Data Driven Statistics

Parts I–II: Review and revise your individual
project from last week. 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

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 or 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%.