Question

The following data, recorded in days, represent the length of time to recovery for patients randomly treated with one of two medications to clear up severe bladder infections: Assume that the recovery times are normally distributed. Medication 1 = n =13 , XBAR = 17 , sample variance = 1.5 Medication2 = n = 10 , XBAR = 19 , sample variance= 1.8 (a) Is there a difference in the mean recovery times for the two medications? Test at the 5% level of significance, assuming equal variances. (b) Is the assumption of equal variances made in (a) valid. Test at the 5% level of significance.

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

(2) Rejection Region

Based on the information provided, the significance level is α = 0.05, and the degrees of freedom are df = 21. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is tc ​= 2.08, for α=0.05 and df = 21.

(3) Test Statistics

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

(4) Decision about the null hypothesis

Since it is observed that |t| = 3.725 > tc ​= 2.08, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0013, and since p = 0.0013 < 0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

at the 0.05 significance level. there is a difference in the mean recovery times for the two medications

b) From above hypothesis results, yes it is valid