Money and capital markets

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fat

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

Question

You are hired as a statistical analyst for Silver’s Gym, and your boss wants to examine the relationship between body fat and weight in men who attend the gym. After compiling the data for weight and body fat of 252 men who attend Silver’s Gym, you find it relevant to examine the statistical measures and to perform hypothesis tests and regression analysis to help make general conclusions for body fat and weight in men.
Part I: Statistical Measures
Statistics is a very powerful topic that is used on a daily basis in many situations. For example, you may be interested in the age of the men who attend Silver’s Gym. You could not assume that all men are the same age. Thus, it would be an inaccurate measure to state that "the average age of men who attend Silver’s Gym is the same age as me."
Averages are only one type of statistical measurements that may be of interest. For example, your company likes to gauge sales during a certain time of year and to keep costs low to a point that the business is making money. These various statistical measurements are important in the world of statistics because they help you make general conclusions about a given population or sample.
To assist in your analysis for Silver’s Gym, answer the following questions about the Body Fat Versus Weight data set:
Click here to download the Body Fat Weight data set.
Calculate the mean, median, range, and standard deviation for the Body Fat Versus Weight data set. Report your findings, and interpret the meanings of each measurement. Notice you are to calculate the mean, median, range, and standard deviation for the body fat and for the weight.
The measures of central tendency are important in real-world situations.
What is the importance of finding the mean/median? Why might you find this information useful?
In some data sets, the mean is more important than the median. For example, you want to know your mean overall grade average because the median grade average would be meaningless. However, you might be interested in a median salary to see the middle value of where salaries fall. Explain which measure, the mean or the median, is more applicable for this data set and this problem.
What is the importance of finding the range/standard deviation? Why might you find this information useful?
Part II: Hypothesis Testing
Organizations sometimes want to go beyond describing the data and actually perform some type of inference on the data. Hypothesis testing is a statistical technique that is used to help make inferences about a population parameter. Hypothesis testing allows you to test whether a claim about a parameter is accurate or not.
Your boss makes the claim that the average body fat in men attending Silver’s Gym is 20%. You believe that the average body fat for men attending Silver’s Gym is not 20%. For claims such as this, you can set up a hypothesis test to reach one of two possible conclusions: either a decision cannot be made to disprove the body fat average of 20%, or there is enough evidence to say that the body fat average claim is inaccurate.
To assist in your analysis for Silver’s Gym, consider the following steps based on your boss’s claim that the mean body fat in men attending Silver’s Gym is 20%:
First, construct the null and alternative hypothesis test based on the claim by your boss.
Using an alpha level of 0.05, perform a hypothesis test, and report your findings. Be sure to discuss which test you will be using and the reason for selection. Recall you found the body fat mean and standard deviation in Part I of the task.
Based on your results, interpret the final decision to report to your boss.

Part I

BODYFAT WEIGHT
Mean 18.9 178.9
Median 19.0 176.5
Standard Deviation 7.8 29.4
Minimum 0.0 118.5
Maximum 45.1 363.2
Range 45.1 244.7
Count 252 252
Interpretations
Mean: This is the most important measure of centre tendency. It is calculated as the sum of of all the observations divided by the number of observation. The mean value of the Bodyfat is 18.9 and the weight is 178.9 for the men in Silver Gym
Median: It is the most middle value of the data set for the men in Silver gym. In other words, here data is ranked either in ascending or descending order, and the two middle numbers are averaged. Here, the median value of the bodyfat is 19.0 whereas the weight of the men attending Silver gym is 176.5
Standard deviation : It is the measure of the variability in the data set. Bascially, it tell us how much spread is there in the data set. The standard deviation for bodyfat is 7.8 and for weight is 29.4
About 95% of the values fall between two SD of the mean. In this case, that means bodyfat tends to fall between 3.4 and 34.4%
About 95% of people have bodyweights between 120.1 and 237.7 lbs
Range: The range is the difference between the minimum and maximum values. Range of bodyfat is 45.1 and the weight is 244.7
The mean and median are important because they are two ways of showing the typical value. 
The mean can be more important when we want to represent all the variability, and there are no significant outliers. 
The median is a better measure when the data is skewed or there are a lot of outliers. 
For example, if we tried to show the typical income of Americans by using an average, we'd end up averaging in the very high incomes of movie stars and CEOs. Since those salaries are very non-typical, a median would be a better measure. 
The range and standard deviation are important because they show the variability. For example, if the data in this set were sampled from Olympic swimmers, we'd find that the range of bodyfat was probably no more than 5 - 10% 
The standard deviation is important because it helps us make inferences about the data. With the standard deviation we can determine how likely it is that a value falls within a certain range. 

Part II

Statement: The doctors office claims that the mean body fat in men is 20%.
Null hypthesis: H0: Mean bodyfat = 20 
Alternative hypthesis: Ha: Mean bodyfat ≠ 20 
For the purpose of hypothesis testing, I will use Z-test for means. This is becauset the sample size is fairly large (N>30)
Critical value of z two-tailed at alpha = 0.05   -1.96
z = (xbar - µ)/(σ/√n) 
z = (18.9 - 20 )/(7.8/√252) 
z = -1.1 / (7.8/15.8745) 
z = -2.2387 
Interpretation: Since the Z < -1.96, there is sufficient evidence to reject the null hypothesis. This means the claim made by doctors office that the mean bodyfat is 20% is not true.
I'd tell my boss that the hypothesis test showed that the mean was significantly different from 20%, and that it's actually lower than that.