Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type. 5.2 7.2 7.3 6.3 8.1 6.8 7.0 7.1 6.8 6.5 7.0 6.3 7.9 9.0 8.0 8.7 7.8 9.7 7.4 7.7 9.7 7.8 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 7.0 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.0 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.5 Prior to obtaining data, denote the beam strengths by X₂. .X and the cylinder strengths by Y₁... ₂ (a) Use rules of expected value to show that X-Y is an unbiased estimator of ₂-₂- OCX-5-₂-H₂ O EX-=(EX-E)² -₂ -₂ EX-= m(EX) - () = μ₁ - O EX-VE(X) - E(X) = 1₂ - 1₂ EX-= EX-E=H₂-H₂ Calculate the estimate for the given data. (Round your answer to three decimal places.) -0.506 ✔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)=v(x) + vin = 0,² +0,² . 07-7=√(x-3) n₂ n₂ 2₂ %₂² Y-Suppose that the x's constitute a random sample from a distribution with mean , and standard deviation d, and that the Y's form a random sample (independent of the X,'s) from another distribution with mean ₂ and standard deviation Compute the estimated standard error. (Round your answer to three decimal places.) 0.561 X MPa (c) Calculate a point estimate of the ratio a/d of the two standard deviations. (Round your answer to three decimal places.) 0.818 1x (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.) 7.0141 x MP₂²

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Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
8.1 6.8 7.0 7.1 6.8 6.5 7.0 6.3 7.9 9.0
7.4 7.7 9.7 7.8 7.7 11.6 11.3 11.8 10.7
5.2 7.2 7.3 6.3
8.0 8.7 7.8 9.7
The data below give accompanying strength observations for cylinders.
7.0 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3
7.0 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.5
Prior to obtaining data, denote the beam strengths by X₁,..., Xm and the cylinder strengths by Y₁,..., Y. Suppose that the X's constitute a random sample from a distribution with mean μ₁ and standard deviation and that the Y's form a random sample (independent of the X,'s) from another distribution with mean ₂ and standard deviation
02.
(a) Use rules of expected value to show that X - Y is an unbiased estimator of μ₁ −μ₂.
E(X) - E(X)
nm
O E(X) = (E(X) - E(² =H₁ - H²₂
O E(X-X) =
= μ₁ - H₂
¸ E(X − ¯) = nm( E(X) – E(\)) = μ₁ −
-
O
1₂
O E(X)=√√E(X) - E(Y) = µ₁ −μ₂
ⒸE(X) = E(X) - E(X) = μ₁ −μ²₂
Calculate the estimate for the given data. (Round your answer to three decimal places.)
-0.506
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-X) = V(X) + V(
2
2
= 0 x ² + 0 ₂ ²
oxy=√V(XY)
n₁
1
n₂2
n₂
0₂²
Compute the estimated standard error. (Round your answer to three decimal places.)
0.561
X MPa
(c) Calculate a point estimate of the ratio ₁/₂ of the two standard deviations. (Round your answer to three decimal places.)
0.818
X
(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.)
7.0141
X MPa²
Transcribed Image Text:Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type. 8.1 6.8 7.0 7.1 6.8 6.5 7.0 6.3 7.9 9.0 7.4 7.7 9.7 7.8 7.7 11.6 11.3 11.8 10.7 5.2 7.2 7.3 6.3 8.0 8.7 7.8 9.7 The data below give accompanying strength observations for cylinders. 7.0 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.0 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.5 Prior to obtaining data, denote the beam strengths by X₁,..., Xm and the cylinder strengths by Y₁,..., Y. Suppose that the X's constitute a random sample from a distribution with mean μ₁ and standard deviation and that the Y's form a random sample (independent of the X,'s) from another distribution with mean ₂ and standard deviation 02. (a) Use rules of expected value to show that X - Y is an unbiased estimator of μ₁ −μ₂. E(X) - E(X) nm O E(X) = (E(X) - E(² =H₁ - H²₂ O E(X-X) = = μ₁ - H₂ ¸ E(X − ¯) = nm( E(X) – E(\)) = μ₁ − - O 1₂ O E(X)=√√E(X) - E(Y) = µ₁ −μ₂ ⒸE(X) = E(X) - E(X) = μ₁ −μ²₂ Calculate the estimate for the given data. (Round your answer to three decimal places.) -0.506 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-X) = V(X) + V( 2 2 = 0 x ² + 0 ₂ ² oxy=√V(XY) n₁ 1 n₂2 n₂ 0₂² Compute the estimated standard error. (Round your answer to three decimal places.) 0.561 X MPa (c) Calculate a point estimate of the ratio ₁/₂ of the two standard deviations. (Round your answer to three decimal places.) 0.818 X (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.) 7.0141 X MPa²
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