ONLY THE LAST ONEConsider the accompanying data on flexural strength (MPa) for concrete beams of a certain type. 5.9 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 8.4 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. 6.5 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 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 byX 1, . . . ,Xm and the cylinder strengths byY 1, . . . ,Yn . Suppose that theXi 's constitute a random sample from a distribution with mean μ1 and standard deviation σ1 and that theYi 's form a random sample (independent of theXi '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. E(X − Y) = E(X) − E(Y) = μ1 − μ2 E(X − Y) = E(X) − E(Y) 2 = μ1 − μ2 E(X − Y) = nm E(X) − E(Y) = μ1 − μ2 E(X − Y) = E(X) − E(Y) nm = μ1 − μ2 E(X − Y) = E(X) − E(Y) = 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). V(X − Y) = V(X) + V(Y) = σX 2 + σY 2 = σ21 n1 + σ2 2 n2 σX − Y = V(X − Y) = σ21 n1 + σ2 2 n2 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 differenceX −Y between beam strength and cylinder strength. (Round your answer to two decimal places.) MPa2

ONLY THE LAST ONE Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type. 5.9 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 8.4 8.7 7.8 9.7 7.4 7.7 9.7...

ONLY THE LAST ONE

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

5.9	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
8.4	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.

6.5	5.8	7.8	7.1	7.2	9.2	6.6	8.3	7.0	8.3
7.8	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 byX
₁, . . . ,X_m
and the cylinder strengths byY
₁, . . . ,Y_n
. Suppose that theX_i
's constitute a random sample from a distribution with mean μ₁ and standard deviation σ₁ and that theY_i
's form a random sample (independent of theX_i
'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.

E(X − Y) = E(X) − E(Y) = μ₁ − μ₂

E(X − Y) =

E(X) − E(Y)

	2

= μ₁ − μ₂

E(X − Y) = nm

E(X) − E(Y)

= μ₁ − μ₂

E(X − Y) =

E(X) − E(Y)

= μ₁ − μ₂

E(X − Y) =

E(X) − E(Y)

= 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).

V(X − Y)

V(X) + V(Y)

σ_X
² + σ_Y
²

σ21

n₁

σ₂
²

σ_X − Y

V(X − Y)

σ21

n₁

σ₂
²

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 differenceX −Y between beam strength and cylinder strength. (Round your answer to two decimal places.)
MPa²

Jun 01, 2022

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ONLY THE LAST ONE Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type. 5.9 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 8.4 8.7 7.8 9.7 7.4 7.7 9.7...

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