Question

Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type

6.0 7.2 7.3 6.3 8.1 6.8 7.0 7.5 6.8 6.5 7.0 6.3 7.9 9.0 9.0 8.7 7.8 9.7 7.4 7.7 9.7 8.2 7.7 11.6 11.3 11.8 10.7

The data below give accompanying strength observations for cylinders.

6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.5 7.5 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.9

Prior to obtaining data, denote the beam strengths by X1, . . . , Xm and the cylinder strengths by Y1, . . . , Yn.

Suppose that the Xi's constitute a random sample from a distribution with mean μ1 and standard deviation σ1 and that the Yi's form a random sample (independent of the Xi's) from another distribution with mean μ2 and standard deviation σ2.

(a) Use rules of expected value to show that X − Y is an unbiased estimator of μ1 − μ2.

Calculate the estimate for the given data. (Round your answer to three decimal places.) MPa

(b) Use rules of variance to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a). Compute the estimated standard error. (Round your answer to three decimal places.) MPa

(c) Calculate a point estimate of the ratio σ1/σ2 of the two standard deviations. (Round your answer to three decimal places.)

(d) Suppose a single beam and a single cylinder are randomly selected. Calculate a point estimate of the variance of the difference X − Y between beam strength and cylinder strength. (Round your answer to two decimal places.) MPa2

Answer 1