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?

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.

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.

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.

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.