In most research, the independent/predictor variable and dependent/response variable are determined before the start of the study. They may be identified by intuition, theory, or experiment. For example, if one variable is gender and the other is memory for phone numbers, intuition would indicate that the predictor variable is gender because it does not change. A theory which says that nightmares cause bedwetting makes the statement that nightmares are the predictor variable and bedwetting the response. If there is still doubt about which variable is which, conducting an experiment might help. But it isn’t always easy to tell whether one variable actually causes or changes another one.
In the Body Fat Versus Weight data set, body fat is the predictor variable and weight is the response variable. Although it would be possible to reverse them, body fat is a better choice for independent variable because it has a true zero.
The scatter plot on the attached Excel spreadsheet suggests that there is a positive correlation between body fat and weight — that is, as body fat increases, weight increases. This is apparent because y generally moves higher (up) as the x value moves higher (to the right).
Using Excel, r was calculated to be 0.62 (r2=0.38). This agrees with the scatter plot, which shows a positive correlation. The square of the correlation coefficient (r2) determines the amount of variance in common between x and y. In this case, 38% of the variance of x and y is in common (Rummel, 2013). A correlation coefficient of 0.62 indicates the correlation is moderately strong (BMJ,2013).
The regression line shown on the Excel spreadsheet is a fairly good fit for this data, since the data points cluster around the line, with the exception of a few outliers. The equation for this line is
y = 2.32x + 134.89
The slope of the line is 2.32, and the y-intercept (the point at which x=0) is 134.89. The slope is a measure of how quickly the y values increase as x values increase. If x (body fat) is zero, then 2.32x is also 0, and the predicted weight would be 134.89. According to the equation, x has a true zero.
Summary of Results
Based on the body fat and weight data obtained from the 252 men that attend Silver’s Gym, the mean body fat percentage is 18.9% (SD=7.8%) and the mean weight is 178.92 lbs. (SD=20.39 lbs.). The hypothesis that the mean body fat percentage was 20% was rejected based on a z-test (α=0.05). The 95% confidence interval is 17.94% – 19.86 %; since 20% is outside this range, the true mean of the body fat percentage is significantly different from 20%. Body fat percentage and weight are positively correlated, since they move up or down together, and the correlation is moderately strong. The equation for the trend line indicates that a person with a body fat percentage of 0 would be expected to have a weight of 134.89 lbs.
- BMJ. (2013). Correlation and Regression. Retrieved from http://www.bmj.com/about-bmj/resources-readers/publications/statistics-square-one/11-correlation-and-regression.
- Rummel, R.J. (2013). Understanding Correlation. Retrieved from http://www.mega.nu/ampp/rummel/uc.htm#C3