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statistics principles and methods
Probability Statistics And Reliability For Engineers And Scientists 3rd Edition Bilal M. Ayyub, Richard H. McCuen - Solutions
11.10. An analytical technique is supposed to have a standard deviation error of measurement (on homogeneous material) of 2 ppm. Set s1¼1.5ppm and s2¼2.5ppm and both risks at 0.05. Determine an appropriate test for the proposed technique.
11.11. For fuses as in Problem 11.1, considering sx as unknown, devise a sampling plan for measurements so that if po¼p1¼0.005, Pa¼0.95 while if po¼p2¼0.02, Pa¼0.05.
11.12. For the contents weights in Problem 11.3, considering sx as unknown, devise a sampling plan for measurements so that if po¼p1¼0.005, Pa¼0.95 while if po¼p2¼0.05, Pa¼0.05.
11.13. Verify the risks given in the one-way check of a process setting in Section 11.7.2.
11.14. Verify the risks given in the two-way check of a process setting in Section 11.7.1.Assuming that ANSI=ASQ Z1.9 is available, find a sampling plan for the following requirements, assuming normal distribution of the x’s.
11.15. Lot size, N¼1000, inspection level IV, one specification, U, s unknown, normal inspection, using sample s, AQL¼0.65%.
11.16. Lot size, N¼3000, inspection level IV, two specifications, s unknown, normal inspection using sample R’s, AQL¼0.40%.
12.1. To illustrate the approach to normality as the number of components increases, perform this experiment on dice totals. For a single die, the faces may show 1, 2, 3, 4, 5, 6 with equal probability, that is, 1=6 each. This is a flat-topped uniform distribution. Now throw three dice and count
12.2. Consider shafts with specification limits 2.0002 and 2.0019 in. The matching bearings also carry specification limits of 2.0009 and 2.0033 in. Note the sizable overlapping of the two sets of limits. Suppose that these limits are met in a3s sense with normal distributions. Make a guess as to
12.3. A pin shows good control on OD ¼ outside diameter with y-double bar¼0.21440, R-bar¼0.00032 in. for samples of n¼5. A mating collar also shows good control on ID, with x-double bar¼0.21503, R-bar¼0.00040 in. and n¼5.Estimate the process standard deviations for each. Let the diametral
12.4. In manufacture of lamps, the clearance w¼xy is of interest, where x¼distance from rim of base to the glass,and y¼distance from rim of base to top of inner shell. For x’s, large samples gave estimated m¼1.0499, s¼ 0.0428 in., whereas for y’s m¼0.9079, s¼0.0120 in. Good normality
12.5. A brass washer and a mica washer are to be assembled one on top of the other. For the former m¼0.1155, s¼0.00045 in., whereas for the latter m¼0.0832, s¼0.00180 in. Find m and s for the combined thickness. What assumptions were needed? Assuming normality, set 3u limits for the combined
12.6. A general concept in interpreting measurements of material is the following: We let w¼the observed measurement, x ¼ the ‘‘true’’ measurement, and y¼the error of measurement(which may be þ or ). Then w¼ xþy. Since x and y are independent we may use (12.9) with k ¼ 2. Also if
12.7. At one time 200ohm resistors, made to ¼ 0.75%, that is, to 200 1.5 ohms were quite costly, due in part to the necessity of sorting out those outside the limits. But production was able to manufacture 100ohm resistors to 1% without sorting and with good normality of distribution. Take
12.8. Five pieces are assembled on a shaft, two of the first kind with m¼2.0140, s¼0.0014, two of a second kind with m¼3.2061, s¼0.0012, and one of a third kind with m¼3.8402, s¼0.0022 (all in inches). For total length w, use w¼ x1þx2þþx5, since random assembly is to be used.What are m
12.9. In packaged weight control, it is not very convenient to obtain directly the weight of the net contents. Instead, we may proceed indirectly by weighing empty containers x, covers y, and total filled packages z. Letting the net weight of contents be w, we thus have w ¼zxy. Now we cannot
12.10. The following is an example of aid to a materials review board. In a plant manufacturing soap and cosmetics, a question arose as to whether to use a lot of caps for cologne bottles. Accordingly tests were made on the distribution of cap strength (the torque which would break the cap) and the
13.1. The data given below are for thicknesses in 0.00001 in.of nonmagnetic coatings of galvanized zinc on 11 pieces of iron and steel. The destructive (stripping) thickness is y; the nondestructive (magnetic) thickness is x:x 105 120 85 121 115 127 630 155 250 310 443 y 116 132 104 139 114 129 720
13.2. The following data are for pounds tension of piston rings before and after ferroxing as in Table 13.1, but with an inverted V bar suspension with the gaps up, first 20 rings out of 300:x 4.3 5.4 4.9 5.5 4.8 4.4 4.5 4.8 4.9 4.4 y 4.8 5.9 5.5 6.0 5.4 5.1 5.2 5.4 5.5 4.8 x 4.3 4.6 5.0 4.4 4.8
13.3. Tons rejected y vs. bags of coke x. Estimate y for x¼2.
13.4. Tons rejected y vs. sealing time in seconds x.Estimate y for x¼25.
13.5. Thirty-five springs were made at each of the following temperatures: 300, 350, 400, . . . , 600, so that n¼245. The dependent variable was initial tension of spring in pounds. Calculations gave r¼0.9346, bo¼7.500, b1¼0.004398, P(y – y-bar)2¼54.26. Test r for significance.Find syx
13.6. In a research study of bituminous road mixes, seven samples were tested for x¼percent water absorbed vs. y¼percent asphalt absorbed. We find PP x¼43.18,¼36.23, P x2¼285.7292, P y2¼207.5937, P xy¼242.6675.a. Find r and test it for significance.b. Find bo and b1 and write the estimation
13.7. For 49 pieces of 1=6-in. poplar (wood) about 110 51 in., a study was made of wet, x, vs. dry, y, lengths.This was to study the shrinkage so as to control y to specification needs. The following results were found:P P x¼5523, y¼5162 in, and n Px2(P x)2¼19,600, n Py(P y)2¼23,672, n Pxy(P
14.1. po¼0.10.find for the respective geometric distributions P(1), P(2), P(3), P(4), m, and s:
14.2. po¼0.05.find for the respective geometric distributions P(1), P(2), P(3), P(4), m, and s:
14.3. In an audit of missile systems, we want 90% confidence of a reliability of at least 0.90. What size of random sample of missile systems do we test for no failures to secure the desired confidence?
14.4. We want to prove 0.99 minimum reliability with 98% confidence. What sample size do we require with no failures to establish this degree of protection?
14.5. Suppose that we observe one failure in 200 trials or tests. Find the minimum established limit for reliability with confidence 0.90, 0.975, 0.995, and interpret the results.
14.6. Suppose that we observe one failure in 500 trials or tests. Find the minimum established limit for reliability with confidence 0.90, 0.975, 0.995, and interpret the results.Find also for confidence 0.95.
14.7. If the average length of life for an exponential distribution is 2000 hr, find h(x). What number of the 1200 components remaining on test, not having failed, may be expected to fail in the next 10 hours? (Hint: Let dx¼10 hr.)
14.8. Two relays each expected to open under certain dangerous conditions are placed in series in a circuit. Their reliabilities are 0.99 and 0.995 and they act independently.Find the reliability of the system of the two.
14.9. Two relays each expected to close under certain dangerous conditions are placed in parallel in a circuit. Their reliabilities are 0.98 and 0.99. Find the reliability of the system of the two. Assume they act independently.
14.10. An assembly consists of three essential components, A, B and C. The reliabilities for each of these respectively is 0.99, 0.90 and 0.98. Because the reliability of B is so low, the design includes two B components in parallel.Assuming that the components act independently, what is the
If the annual sales of chewing gum were highly correlated with annual bank robberies, what would the correlation coefficient reflect?
For each part, identify a variable that has a cause-and-effect relationship with the following variable:(a) the efficiency of a hydraulic pump; (b) erosion of a river bed; (c) the power output of a wind machine; and (d) the corrosion rate of steel.
Plot the following data, assess the degree of systematic variation, and assess the appropriateness of using a linear model:X 0.7 1.3 2.8 5.1 7.3 8.9 11.5 Y 1.1 3.8 7.2 7.9 9.7 8.8 9.6
Plot the following data, and assess the quality of the data for prediction:X 0.7 0.8 1.3 1.2 1.7 1.9 4.1 Y 1.0 1.8 1.2 2.1 1.0 2.8 5.7
The erosion of soil (E) from construction sites is measured along with the average slope (S) of the site. Graph the following data and comment on the relationship between erosion and slope.S (%) 1.2 1.6 2.4 3.2 3.6 4.1 4.9 E (tons/ac/year) 38 78 55 84 52 111 94
The sediment trap efficiency (Et, %) of a wetland depends in part on the average depth (D, ft) of the wetland. Graph the following measurements on Et and D and then discuss the importance of depth in controlling the trapping of soil in the wetland:D 0.7 1.3 1.9 2.2 2.6 2.9 3.3 Et 34 55 49 68 44 60
Expand Equation 12.2 to show that the TV is the sum of the explained and UV.
Using the following values of X and Y, show that the TV equals the sum of the EV and the UV:X 2 5 7 6 2 Y 1 3 5 7 9 Assume that the regression equation relating Y and X is given by Yˆ = 4.5849 + 0.09434X
Using the following values of X and Y, show that the TV equals the sum of the EV and the UV:X 1 2 3 4 5 6 Y 2 2 3 5 4 5 Assume that the regression equation relating Y and X is given by Yˆ = 1.2 + 0.657X
For the data of Problem 12.6, compute the TV and the components UV and EV assuming(a) Et = 35 + 15D. (b) Discuss the resulting separation of variation; (c) If the model is biased, discuss the potential effect of the bias; and (d) Compute TV, UV, and EV assuming ˆE D t = 30 and compare the results
For the data of Problem 12.5, compute TV, UV, and EV if the model is E = 73.14 tons/ac/year.Discuss the results.
Derive Equation 12.5 from Equation 12.4.
For the following observations on X and Y, compute the correlation coefficient.X –3 –2 –1 0 1 2 Y 2 2 3 0 –2 –1
For the data given in Problem 12.8, compute the correlation coefficient using the standard normal transformation definition of Equation 12.5.
For the data in Problem 12.9, compute the correlation coefficient.
Compute the correlation coefficient for the data set of Problem 12.5.
Compute the correlation coefficient for the data set of Problem 12.6.
Given the following observations on X and Y, (a) construct a graph of X versus Y, and (b) compute the Pearson correlation coefficient:X 1 2 3 4 5 Y 1 1 2 4 4
Compute the correlation coefficient for the data of Problem 12.4.
Given the following paired observations on variables X and Y, and the values of the standardized variables ZX and ZY, compute the Pearson correlation coefficient between each of the following pairs:(a) X and Y; (b) ZX and ZY; (c) ZX and X; (d) ZX and Y; (e) X and Y – 3; and (f) X and 2Y. Briefly
Given the following paired observations on variables A and B, and the values of the standardized variables, ZA and ZB, compute the Pearson correlation coefficient between each of the following pairs: (a) A and B; (b) ZA and ZB; (c) A and ZB; (d) A and B + 2; (e) B and 3A; (f) Briefly discuss the
Engineering analyses often involve small samples, often fewer than five. Explain why a high sample correlation coefficient obtained with a small sample should not necessarily be taken as an indication of a strong relationship between two variables.
Explain why the critical correlation coefficient for testing H0: r = 0 decreases as the sample size increases.
Discuss, possibly with sketches, how the distribution of the correlation coefficient varies for population values of ρ from 0 to −1.
A sample of four yields a correlation coefficient of 0.915. Is it safe to conclude that the two variables are related?
A sample of 5 yields a correlation coefficient of −0.82. Is it reasonable to conclude that the two variables are related?
A sample of 7 yields a correlation coefficient of 0.78, while a second sample of 9 on the same two random variables yields a correlation coefficient of 0.61. What conclusions can be drawn about the cause-and-effect relationship between the two variables?
Seven students in an introduction to statistics course received the following grades on the first two tests. Does the evidence suggest that knowledge of the material covered on the first test helped the students on the second test? Use a 5% level of significance.Test 1 98 94 93 90 87 85 84 Test 2
Given the following observations on random variable X and Y, (a) calculate the correlation coefficient;(b) using a level of significance of 5%, test the null hypothesis that the two variables are uncorrelated;(c) using a level of significance of 1%, test the null hypothesis that the population
Test the correlation coefficient for the data of Problem 12.5 for statistical significance. See Problem 12.16.
Test the correlation coefficient for the data of Problem 12.6 for statistical significance. See Problem 12.17.
The correlation coefficient computed from a sample of 15 observations equals 0.431. Test the null hypothesis using a level of significance of 5% that the correlation of the population is (a) 0.0; and (b) 0.65.
A sample of 12 observations on X and Y yields a correlation coefficient of 0.582. Using a level of significance of 5%, test the null hypothesis that the correlation coefficient of the population is (a) 0.0;and (b) –0.1. Discuss.
A sample of 19 measurements on X and Y yields a correlation coefficient of −0.35. (a) Is it reasonable to assume that the two variables are related? (b) Is it reasonable to assume that the population correlation coefficient is more negative then −0.7?
A sample of 16 measurements on X and Y yields a correlation coefficient of 0.44. (a) Is it reasonable to assume that the two variables are related? (b) Is it reasonable to assume that the population correlation coefficient is at least 0.8?
For large sample sizes, say 30 or more, low critical values of correlation coefficients are expected when testing the null hypothesis that ρ = 0. A low sample value that exceeds the critical value still represents a small fraction of explained variance, that is, low R2. Is a regression equation
The design engineer believes that a correlation of 0.8 is needed in the population to have the necessary prediction accuracy. A sample of 22 measurements on Y and X yields a sample correlation coefficient of 0.49. Is it safe to assume that the population correlation coefficient is at least 0.8? Use
What is the smallest sample size for which a correlation coefficient of 0.61 is statistically significant at 5%, with a two-tailed alternative hypothesis?
Using the data of Problem 12.4, fit the zero-intercept model of Equation 12.22.
Using the data of Problem 12.28, fit the zero-intercept model of Equation 12.22. Predict values of test 2 using values of test 1. Discuss the results.
Discuss whether or not a zero-intercept model would be appropriate for the case of Problem 12.6.
Using the data of Problem 12.5 fit a zero-intercept model and discuss its rationality.
Given the following four pairs of observations X 1 1 2 2 ?Y 1 2 1 2 ?Compute the correlation coefficient and the regression coefficients if (a) the fifth pair is (Y = 8, X = 8);(b) the fifth pair is (Y = 2, X = 8). (c) Plot both sets of points and draw the regression lines. What general
Given the following pairs of observations X 1 2 2 3 ?Y 4 3 5 4 ?Compute the correlation coefficient and the regression coefficients if the fifth pair is (a) (X = 8, Y = 8); (b) the fifth pair is (X = 8, Y = 0). (c) Plot both sets of points and draw the regression lines.What general observations do
Show that Equation 12.38 results in a larger sum of squares of the errors for the data of Table 12.2 than the value obtained with Equation 12.37.
Using the following values of X and Y, (a) compute the values of the regression model of Equation 12.28:X 1 3 5 Y 5 2 1(b) Compute the correlation coefficient. (c) Test the statistical significance of the correlation coefficient.(d) What is the fraction of EV?
Given the following paired observations on Y and X, (a) graph Y versus X; (b) calculate the correlation coefficient; (c) determine the slope and intercept coefficients of the linear regression line; (d) show the regression line on the graph of part (a); (e) compute the predicted value of Y for each
Use the data of Problem 12.5 to fit the coefficients of Equation 12.28. Also compute the goodnessof-fit statistics (R,R2,Se,Sy) and explain the meaning of each relative to the issue. Also discuss the rationality of the regression coefficients.
Use the data of Problem 12.6 to fit the coefficients of the linear model (Equation 12.28). Compute and discuss the goodness of fit statistics.
Using the data of Problem 12.20, compute the standard error of estimate Se the standard error ratio, Se/Sy, R, and R2. Interpret these statistics.
Using the data of Problem 12.20, compare the standard error of estimate based on Equation 12.40 with the correct value of Equation 12.45. Discuss the cause of the difference.
Using the data of Problem 12.20, show that the SST of Equation 12.47 is equal to the TV of Equation 12.2 and that the SSR of Equation 12.48 is equal to the explained sum of squares of Equation 12.2.
Using the data set of Problem 12.29, compute the regression coefficients for the linear bivariate model with X as the predictor variable. Also, determine the standard error of estimate, the R and R2 values, and perform an ANOVA (use a 5% level of significance). Discuss the results.
Compute the standardized partial regression coefficient t (Equation 12.51) for the data of Problem 12.20, and compare the value with the correlation coefficient computed in Problem 12.20.
Show that the standardized partial regression coefficient (t) is equal to the correlation coefficient (R)for in bivariate regression analysis.
If the correlation coefficient is equal to 0.71 and the slope coefficient b1 is equal to 46.3, which variable, X or Y, has the larger variance? Explain.
Show that the ANOVA F test on H0 :b1 = 0 is identical to the two-tailed hypothesis test on the correlation coefficient, H0 :r = 0
State the four assumptions that underlie the linear regression analysis and identify statistical methods that could be used to test the assumptions.
Compute the standard errors of the regression coefficients and the two-sided 95% confidence intervals for the regression coefficients of Problem 12.47.
Using the data of Problem 12.29 and the regression coefficients of Problem 12.53, compute standard errors of the regression coefficients and the two-sided, 90% confidence intervals for the coefficients.
The ratio of the standard error of the slope coefficient (Equation 12.57) to the slope coefficient b1 is a measure of the relative accuracy of the coefficient, with a value below 30% indicating good accuracy and a value greater than 50% indicating poor accuracy. Evaluate the relative accuracy of
Perform hypothesis tests for the significance of the regression coefficients for Problem 12.47. Use a 5% level of significance and alternative hypothesis that the coefficients are different from zero.
Perform hypothesis tests for the significance of the regression coefficients for Problem 12.53. Use a 10% level of significance and alternative hypothesis that the coefficients are different from zero.
Compute a two-sided, 95% confidence interval for the line as a whole using the data for Problem 12.47. Provide values of X – 2SX, X – SX, X + SX, and X + 2SX.
Compute a two-sided, 95% confidence interval for the line as a whole using the data for Problem 12.29. Provide values of X – 2SX, X – SX, X + SX, and X + 2SX.
Using the data from Problem 12.47, compute the two-sided, 95% confidence interval for the mean value of Y at X = 5. Also, compute a similar interval at X = 11. Compare the two intervals and explain the difference in their widths.
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