Based solely on these statistical tests, we can conclude that exercise and a reduction in cholesterol levels are statistically significantly related.
Regression Analysis of Exercise and Cholesterol Reduction
Data:
Patient Exercise Reduction
1 248 110
2 386 114
3 113 43
4 386 99
5 496 152
6 192 22
7 399 132
8 296 45
9 238 62
10 193 1
11 260 37
12 482 97
13 227 78
14 95 21
15 128 -21
Regression Model:
Reduction = β₀ + β₁ * Exercise + ε
Coefficients:
Y-intercept (β₀) = 21.434
Coefficient for exercise (β₁) = 0.2264 (rounded to 4 decimals)
F-statistic:
F = 45.75
p-value for Regression F-test:
p-value ≈ 0.0000 (rounded to 4 decimals)
p-value for t-test of Exercise coefficient:
p-value ≈ 0.0000 (rounded to 4 decimals)
Alternate Hypothesis for t-test:
The coefficient for exercise (β₁) is not equal to zero. This implies that there is a linear relationship between exercise and reduction in cholesterol levels.
Conclusion:
The F-test with a tiny p-value rejects the null hypothesis of an invalid regression model (p < 0.05). This suggests a significant link between exercise and reduced cholesterol. The t-test, also with a small p-value, supports the non-zero coefficient for exercise, affirming the relationship.
Therefore, based solely on these statistical tests, we can conclude that exercise and a reduction in cholesterol levels are statistically significantly related.
The Complete Question
Exercise is often recommended by medical doctors to address health problems. The reduction of cholesterol appears to be a benefit of exercise. To study the relationship more carefully, a medical doctor took a random sample of 15 patients who do not exercise and measured their cholesterol levels. All patients then started a regular exercise program. At the end of 3 months, each patient reported the average time per week they exercised (in minutes) and their cholesterol level.
The medical doctor then calculated the reduction of their cholesterol levels from before the patients began their exercise program (below). Analyze these data to determine if exercise & a reduction in cholesterol level are related.
Patient Exercise Reduction
1 248 110
2 386 114
3 113 43
4 386 99
5 496 152
6 192 22
7 399 132
8 296 45
9 238 62
10 193 1
11 260 37
12 482 97
13 227 78
14 95 21
15 128 -21
What are the coefficients of the regression model? (report to 4 decimals): Y-intercept? Coefficient for exercise?
What is the value of F-statistic for the regression?
What is the p-value for the regression F-test? (report to 4 decimals)
What is the p-value for the t-test of the coefficient for exercise?
In general: what is the alternate hypothesis for the t-test of the coefficient of an independent variable in a regression analysis? (i.e. what are we seeking to prove with the t-test)? Sentence form.
Based solely on the F-test and t-test, the regression model is valid or not valid? And we conclude that exercise & a reduction in cholesterol are related or not related?