Question

Consider a study to determine the compressive strength of concrete blocks used for paving Two methods of manufacturing concrete blocks are under investigation- method 1 (which is more expensive) and method 2. Research indicates that the greater the compressive strength of concrete, the greater the durability of a concrete block. Therefore the question is which one, if either, of the two methods produces greater compressive strength. In addition, the manufacturer considers that an average difference between the methods of less than 500 psi (pounds per square inch) would be of no consequence, whereas a difference of 500 psi or more could result in considerable benefits. Some possible outcomes of the study are Case 1 Case 2 Case 3 Case 4 Case 5 se(x,-X,) 325.61 1.86 10.24 7.07 14.14 P-value 0.6861 0.0000 0.2853 0.0000 0.0000 95% CI (-778, 514) 53, 61) 9, 31) 573, 601) -132 57 587 85 (-113, -57 Note: The hypotheses associated with the quoted P-values are There is no difference in the underlying mean compressive strength of concrete blocks when using the two methods of manufacture. H, : There is a difference in the underlying mean compressive strength of concrete blocks when using the two methods of manufacture. (a) (i) What is the hypothesised value? (i) In which direction and how far away, in terms of standard errors, is the estimated 1 mark ] difference in Case 1 (xi-, =-132 ) from the hypothesised value? [ 2 marks ] (b) which case(s) demonstrates, at the 5% level, that the difference in sample means, x- is statistically significant? For which casels) are we able to claim that the difference in the underlying mean compressive strength of concrete blocks between the methods of manufacture (i) has practical significance? (i) does not have practical significance? [ 1 mark ] (c) 1 mark ] I 1 mark ] (d) In which case(s) have we learned nothing useful about the difference in the underlying mean compressive strength of concrete blocks between the methods of manufacture? (e) Suppose the actual outcome for the study is se(x,-x,) p-value 95% ci Case 6 Write three to five sentences interpreting this output. You need to refer to statistical Give a reason(s) for your choice -598 1.02 0.0000 -600, -596) significance and practical significance. Which method of manufacture would you recommend? 1 3 marks]

Answer 1

(a)
(i)
Null hypothesis H0: Mean difference in compressive strength is 0
So, the hypothesized value is 0.

(ii)
Test statistic = -132/325.61 = -0.405393
Since the test statistic is negative, the estimated difference is 0.405393 standard errors towards the left from the hypothesized mean.

(b)
P-value for case 2, 4 and 5 are less than 0.05 significance level. Thus, case 2, 4 and 5 demonstrates that the sample mean difference is statistically significant.

(c)
(i)
It is given in the problem, average difference of more 500 psi would be of no use for the manufacturer. The average difference is more than 500 psi for case 4.
Thus, for case 4, we can claim that the mean difference is of practical significance.

(ii)
All other cases, 1, 2, 3 and 5 do not have the practical significance.

(d)
Case 1 and 3 does not show any statistical and practical significance.

(e)
Since p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that there is a difference in the underlying mean compressive strength of concrete blocks when using the two methods of manufacture.
Thus, Case 6 is of statistical significance.
The average difference is more than 500 psi for case 6.
Thus, for case 6, we can claim that the mean difference is of practical significance.
Since, the mean difference is negative, we can conclude that mean strength of method 2 is greater than that of method 1.
Thus, the manufacturer should be recommended for method 2.