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
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
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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I just need help with parts b2), c) and d)
Thank you
![**Title: Analyzing Flexural Strength Data for Concrete Beams**
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type:
Beam Data (X's):
5.2, 7.2, 7.3, 6.3, 6.1, 8.8, 7.0, 7.1, 6.8, 7.0, 6.3, 7.9, 9.0
Cylinder Data (Y's):
8.0, 8.7, 8.9, 7.7, 7.4, 7.9, 7.7, 7.8, 7.7, 8.6, 11.3, 11.8, 10.7
The data are paired observations of beam and cylinder strengths.
**Objective**:
Determine if the average of the beam strengths provides an unbiased estimate of the difference between the mean strengths of beams and cylinders.
### Part (a): Unbiased Estimator
The goal is to use expected values to show that the estimator \( \overline{X} - \overline{Y} \) is unbiased for \( \mu_1 - \mu_2 \):
- \( E(\overline{X}) = \mu_1 \)
- \( E(\overline{Y}) = \mu_2 \)
- Therefore, \( E(\overline{X} - \overline{Y}) = \mu_1 - \mu_2 \)
A correct setup affirming that the estimator \( \overline{X} - \overline{Y} \) is unbiased is:
\[ E(\overline{X}) - E(\overline{Y}) = \mu_1 - \mu_2 \]
This confirms the unbiased nature of the estimator.
**Estimate Calculation**:
Round your answer to three decimal places.
\[ \hat{\mu_1 - \mu_2} = \text{5.506} \, \text{MPa} \]
### Part (b): Variance and Standard Error
Derive the variance and standard deviation (standard error) for the estimator \( \overline{X} - \overline{Y} \).
**Variance Expression**:
\[ Var(\overline{X} - \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b236b3-1e1c-48f8-a6f5-9c3f49a681da%2F41cfae13-1850-4bd1-85f4-ea2098931b8a%2F6j0k6ta_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Flexural Strength Data for Concrete Beams**
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type:
Beam Data (X's):
5.2, 7.2, 7.3, 6.3, 6.1, 8.8, 7.0, 7.1, 6.8, 7.0, 6.3, 7.9, 9.0
Cylinder Data (Y's):
8.0, 8.7, 8.9, 7.7, 7.4, 7.9, 7.7, 7.8, 7.7, 8.6, 11.3, 11.8, 10.7
The data are paired observations of beam and cylinder strengths.
**Objective**:
Determine if the average of the beam strengths provides an unbiased estimate of the difference between the mean strengths of beams and cylinders.
### Part (a): Unbiased Estimator
The goal is to use expected values to show that the estimator \( \overline{X} - \overline{Y} \) is unbiased for \( \mu_1 - \mu_2 \):
- \( E(\overline{X}) = \mu_1 \)
- \( E(\overline{Y}) = \mu_2 \)
- Therefore, \( E(\overline{X} - \overline{Y}) = \mu_1 - \mu_2 \)
A correct setup affirming that the estimator \( \overline{X} - \overline{Y} \) is unbiased is:
\[ E(\overline{X}) - E(\overline{Y}) = \mu_1 - \mu_2 \]
This confirms the unbiased nature of the estimator.
**Estimate Calculation**:
Round your answer to three decimal places.
\[ \hat{\mu_1 - \mu_2} = \text{5.506} \, \text{MPa} \]
### Part (b): Variance and Standard Error
Derive the variance and standard deviation (standard error) for the estimator \( \overline{X} - \overline{Y} \).
**Variance Expression**:
\[ Var(\overline{X} - \
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