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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 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.8 8.7 7.8 9.7 7.4 7.7 9.7 8.1 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 6.7 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.8 7.9 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.7 Prior to obtaining data, denote the beam strengths by X1, ..., Xm and the cylinder strengths by Y1, ..., Yp. Suppose that the X/s constitute a random sample from a distribution with mean uj and standard deviation a, and that the Y's form a random sample (independent of the X/s) from another distribution with mean u2 and standard deviation g2. (a) Use rules of expected value to show that X - Y is an unbiased estimator of u1 - H2. O E(X – Y) = E(X) – E(Y) = H1 - 42 nm O E(X - Y) = v E(X) – E(Y) = H1 – H2 E(X – Y) = E(X) – E(Ÿ) = µ1 – H2 E(X – 7) = (E(X) – E(Y)°. = 41 - 42

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 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.8 8.7 7.8 9.7 7.4 7.7 9.7 8.1 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 6.7 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.8 7.9 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.7 Prior to obtaining data, denote the beam strengths by X1, ..., Xm and the cylinder strengths by Y1, ..., Yp. Suppose that the X/s constitute a random sample from a distribution with mean uj and standard deviation a, and that the Y's form a random sample (independent of the X/s) from another distribution with mean u2 and standard deviation g2. (a) Use rules of expected value to show that X - Y is an unbiased estimator of u1 - H2. O E(X – Y) = E(X) – E(Y) = H1 - 42 nm O E(X - Y) = v E(X) – E(Y) = H1 – H2 E(X – Y) = E(X) – E(Ÿ) = µ1 – H2 E(X – 7) = (E(X) – E(Y)°. = 41 - 42

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.3: Measures Of Spread
Problem 1GP
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(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) + V(Y) = ox? + oy? %D + ni ox - Y = V V(X ) 02? + n1 Compute the estimated standard error. (Round your answer to three decimal places.) MPа (c) Calculate a point estimate of the ratio 01/02 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.) MPa?
Transcribed Image Text:(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) + V(Y) = ox? + oy? %D + ni ox - Y = V V(X ) 02? + n1 Compute the estimated standard error. (Round your answer to three decimal places.) MPа (c) Calculate a point estimate of the ratio 01/02 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.) MPa?
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 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.8 8.7 7.8 9.7 7.4 7.7 9.7 8.1 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 6.7 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.8 7.9 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.7 Prior to obtaining data, denote the beam strengths by X1, . . . , Xm and the cylinder strengths by Y1, . . ., Yn. Suppose that the X;'s constitute a random sample from a distribution with mean u1 and standard deviation o1 and that the Y;'s form a random sample (independent of the X;'s) from another distribution with mean µz and standard deviation o2. (a) Use rules of expected value to show that X - Y is an unbiased estimator of u1 - µ2. E(X - Y) E(X) – E(Y) = µi 42 nm E(X – Y) = V E(X) – E(Y) = µ1 M2 E(X – Y) E(X) – E(Y) = H1 – H2 2 O E(X - ) = (EX) – E() = µ1 - µ2 O E(X – ) = nm E(X) – E(Y) = H1 - H2 Calculate the estimate for the given data. (Round your answer to three decimal places.) MPa
Transcribed Image Text: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 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.8 8.7 7.8 9.7 7.4 7.7 9.7 8.1 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 6.7 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.8 7.9 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.7 Prior to obtaining data, denote the beam strengths by X1, . . . , Xm and the cylinder strengths by Y1, . . ., Yn. Suppose that the X;'s constitute a random sample from a distribution with mean u1 and standard deviation o1 and that the Y;'s form a random sample (independent of the X;'s) from another distribution with mean µz and standard deviation o2. (a) Use rules of expected value to show that X - Y is an unbiased estimator of u1 - µ2. E(X - Y) E(X) – E(Y) = µi 42 nm E(X – Y) = V E(X) – E(Y) = µ1 M2 E(X – Y) E(X) – E(Y) = H1 – H2 2 O E(X - ) = (EX) – E() = µ1 - µ2 O E(X – ) = nm E(X) – E(Y) = H1 - H2 Calculate the estimate for the given data. (Round your answer to three decimal places.) MPa
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