The article from which the data in Exercise 1 was extracted also gave the accompanying s...

The article from which the data in Exercise 1 was extracted also gave the accompanying strength observations for cylinders:

Prior to obtaining data, denote the beam strengths by X₁, . . . , X_m and the cylinder strengths by Y₁, . . . , Y_n. Suppose that the X_i’s constitute a random sample from a distribution with mean 1 and standard deviation s1 and that the Y_i’s form a random sample (independent of the X_i’s) from another distribution with mean μ₂ and standard deviation σ₂.

a. Use rules of expected value to show that . Calculate the estimate for the given data.

b. Use rules of variance from Chapter 5 to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a), and then compute the estimated standard error.

c. Calculate a point estimate of the ratio σ₁/σ₂ of the two standard deviations.

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.

Reference Exercise 1

The accompanying data on flexural strength (MPa) for concrete beams of a certain type was introduced in Example 1.2.

a. Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion, and state which estimator you used. [Hint: ∑_xi = 219.8.]

b. Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%, and state which estimator you used.

c. Calculate and interpret a point estimate of the population standard deviation s. Which estimator did you use? [Hint: : ∑_x²_i = 1860.94]

d. Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a “success” if it exceeds 10.]