Answer to Question #280188 in Statistics and Probability for ann

Question

 Among 157 African-American men, the mean systolic blood pressure was 146 mm Hg with a standard deviation of 27. We wish to know if on the basis of these data, we may conclude that the mean systolic blood pressure for a population of African-American is greater than 140. Use α=0.01. 

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\leq 140$

$H_1:\mu>140$

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.01,$ $df=n-1=157-1=156$ degrees of freedom, and the critical value for a right-tailed test is $t_c =2.350489.$

The rejection region for this right-tailed test is $R = \{t: t > 2.350489\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{146-140}{27/\sqrt{157}}\approx2.784436$$

Since it is observed that $t = 2.784436>2.350489=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, $df=156, t=2.784436,$ is $p = 0.003013,$

and since $p = 0.003 013< 0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is greater than $140,$ at the $\alpha = 0.01$ significance level.

Therefore, there is enough evidence to conclude that the mean systolic blood pressure for a population of African-American is greater than $140,$ at the $\alpha = 0.01$ significance level.