The mean BMI in patients free of diabetes was reported as 28.2. The investigator conducting the study described in Problem 1 hypothesizes that the BMI in patients free of diabetes is higher. Based on the data in Probiem 1 is there evidence that the BMI is significantly higher than 28.2? Use a 5% level of significance.

Problem 1

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


Given that the sample size $n = 12$, sample mean $\overline{x}= 31.667$ and sample standard deviation $s = 5.883$.

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : \mu = 28.2$ against $H_1 : \mu > 28.2$ ($\text{right-tailed}$)

Step 2 Test Statistic

The test statistic is

$$ \begin{aligned} t& =\frac{\overline{x} -\mu}{s/\sqrt{n}} \end{aligned} $$
which follows $t$ distribution with $n-1$ degrees of freedom.

Step 3 Significance Level

The significance level is $\alpha = 0.05$.

Step 4 Critical Value(s)

As the alternative hypothesis is $\text{right-tailed}$, the critical value of $t$ $\text{is}$ $1.796$.

t-critical right-tailed
t-critical right-tailed

The rejection region (i.e. critical region) is $\text{t > 1.796}$.

Step 5 Computation

The test statistic under the null hypothesis is

$$ \begin{aligned} t&=\frac{ \overline{x} -\mu_0}{s/\sqrt{n}}\\ &= \frac{31.667-28.2}{5.883/ \sqrt{12 }}\\ &= 2.041 \end{aligned} $$

Step 6 Decision (Traditional Approach)

The test statistic is $t =2.041$ which falls $\text{inside}$ the critical region, we $\text{reject}$ the null hypothesis.


Step 6 Decision ($p$-value Approach)

This is a $\text{right-tailed}$ test, so the p-value is the area to the right of the test statistic ($t=2.041$). The p-value = $0.033$.

The p-value is $0.033$ which is $\text{less than}$ the significance level of $\alpha = 0.05$, we $\text{reject}$ the null hypothesis.

We conclude that the BMI is significantly higher than 28.2 at a 5% level of significance.