Answer to Question #132844 in Statistics and Probability for rix_Rachel

Question

Two different types of drugs A and B were tried on different patients for increasing weight. 5 persons were given A and 7 persons were given B. Increase in weights(in. kg) are given below:
A: 3.6 5.5 5.9 4.1 1.4
B: 4.5 3.6 5.5 6.8 2.7 3.6 5.0
Do the drugs differ significantly with regard to their effect in increasing weight?

----- This question is from the Test of Hypothesis -----

Answer 1

Using TI84, the provided sample means are shown below:

$\bar X_1 = 4.1$ $\bar X_2$ =4.529

Also, the provided sample standard deviations are :$s _1 ​ =1.785; s_2 = 1.378$

and the sample sizes are  $n_1 = 5 and$ $n_2 = 7$

The following null and alternative hypotheses need to be tested:

$Ho: \mu_1= \mu_2$

$Ha: \mu_1≠ \mu_2$

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df = $df=\frac{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}{\frac{\frac{s_{1}^{2}}{n_1}}{n_1-1}+\frac{\frac{s_{2}^{2}}{n_2}}{n_2-1}}$ =$df=\frac{\frac{1.785^{2}}{5}+\frac{1.378^{2}}7}{\frac{\frac{1.785^{2}}5}{5-1}+\frac{\frac{1.378^{2}}{7}}{7-1}}$= 7.254

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

$t = \frac{\overline{X1}-\overline{X2}}{\sqrt{\frac{(s1^2)}{n1}+\frac{s2^2}{n2}}}$ = $\frac{4.1-4..529}{\sqrt{\frac{(1.7851^2)}{5}+\frac{1.378^2}{7}}}$ = -0.45

Hence, it is found that the critical value for this two-tailed test is tc

​=2.348, for α=0.05 and df = 7.254, using t table the rejection region for this two-tailed test is R={t:∣t∣>2.348}.since critical value is greater than test statistic ,it is concluded that the null hypothesis is not rejected.

t is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1 is different than μ2​, at the 0.05 significance level.