131.50
28
22.4
148.00
29
4.7
133.25
30
9.4
160.75
31
12.3
182.00
32
6.5
160.25
33
13.4
168.00
34
20.9
218.50
35
31.1
247.25
36
38.2
191.75
37
23.6
202.25
38
27.5
196.75
39
33.8
363.15
40
31.3
203.00
41
33.1
262.75
42
31.7
205.00
43
30.4
217.00
44
30.8
212.00
45
8.4
125.25
46
14.1
164.25
47
11.2
133.50
48
6.4
148.50
49
13.4
135.75
50
5.0
127.50
51
10.7
158.25
52
7.4
139.25
53
8.7
137.25
54
7.1
152.75
55
4.9
136.25
56
22.2
198.00
57
20.1
181.50
58
27.1
201.25
59
30.4
202.50
60
24.0
179.75
61
25.4
216.00
62
28.8
178.75
63
29.6
193.25
64
25.1
178.00
65
31.0
205.50
66
28.9
183.50
67
21.1
151.50
68
14.0
154.75
69
7.1
155.25
70
13.2
156.75
71
23.7
167.50
72
9.4
146.75
73
9.1
160.75
74
13.7
125.00
75
12.0
143.00
76
18.3
148.25
77
9.2
162.50
78
21.7
177.75
79
21.1
161.25
80
18.6
171.25
81
30.2
163.75
82
26.0
150.25
83
18.2
190.25
84
26.2
170.75
85
26.1
168.00
86
25.8
167.00
87
15.0
157.75
88
22.6
160.00
89
8.8
176.75
90
14.3
176.00
91
20.2
177.00
92
18.1
179.75
93
9.2
165.25
94
24.2
192.50
95
9.6
184.25
96
17.3
224.50
97
10.1
188.75
98
11.1
162.50
99
17.7
156.50
100
21.7
197.00
101
20.8
198.50
102
20.1
173.75
103
19.8
172.75
104
21.9
196.75
105
24.7
177.00
106
17.8
165.50
107
19.1
200.25
108
18.2
203.25
109
17.2
194.00
110
21.0
168.50
111
19.5
170.75
112
27.1
183.25
113
21.6
178.25
114
20.9
163.00
115
25.9
175.25
116
16.7
158.00
117
19.8
177.25
118
14.1
179.00
119
25.1
191.00
120
17.9
187.50
121
27.0
206.50
122
24.6
185.25
123
14.8
160.25
124
16.0
151.50
125
14.0
161.00
126
17.4
167.00
127
26.4
177.50
128
17.4
152.25
129
20.4
192.25
130
15.0
165.25
131
18.0
171.75
132
22.2
171.25
133
23.1
197.00
134
25.3
157.00
135
23.8
168.25
136
26.3
186.00
137
21.4
166.75
138
28.4
187.75
139
21.8
168.25
140
20.1
212.75
141
24.3
176.75
142
18.1
173.25
143
22.7
167.00
144
9.9
159.75
145
10.8
188.15
146
14.4
156.00
147
19.0
208.50
148
28.6
206.50
149
6.1
143.75
150
24.5
223.00
151
9.9
152.25
152
19.1
241.75
153
10.6
146.00
154
16.5
156.75
155
20.5
200.25
156
17.2
171.50
157
30.1
205.75
158
10.5
182.50
159
12.8
136.50
160
22.0
177.25
161
9.9
151.25
162
14.8
196.00
163
13.3
184.25
164
15.2
140.00
165
26.5
218.75
166
19.0
217.00
167
21.4
166.25
168
20.0
224.75
169
34.7
228.25
170
16.5
172.75
171
4.1
152.25
172
1.9
125.75
173
20.2
177.25
174
16.8
176.25
175
24.6
226.75
176
10.4
145.25
177
13.4
151.00
178
28.8
241.25
179
22.0
187.25
180
16.8
234.75
181
25.8
219.25
182
0.0
118.50
183
11.9
145.75
184
12.4
159.25
185
17.4
170.50
186
9.2
167.50
187
23.0
232.75
188
20.1
210.50
189
20.2
202.25
190
23.8
185.00
191
11.8
153.00
192
36.5
244.25
193
16.0
193.50
194
24.0
224.75
195
22.3
162.75
196
24.8
180.00
197
21.5
156.25
198
17.6
168.00
199
7.3
167.25
200
22.6
170.75
201
12.5
178.25
202
21.7
150.00
203
27.7
200.50
204
6.8
184.00
205
33.4
223.00
206
16.6
208.75
207
31.7
166.00
208
31.5
195.00
209
10.1
160.50
210
11.3
159.75
211
7.8
140.50
212
26.4
216.25
213
19.3
168.25
214
18.5
194.75
215
19.3
172.75
216
45.1
219.00
217
13.8
149.25
218
8.2
154.50
219
23.9
199.25
220
15.1
154.50
221
12.7
153.25
222
25.3
230.00
223
11.9
161.75
224
6.1
142.25
225
11.3
179.75
226
12.8
126.50
227
14.9
169.50
228
24.5
198.50
229
15.0
174.50
230
16.9
167.75
231
11.1
147.75
232
16.1
182.25
233
15.5
175.50
234
25.9
161.75
235
25.5
157.75
236
18.4
168.75
237
24.0
191.50
238
26.4
219.15
239
12.7
155.25
240
28.8
189.75
241
17.0
127.50
242
33.6
224.50
243
29.3
234.25
244
31.4
227.75
245
28.1
199.50
246
15.3
155.50
247
29.1
215.50
248
11.5
134.25
249
32.3
201.00
250
28.3
186.75
251
25.3
190.75
252
30.7
207.50
– mongo0517 days ago