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(a) What is X-1, 1-a where n = 20 and a = 0.05? (b) What is the one-sided 95% confidence limit for o?

(a) What is X-1, 1-a where n = 20 and a = 0.05? (b) What is the one-sided 95% confidence limit for o?

MATLAB: An Introduction with Applications
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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1. The 28-day compressive strength test data (in MPa) from the first 20 concrete cylinders from Assignment 10 are shown below: 60.5 60.9 59.8 53.4 56.9 67.3 68.9 49.9 57.8 60.9 61.9 67.2 64.9 63.4 60.5 68.1 68.3 65.7 61.5 60.0 Group 1 Group 2 Group 3 Using just these 20 tests, the overall mean strength is 61.89 MPa with sample standard deviation s is 5.045 MPa. Assuming the strengths are normally distributed, we want to solve for the one-sided 95% confidence limit on the standard deviation o. In other words, we know s based on our first 20 tests. We want to find o such that: (n-1)s² P[0² < = 1 − α = 0.95 using the x² Chi-squared distribution 2 Xn-1, 1-al (a) What is X-1, 1-a where n = 20 and a = 0.05? (b) What is the one-sided 95% confidence limit for o? 2. Data from a permeability test on sand gave the following gradient-flow rate results (this is a test that you will conduct in the Geotechnical laboratory course in spring of Junior year): gradient i 0.0 0.22 0.41 0.59 0.80 1.01 1.09 flow q (mm/s) 0.0 0.38 0.81 1.15 1.54 1.91 2.25 (a) Using the equations developed in class, solve for the "best fit" straight line through the data using least- squares regression, by hand, presuming i is the independent variable (ie use the equations based on the sums Ex, Ey, Exy, Ex², N). (b) Solve for the best fit straight line again using the statistical form of the solution, with slope = Sxy/Sxx and y-intercept = (Mean y) - slope* (mean x). Compare with (a). (c) It is known from Darcy's law in soil mechanics that the flow rate is directly proportional to the gradient and MUST be zero when the gradient is zero, ie q = kxi where k is a constant. How can we determine the best fit slope k and ensure that the line goes through the origin (0,0)? (d) Compute an unbiased estimate of the variance of the error, s² = (1 Sxy² SxxSyy. (e) Using a spreadsheet program, plot the data along with the best fit straight line. (f) Using the results from (a) what are the 95% confidence limits on the mean of the flow rate q at i = 0.5? (g) Using the results from (a) what are the 90% confidence limits on a future prediction of q at i = 0.5? n n-2 Syy 3. Given a set of N data pairs (x₁, yi), how would you solve for the unknown coefficients a, b and c for the polynomial of the form y = a + bx³ + cx5 which best fits the data (in a least-squares sense). 4. Given a set of N data (xi, Yi, Zi) where x; and y¡ are independent variables, presumed to be known precisely, and where zi is the dependent variable, how would you solve for the coefficients a, b and c for the polynomial of the form z = a + bx + cy which best fits the data (in a least-squares sense).
Transcribed Image Text:1. The 28-day compressive strength test data (in MPa) from the first 20 concrete cylinders from Assignment 10 are shown below: 60.5 60.9 59.8 53.4 56.9 67.3 68.9 49.9 57.8 60.9 61.9 67.2 64.9 63.4 60.5 68.1 68.3 65.7 61.5 60.0 Group 1 Group 2 Group 3 Using just these 20 tests, the overall mean strength is 61.89 MPa with sample standard deviation s is 5.045 MPa. Assuming the strengths are normally distributed, we want to solve for the one-sided 95% confidence limit on the standard deviation o. In other words, we know s based on our first 20 tests. We want to find o such that: (n-1)s² P[0² < = 1 − α = 0.95 using the x² Chi-squared distribution 2 Xn-1, 1-al (a) What is X-1, 1-a where n = 20 and a = 0.05? (b) What is the one-sided 95% confidence limit for o? 2. Data from a permeability test on sand gave the following gradient-flow rate results (this is a test that you will conduct in the Geotechnical laboratory course in spring of Junior year): gradient i 0.0 0.22 0.41 0.59 0.80 1.01 1.09 flow q (mm/s) 0.0 0.38 0.81 1.15 1.54 1.91 2.25 (a) Using the equations developed in class, solve for the "best fit" straight line through the data using least- squares regression, by hand, presuming i is the independent variable (ie use the equations based on the sums Ex, Ey, Exy, Ex², N). (b) Solve for the best fit straight line again using the statistical form of the solution, with slope = Sxy/Sxx and y-intercept = (Mean y) - slope* (mean x). Compare with (a). (c) It is known from Darcy's law in soil mechanics that the flow rate is directly proportional to the gradient and MUST be zero when the gradient is zero, ie q = kxi where k is a constant. How can we determine the best fit slope k and ensure that the line goes through the origin (0,0)? (d) Compute an unbiased estimate of the variance of the error, s² = (1 Sxy² SxxSyy. (e) Using a spreadsheet program, plot the data along with the best fit straight line. (f) Using the results from (a) what are the 95% confidence limits on the mean of the flow rate q at i = 0.5? (g) Using the results from (a) what are the 90% confidence limits on a future prediction of q at i = 0.5? n n-2 Syy 3. Given a set of N data pairs (x₁, yi), how would you solve for the unknown coefficients a, b and c for the polynomial of the form y = a + bx³ + cx5 which best fits the data (in a least-squares sense). 4. Given a set of N data (xi, Yi, Zi) where x; and y¡ are independent variables, presumed to be known precisely, and where zi is the dependent variable, how would you solve for the coefficients a, b and c for the polynomial of the form z = a + bx + cy which best fits the data (in a least-squares sense).
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