The following are body mass index (BMD) scores measured in 12 patients who are free of diabetes and participating in a study of risk factors for obesity. Body mass index is measured as the ratio of weight in kilograms to height in meters squared.

Generate a 95% confidence interval estimate of the true BMI.

25 27 31 33 26 28 38 41 24 32 35 40

#### Solution

Given that sample size `$n = 12$`

, sample mean `$\overline{X}= 31.667$`

, sample standard deviation `$s = 5.883$`

.

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

##### Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is `$1-\alpha = 0.95$`

. Thus, the level of significance is `$\alpha = 0.05$`

.

##### Step 2 Given information

Given that sample size `$n =12$`

, sample mean `$\overline{X}=31.6667$`

, sample standard deviation `$s=5.883$`

.

##### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the population mean $\mu$ is

` $$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$ `

where `$E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}$`

, and `$t_{\alpha/2, n-1}$`

is the $t$ value providing an area of $\alpha/2$ in the upper tail of the students' $t$ distribution.

##### Step 4 Determine the critical value

The critical value of $t$ for given level of significance and $n-1$ degrees of freedom is `$t_{\alpha/2,n-1}$`

.

Thus `$t_{\alpha/2,n-1} = t_{0.025,12-1}= 2.201$`

.

##### Step 5 Compute the margin of error

The margin of error for mean is

` $$ \begin{aligned} E & = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}\\ & = 2.201 \frac{5.883}{\sqrt{12}} \\ & = 3.738. \end{aligned} $$ `

##### Step 6 Determine the confidence interval

$95$% confidence interval estimate for population mean is

` $$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 31.667 - 3.738 & \leq \mu \leq 31.667 + 3.738\\ 27.929 &\leq \mu \leq 35.404. \end{aligned} $$ `

Thus, `$95$`

% confidence interval estimate for population mean is `$(27.929,35.404)$`

.

#### Further Reading

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