Statistical analysis of the Relationship between Body Fat and Weight

**Part 1: Statistical Measures**

Statistics avails a number of tools necessary in the analysis of most phenomenons in the daily cycles. The analysis will look at the age of the men who are members of Silver’s Gym. It would be rather misleading to make an assumption that all men have a similar age. This situation prove that it is indeed an inaccurate to lay a procedure suggesting that the age of male attendees at the gym is similar on average. There are variables rather, that in analysis of gym tend to be normally distributed across all ages. These give a clear picture with regards to continuous probability distribution. An example is the relationship that exists between the weight of individuals and their levels of body fat.

In the analysis of the relationship between body fat and weight on men attending the gym, we focus of some key statistical measures; these include the median, mean, range, and standard deviation for the data set.

**Values and Interpretations**

**Mean**

This is a measure of the average the total observations. In silver gyms the total observations are 252, the total body fat and total weights divided by the total observations gives the means of.

Body fat = 18.89841

Weight = 178.90377

**Median**

The median involves arranging the data from the smallest to the largest, the data set that stands in the middle cutting the total observation into two halves is the median. In even values of total observations as the case with our gym figure the middle two values are used to determine the median. This gives us a result of

Body fat = 19

Weight = 176.5

**Range**

This can be determined from the observations of the weights and body fats as the difference between the highest and lowest values in the observations.

Body fats = 3

Weight = 117

**Standard Deviations**

Determining the range encompasses the variability in the given data set. In other words it is a measure of how spread the values are. It can be determined by a formula

This gives us a std dev. of;

Body fat = 7.765

Weight = 29.386

**Central tendency**

Measure of central tendency is a vital aspect of day to day real life statistics. In finding central tendency in mean gives us one representative value for the entire observations. In this way representing many related values as one value. In finding median, we are able to determine the central value where the observations lie. Mean and median are also very useful as a means of condensing the data often vast and classified into single values that can help answer questions on the entire observations and reflect their distributions.

In the observations regarding the gym values, the mean would give us the actual average percentage of body fat given the weights. The median on the other hand would help us identify the central weights and body fats for the individuals. It is therefore evident that in our bid to determine the relationship between the body fats and weights of the men, of key consideration will be the means.

**Part II: Hypothesis Testing**

From the claim by the Boss that the average fat in men attending the gym is 20% we can structure our hypothesis as;

Null hypothesis

The average body fat relative to the weights of men attending the gym is not 20%.

Alternative hypothesis

The average body fat relative to the weights of men attending the gym is 20%.

Taking the critical value approach, the probability that the test statistic end up in the desired interval is the alpha level. Now basing our statistics on decision rule, the confidence interval is given by 100(1-alpha). This is the interval of values likely to include the parameter. This alpha level gives 95 percent. Given this the number of men with fats at 20% level of their body weight have to be 95% to prove the null hypothesis invalid.

On taking the mean values; Body fat = 18.89841 and Weight = 178.90377 and determining the percent of body fats.

% fats =18.89841/178.90377 ×100

= 10.56345

On this representation the number of men with 20% of fats in their weights evidently is not 95% of the 252 observations. This confirms the validity of the null hypothesis and rejects the alternative hypothesis.

The final decision based on the interpretation of the analysis is that, the fat level in the weights of the men attending the gym stands lower than the postulated 20%. The value on average stands at 10.56345%.

**References**

- Sharma, N. Measure of Central Tendency(Mean, Median and Mode). Shake hand with Life, 2006.
- Aha journals (n.d) www.circ.ahajournals.org/content/114/10/1078
- Quora (n.d) www.quora.com/Statistical-Hypothesis-Testing