The following data were collected in a clinical trial to compare a new drug to a placebo for its effectiveness in lowering total serum cholesterol. Generate a 95% confidence interval for the difference in mean total cholesterol levels between treatments.

. | New Drug (n=75) | Placebo (n = 75) | Total Sample (n = 150) |
---|---|---|---|

Mean (SD) Total Serum Cholesterol | 185.0 (24.5) | 204.3 (21.8) | 194.7 (23.2) |

% of patients with Total Cholesterol < 200 | 78.0% | 65.0 % | 71.5 % |

#### Solution

Given that `$n_1 = 75$`

, `$\overline{X}_1 =185$`

, `$s_1 = 24.5$`

, `$n_2 =75$`

, `$\overline{X}_2 =204.3$`

and `$s_2 = 21.8$`

.

**Specify the confidence level $(1-\alpha)$**

The confidence level is `$1-\alpha = 0.95$`

, thus `$\alpha = 0.05$`

.

**Given information**

Given that `$n_1 = 75$`

, `$\overline{X}_1= 185$`

, `$s_1 = 24.5$`

.

`$n_2 = 75$`

, `$\overline{X}_2= 204.3$`

, `$s_2 = 21.8$`

.

**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 `$t_{\alpha/2,n_1+n_2-2} = t_{0.025,148} = 1.976$`

.

**Compute the margin of error**

The margin of error for proportions is

` $$ \begin{aligned} E & = t_{\alpha/2,n_1+n_2-2} \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ & = 1.976 \sqrt{\frac{24.5^2}{75}+\frac{21.8^2}{75}}\\ & = 7.483. \end{aligned} $$ `

**Determine the confidence interval**

`$95$%`

confidence interval estimate for the difference `$(\mu_1-\mu_2)$`

is

` $$ \begin{aligned} (\overline{X} -\overline{Y})- E & \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E\\ (185-204.3) - 7.483 & \leq (\mu_1-\mu_2) \leq (185-204.3) + 7.483\\ -26.783 & \leq (\mu_1-\mu_2) \leq -11.817. \end{aligned} $$ `

Thus, `$95$%`

confidence interval for the difference in mean total cholesterol levels between treatments `$(\mu_1-\mu_2)$`

is `$(-26.783,-11.817)$`

.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators