Superplasticized concrete is formed by adding chemicals to conventional concrete

Question

Superplasticized concrete is formed by adding chemicals to conventional concrete to make it more fluid so that it can be placed more easily. Suppose that a sample of 50 new construction projects in the Dallas-Fort Worth area yields 15 that are using this type of concrete. A sample of 60 new projects in the Boston area also yields 15 superplasticized concrete.

(a) Let p1 and p2 denote the propprtion of new construction projects in Dallas-Fort Worth and Boston, respectively, that are using superplasticized concrete. Find point estimates for p1, p2 and p1 - p2.

(b) Find a 95% confidence interval for p1 - p2.

(c) Would you be surprised to hear someone claim that the proportion of Dallas-Fort Worth projects using this type of concrete is clearly larger that that in the Boston area? Explain, based on the confidence interval of part (b)

Explanation / Answer

a) p1 = 15/50 = 0.3 and p2 = 15/60 = 0.25

Then point estimate for p1 p2 = 0.3 - 0.25 = 0.05

b) Var(p1) = (0.3*0.7)/50 = 0.0042 and Var(p2) = (0.25*0.75)/60 = 0.003125.

Var (p1-p2) = Var(p1) + Var(p2) = 0.0042 + 0.003125 = 0.007325. Hence SE(p1-p2) = 0.007325^0.5 = 0.0856.

95% confidence interval for (p1-p2) = (p1-p2) +/-1.96*SE(p1-p2) = 0.05 +/- 1.96*0.0856 = (-0.118, 0.218).

c) As 0 is included in the 95% confidence interval, we cannot reject the null hypothesis and so we accept it.

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