Answer to Question #257992 in Statistics and Probability for hari

Question #257992
In a certain experiment to compare two types of drugs A and B, the following results of increase in weights (in kg.) were observed in 5 persons. Person number 1 2 3 4 5 Increase weight in kg. Drug A 3.6 5.5 5.9 4.1 1.4 Drug B 4.5 3.6 5.5. 6.8 2.7 (i) Assuming that the two samples of persons are independent; can we conclude that the drug B is better than drug A? (ii) Also, examine the case when the same set of 5 persons was used in both the drugs.
1
Expert's answer
2021-10-29T04:04:36-0400

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 \n_1\n\u200b\t\n =1.785;\ns_2 = 1.378"

and the sample sizes are "n_1 = 5\n\nand" "n_2 = 7"


The following null and alternative hypotheses need to be tested:

"Ho: \\mu_1= \\mu_2"

"Ha: \\mu_1\u2260 \\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.


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