The following table gives information on ages and cholesterol levels for a random sample of 10
men.
Age58694339635247317436
Cholesterol level189235193177154191213165198181
a. Taking age as an independent variable and cholesterol level as a dependent variable,
compute SSxx, SSyy, and SSxy.
b. Find the regression of cholesterol level on age.
c. Briefly explain the meaning of the values of aand bcalculated in part b.
d. Calculate rand r
2
and explain what they mean.
e. Plot the scatter diagram and the regression line.
f. Predict the cholesterol level of a 60yearold man.
g. Compute the standard deviation of errors.
h. Construct a 95% confidence interval for B.
i. Test at the 5% significance level if Bis positive.
j. Using α = .025, can you conclude that the linear correlation coefficient is positive?
To answer these questions, we will need to perform various calculations and analyze the data. Here's a step-by-step guide on how to get the answers:
a. To compute SSxx, SSyy, and SSxy:
1. Calculate the mean of the age (x) and cholesterol level (y) values.
2. Calculate the deviation of each age value from the mean (subtract mean from each value) and square the deviations.
3. Sum up the squared deviations to get SSxx (the sum of squares of the independent variable, age).
4. Repeat steps 2 and 3 for the cholesterol level values to find SSyy (the sum of squares of the dependent variable, cholesterol level).
5. Multiply each age deviation by the corresponding cholesterol level deviation, then sum up these products to find SSxy (the sum of products of deviations).
b. To find the regression of cholesterol level on age:
1. Calculate the slope (b) using the formula: b = SSxy / SSxx.
2. Calculate the intercept (a) using the formula: a = mean(y) - b * mean(x).
3. The regression equation is then: y = a + b * x.
c. The values of a and b in the regression equation represent the intercept and slope, respectively. The intercept (a) indicates the predicted cholesterol level when age (x) is 0. The slope (b) represents the expected change in cholesterol level for a one-unit increase in age.
d. To calculate r (linear correlation coefficient) and r^2 (coefficient of determination):
1. Calculate the standard deviations of x (age) and y (cholesterol level).
2. Calculate the covariance of x and y.
3. Divide the covariance by the product of the standard deviations to obtain r.
4. Square r to get r^2, which represents the proportion of the variance in y that can be explained by x.
(Note: The sign of r indicates the direction of the relationship between x and y.)
e. Plotting a scatter diagram and the regression line:
1. Create a scatter plot with age (x) on the x-axis and cholesterol level (y) on the y-axis.
2. Plot each observation as a point on the graph.
3. Draw the regression line using the calculated intercept (a) and slope (b) from part b.
f. To predict the cholesterol level of a 60-year-old man:
1. Substitute x = 60 into the regression equation: y = a + b * x.
2. Calculate the predicted cholesterol level using the obtained values of a, b, and x.
g. To compute the standard deviation of errors:
1. Calculate the residuals (errors) by subtracting the predicted values from the actual values.
2. Calculate the sum of the squared residuals.
3. Divide the sum of squared residuals by (n - 2), where n is the sample size, to get the mean squared error.
4. Take the square root of the mean squared error to obtain the standard deviation of errors.
h. To construct a 95% confidence interval for B (the slope):
1. Calculate the standard error of the slope, SE(b), using the formula: SE(b) = sqrt(mean squared error) / sqrt(SSxx).
2. Find the t-value for a 95% confidence interval with (n - 2) degrees of freedom.
3. Create the confidence interval using the formula: b ± t-value * SE(b).
i. To test if B is positive at the 5% significance level:
1. Set up the null hypothesis: Ho: B = 0 (no linear relationship).
2. Set up the alternative hypothesis: Ha: B ≠ 0 (there is a linear relationship).
3. Calculate the t-value using the formula: t = b / SE(b).
4. Compare the calculated t-value with the critical t-value at the 5% significance level.
(Note: If the calculated t-value is greater than the critical t-value, we reject the null hypothesis and conclude that B is significantly different from zero, indicating a linear relationship.)
j. To determine if the linear correlation coefficient is positive at α = .025:
1. Set up the null hypothesis: Ho: ρ = 0 (no linear correlation).
2. Set up the alternative hypothesis: Ha: ρ > 0 (positive linear correlation).
3. Calculate the critical value of r at α/2 = .0125 (half of α) and degrees of freedom (n - 2).
4. Compare the calculated r with the critical value.
(Note: If the calculated r is greater than the critical value, we conclude that there is a positive linear correlation at the 0.025 significance level.)