Question

Question 4 (14 marks) Triglycerides are the main component of body fat in humans and animals. The triglyceride level in blood is a common measure in assessing a person's susceptibility to coronary heart disease. To determine whether regular exercise can reduce the triglyceride levels in blood for men, researchers measured the level in seven male volunteers, before and after a 10 week supervised exercise program. The measured blood serum triglycerides levels (mmol/L) are below: Participant 1 2 3 4 5 6 7 Before 0.87 1.13 3.14 2.14 2.98 1.18 1.60 After | 0.57 1.03 | 1.47 | 1.43 | 1.20 1.09 1.51 a) Compute and interpret a 95% confidence interval for the mean change in triglyceride levels due to the exercise program. b) Based on this confidence interval, is there sufficient evidence to conclude that there has been a change in mean triglyceride level? c) Conduct a test of significance to determine if the mean triglyceride level has been reduced by the exercise program. Do the two approaches (95% confidence interval and test at 5%) agree in their conclusions? If not, why not? d) Dome

Answer 1

$\bar d$ = (0.3 + 0.1 + 1.67 + 0.71 + 1.78 + 0.09 + 0.09)/7 = 0.677

s_d = sqrt(((0.3 - 0.677)^2 + (0.1 - 0.677)^2 + (1.67 - 0.677)^2 + (0.71 - 0.677)^2 + (1.78 - 0.677)^2 + (0.09 - 0.677)^2 + (0.09 - 0.677)^2)/7) = 0.6936

a) At 95% confidence interval the critical value is t^* = 2.447

The 95% confidence interval is

$\bar d$ +/- t^* * s_d/$\sqrt n$

= 0.677 +/- 2.447 * 0.6936/$\sqrt 7$

= 0.677 +/- 0.6415

= 0.0355, 1.3185

b) The interval doesn't not contain 0. So there is sufficient evidence to conclude that there has been a change in mean triglyceride level.

c) H₀: $\mu_{d}$ = 0

H₁: $\mu_{d}$ > 0

The test statistic t = ($\bar d$ - D)/(s_d/$\sqrt n$)

= (0.677 - 0)/(0.6936/$\sqrt 7$)

= 2.582

At 0.05 significance level the critical value is t_{0.95, 6} = 1.943

Since the test statistic value is greater than the critical value (2.582 > 1.943), so we should reject the null hypothesis.

At 0.05 significance level, there is sufficient evidence to conclude that the mean triglyceride level has been reduced by the exercise program.

d) Yes, the two approaches 95% confidence interval and test at 5% agree in their conclusions.