An observational study is conducted to investigate the association between age and total serum cholesterol. The correlation is estimated at r = 0.35. The study involves n=125 participants and the mean (std dev) age is 44.3 (10.0) years with an age range of 35 to 55 years, and mean (std dev) total cholesterol is 202.8 (38.4).
Estimate the equation of the line that best describes the association between age (as the independent variable) and total serum cholesterol.
23
To estimate the equation of the line that best describes the association between age (as the independent variable) and total serum cholesterol, you can use simple linear regression. The equation for a simple linear regression model can be represented as:
y = mx + b
where:
- y is the dependent variable (total serum cholesterol in this case)
- x is the independent variable (age in this case)
- m is the slope of the line (the change in y for a unit change in x)
- b is the y-intercept (the value of y when x is 0)
To estimate the equation, you need to find the values of m and b based on the given information. Here's how you can do it step by step:
1. Calculate the mean (average) values of age and total serum cholesterol:
- Mean age (x̄) = 44.3 years
- Mean total serum cholesterol (ȳ) = 202.8
2. Use the formula for the slope (m) of the regression line:
- m = r * (std dev y / std dev x)
- r is the correlation coefficient, given as r = 0.35
- std dev y is the standard deviation of total serum cholesterol, given as 38.4
- std dev x is the standard deviation of age, which we don't have explicitly. But assuming age is normally distributed, you can use the range (55 - 35 = 20) as a rough estimate of the standard deviation.
- Substitute the values into the formula:
- m = 0.35 * (38.4 / 20)
3. Calculate the y-intercept (b):
- b = ȳ - m * x̄
- ȳ is the mean total serum cholesterol, given as 202.8
- x̄ is the mean age, given as 44.3
- m is the slope calculated in step 2
- Substitute the values into the formula:
- b = 202.8 - (m * 44.3)
4. Substitute the calculated values of m and b into the equation y = mx + b to obtain the final equation that describes the association between age and total serum cholesterol.
It's important to note that the quality of the estimate depends on the assumptions made, such as the linearity of the relationship and the normality of the variables. Additionally, since this is an observational study, there could be other confounding factors that influence the association between age and total serum cholesterol that are not accounted for in this analysis.