A study is run comparing HDL cholesterol levels between men who exercise regularly and those who do not. The data are shown below.
| Regular Exercise | N | Mean | Std Dev |
|---|---|---|---|
| Yes | 35 | 48.5 | 12.5 |
| No | 120 | 56.9 | 11.9 |
Generate a 95 % confidence interval for the difference in mean HDL levels between men who exercise regularly and those who do not.
Solution
Given that $n_1 = 35$, $\overline{X}_1 =48.5$, $s_1 = 12.5$, $n_2 =120$, $\overline{X}_2 =56.9$ and $s_2 = 11.9$.
Specify the confidence level $(1-\alpha)$
The confidence level is $1-\alpha = 0.95$, thus $\alpha = 0.05$.
Given information
Given that $n_1 = 35$, $\overline{X}_1= 48.5$, $s_1 = 12.5$.
$n_2 = 120$, $\overline{X}_2= 56.9$, $s_2 = 11.9$.
Specify the formula
$100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is
$$ \begin{aligned} (\overline{X} -\overline{Y})- E \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E. \end{aligned} $$
where $E = t_{\alpha/2,n_1+n_2-2} \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$.
Determine the critical value
The critical value $t_{\alpha/2,n_1+n_2-2} = t_{0.025,153} = 1.976$.
Compute the margin of error
The margin of error for proportions is
$$ \begin{aligned} E & = t_{\alpha/2,n_1+n_2-2} \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ & = 1.976 \sqrt{\frac{12.5^2}{35}+\frac{11.9^2}{120}}\\ & = 4.695. \end{aligned} $$
Determine the confidence interval
$95$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is
$$ \begin{aligned} (\overline{X} -\overline{Y})- E & \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E\\ (48.5-56.9) - 4.695 & \leq (\mu_1-\mu_2) \leq (48.5-56.9) + 4.695\\ -13.095 & \leq (\mu_1-\mu_2) \leq -3.705. \end{aligned} $$
Thus, $95$% confidence interval for the difference in mean HDL levels between men who exercise regularly and those who do not is $(-13.095,-3.705)$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
