Question

Improving health A large company's medical director launches a health promotion campaign to encourage employees to exercise more and eat better foods. One measure of the effectiveness of such a program is a drop in blood pressure. The director chooses a random sample of $50$employees and compares their blood pressures from physical cams given before the campaign and again a year later. The mean change (after - before) in systolic blood pressure for these $50$employees is $-6$and the standard deviation is $19.8$.

(a) Do these data provide convincing evidence of an average decrease in blood pressure among all of the company's employees during this year? Carry out a test at the $\alpha =0.05$significance level.

(b) Can we conclude that the health campaign caused a decrease in blood pressure? Why or why not?

Short answer

(a) Yes, there is sufficient evidence

(b)No

Step-by-step solution

Part (a) Step 1 : Given Information 

Given in the question that,

Population mean$\left(\mu \right)=0$

Sample mean$\left(\overline{x}\right)=-6$

Sample size$\left(n\right)=50$

Sample standard deviation$\left(s\right)=19.8$we have to find Do these data provide convincing evidence of an average decrease in blood pressure among all of the company's employees during this year.

Part (a) Step 2 :Explanation

The formula to compute the test statistic is:

$t=\frac{\overline{x}-\mu }{{\displaystyle \frac{s}{\sqrt{n}}}}$

using the information, the null and alternative hypothesis are:

$H_0:\mu =0$

$H_{\alpha }:\mu <0$

$H_0:\mu <0$

Using the Ti-83 calculator the output is:

The p-value is less than the significance level. Thus, decision is written as the rejection of the null hypothesis

At $5\%$significance level, there are sufficient evidence to support the provided claim.

 Part (b) Step 1: Given Information

Given in the question that,

Population mean$\left(\mu \right)=0$

Sample mean$\overline{x}=-6$

Sample size$\left(n\right)=50$

Sample standard deviation$\left(s\right)=19.8$we have to find Can we conclude that the health campaign caused a decrease in blood pressure.

Part (b) Step 2: Explanation 

Because the outcome of an observational study is unaffected, and the case presented is an observational study. As a result, no conclusion can be formed at this time.