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:
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.
BODYFAT WEIGHT
12.6 154.25
6.9 173.25
24.6 154.00
10.9 184.75
27.8 184.25
20.6 210.25
19.0 181.00
12.8 176.00
5.1 191.00
12.0 198.25
7.5 186.25
8.5 216.00
20.5 180.50
20.8 205.25
21.7 187.75
20.5 162.75
28.1 195.75
22.4 209.25
16.1 183.75
16.5 211.75
19.0 179.00
15.3 200.50
15.7 140.25
17.6 148.75
14.2 151.25
4.6 159.25
8.5 131.50
22.4 148.00
4.7 133.25
9.4 160.75
12.3 182.00
6.5 160.25
13.4 168.00
20.9 218.50
31.1 247.25
38.2 191.75
23.6 202.25
27.5 196.75
33.8 363.15
31.3 203.00
33.1 262.75
31.7 205.00
30.4 217.00
30.8 212.00
8.4 125.25
14.1 164.25
11.2 133.50
6.4 148.50
13.4 135.75
5.0 127.50
10.7 158.25
7.4 139.25
8.7 137.25
7.1 152.75
4.9 136.25
22.2 198.00
20.1 181.50
27.1 201.25
30.4 202.50
24.0 179.75
25.4 216.00
28.8 178.75
29.6 193.25
25.1 178.00
31.0 205.50
28.9 183.50
21.1 151.50
14.0 154.75
7.1 155.25
13.2 156.75
23.7 167.50
9.4 146.75
9.1 160.75
13.7 125.00
12.0 143.00
18.3 148.25
9.2 162.50
21.7 177.75
21.1 161.25
18.6 171.25
30.2 163.75
26.0 150.25
18.2 190.25
26.2 170.75
26.1 168.00
25.8 167.00
15.0 157.75
22.6 160.00
8.8 176.75
14.3 176.00
20.2 177.00
18.1 179.75
9.2 165.25
24.2 192.50
9.6 184.25
17.3 224.50
10.1 188.75
11.1 162.50
17.7 156.50
21.7 197.00
20.8 198.50
20.1 173.75
19.8 172.75
21.9 196.75
24.7 177.00
17.8 165.50
19.1 200.25
18.2 203.25
17.2 194.00
21.0 168.50
19.5 170.75
27.1 183.25
21.6 178.25
20.9 163.00
25.9 175.25
16.7 158.00
19.8 177.25
14.1 179.00
25.1 191.00
17.9 187.50
27.0 206.50
24.6 185.25
14.8 160.25
16.0 151.50
14.0 161.00
17.4 167.00
26.4 177.50
17.4 152.25
20.4 192.25
15.0 165.25
18.0 171.75
22.2 171.25
23.1 197.00
25.3 157.00
23.8 168.25
26.3 186.00
21.4 166.75
28.4 187.75
21.8 168.25
20.1 212.75
24.3 176.75
18.1 173.25
22.7 167.00
9.9 159.75
10.8 188.15
14.4 156.00
19.0 208.50
28.6 206.50
6.1 143.75
24.5 223.00
9.9 152.25
19.1 241.75
10.6 146.00
16.5 156.75
20.5 200.25
17.2 171.50
30.1 205.75
10.5 182.50
12.8 136.50
22.0 177.25
9.9 151.25
14.8 196.00
13.3 184.25
15.2 140.00
26.5 218.75
19.0 217.00
21.4 166.25
20.0 224.75
34.7 228.25
16.5 172.75
4.1 152.25
1.9 125.75
20.2 177.25
16.8 176.25
24.6 226.75
10.4 145.25
13.4 151.00
28.8 241.25
22.0 187.25
16.8 234.75
25.8 219.25
0.0 118.50
11.9 145.75
12.4 159.25
17.4 170.50
9.2 167.50
23.0 232.75
20.1 210.50
20.2 202.25
23.8 185.00
11.8 153.00
36.5 244.25
16.0 193.50
24.0 224.75
22.3 162.75
24.8 180.00
21.5 156.25
17.6 168.00
7.3 167.25
22.6 170.75
12.5 178.25
21.7 150.00
27.7 200.50
6.8 184.00
33.4 223.00
16.6 208.75
31.7 166.00
31.5 195.00
10.1 160.50
11.3 159.75
7.8 140.50
26.4 216.25
19.3 168.25
18.5 194.75
19.3 172.75
45.1 219.00
13.8 149.25
8.2 154.50
23.9 199.25
15.1 154.50
12.7 153.25
25.3 230.00
11.9 161.75
6.1 142.25
11.3 179.75
12.8 126.50
14.9 169.50
24.5 198.50
15.0 174.50
16.9 167.75
11.1 147.75
16.1 182.25
15.5 175.50
25.9 161.75
25.5 157.75
18.4 168.75
24.0 191.50
26.4 219.15
12.7 155.25
28.8 189.75
17.0 127.50
33.6 224.50
29.3 234.25
31.4 227.75
28.1 199.50
15.3 155.50
29.1 215.50
11.5 134.25
32.3 201.00
28.3 186.75
25.3 190.75
30.7 207.50
To calculate the mean, median, range, and standard deviation for the Body Fat Versus Weight data set, follow these steps:
Step 1: Organize the data set
BODYFAT WEIGHT
12.6 154.25
6.9 173.25
24.6 154.00
10.9 184.75
27.8 184.25
20.6 210.25
...
30.7 207.50
Step 2: Calculate the mean for body fat
To calculate the mean, add up all the body fat values and divide by the total number of values:
Mean = Sum of body fat values / Number of values
For example, the mean for body fat is calculated as follows:
Mean = (12.6 + 6.9 + 24.6 + 10.9 + 27.8 + 20.6 + ...) / Total number of values
Step 3: Calculate the median for body fat
To calculate the median, arrange the body fat values in ascending order and find the middle value. If there is an even number of values, take the average of the two middle values.
For example, to find the median for body fat:
1. Sort the body fat values in ascending order: 4.1, 1.9, 0.0, 4.6, 4.7, 5.0, 5.1, 6.1, ...
2. Since there are 252 values, the median will be the 126th value.
Step 4: Calculate the range for body fat
To calculate the range, subtract the smallest value from the largest value:
Range = Largest value - Smallest value
For example, to find the range for body fat:
Range = Largest body fat value - Smallest body fat value
Step 5: Calculate the standard deviation for body fat
The standard deviation measures how spread out the values are from the mean. It can be calculated using the following formula:
Standard Deviation = sqrt((Sum of (x - mean)^2) / (Total number of values - 1))
For example, to calculate the standard deviation for body fat:
1. Calculate the difference between each body fat value and the mean.
2. Square each difference.
3. Sum up all the squared differences.
4. Divide the sum by the total number of values minus 1.
5. Take the square root of the result.
Repeat steps 2-5 for weight to calculate the mean, median, range, and standard deviation for weight.