6· (Three parts; 12 marks in total) An experiment was conducted to test the effe

Question

6· (Three parts; 12 marks in total) An experiment was conducted to test the effectiveness of a new drug for the treatment of high blood pressure. Forty people with high blood pressure were randomly allocated to two groups (20 per group), a control group receiving no drug (just a placebo) and a treatment group receiving the new drug. The systolic blood pressure (SBP) of each person was recorded after six weeks. Incomplete SPSS output of the analysis is shown below. The data are normally distributed. Group Statistics Drug No New Mean Std. Deviation Std. Error Mean 20 149.000 SBP 20 141.500 Independent Samples Test t-test for Equality of Means Levene's Test for Equality of Variances df Mean Sig (2- tailed) Difference Difference Std. Error 99% Confidence Interval of the Lower Upper Equal variances 052820 38 2.937 SBP Equal variances not assumed 37.983 2.937 (a) (4 marks) Suppose you want to test the hypothesis that there is a difference in SBP between the control group and the treatment group, what would be the most appropriate hypothesis test t perform? Describe and check the assumptions of the test you have selected. (b) (5 marks) Since this is a medical issue, which is crucial, test the hypothesis stated in part (a) at the 1% ate test, which you selected in part (a), in order to significance level. Carry out the most determine if there is a difference in SBP between the control and treatment groups. SHOW ALL STEPS of the hypothesis test; however, for the calculation step, make use of the SPSS output. (c) (3 marks) The researcher was certain that it is impossible that the new drug could increase SBP, but that there is a possibility that it could reduce SBP. Therefore, at the 1% significance level, test the hypothesis that mean SBP in the treatment group was lower than in the control group. You do NOT need to perform all steps. Only rewrite the hypotheses and then state the P-value, the strength of the evidence against Ho and the conclusion.

Explanation / Answer

Let X = SBP of Group 1 (Placebo) and Y = SBP of Group 2 (New Drug)

We are given that data are normally distributed.

Let X ~ N(µ1, 12) and Y ~ N(µ2, 22)

We assume that 12 = 22

Back-up Theory

Part (a)

We wish to test if the difference in the SBP between the two groups is significant.

The most appropriate test is: 2-sample t-test with

Null: H0: µ1 = µ2 or µ1 - µ2 = 0 Vs Alternative: HA: µ1 µ2 or µ1 - µ2 0

Test statistic, t = t = (Xbar - Ybar)/{s(2/n)} where

s2 = (s12 + s22)/2;

Xbar and Ybar are sample averages and s1, s2 are sample standard deviations based on n observations each on X and Y.

Assumptions

1) X and Y are normally distributed – this is already given in the question as granted.

2) 12 = 22 = 2, say and 2 is unknown

Assumption (2) is valid since from the SPSS output it is seen the F statistic is only 0.052 which is trivially not significant.

Thus, all assumptions are valid ANSWER

Part (b)

To perform the actual test as given in Part (a),

Test statistic, t = t = (Xbar - Ybar)/SE(difference)

= (149.0 – 141.500)/2.937

= 7.5/2.937

= 2.554

Given level of significance, = 1%, the alternative being two-sided, the critical value = upper 0.5% point of t38 which is found to be 2.712 using Excel Function of t-distribution.

Since tcal < tcrit, the null hypothesis µ1 - µ2 = 0 is accepted.

Conclusion: there is not enough evidence to suggest that the difference in SBP between the two groups is significant. ANSWER

Part (c)

Here, the researchers are particular that the new drug is effective in reducing SBP. Thus, the alternative becomes HA: µ1 > µ2 or µ1 - µ2 > 0 [implying mean SBP under the new drug, µ2 < µ1, mean SBP under the placebo]

Everything remains the same as in Part (b), except that the critical value now would be upper 1% point (since the alternative is one-side) of t38 which is found to be 2.428 using Excel Function of t-distribution.

Since tcal > tcrit, the null hypothesis µ1 - µ2 = 0 is rejcted.

Conclusion: there is enough evidence to suggest that the new drug is effective in bringing down the SBP. ANSWER

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