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Probability Statistics And Reliability For Engineers And Scientists 3rd Edition Bilal M. Ayyub, Richard H. McCuen - Solutions
=+4.35 ● Blood cocaine concentration (in milligrams per liter) was determined for both a sample of individuals who had died from cocaine-induced excited delirium and a sample of those who had died from a cocaine overdose without excited delirium. The accompanying data are consistent with summary
=+Construct a boxplot for each type of box. Use the same scale for all four boxplots. Discuss the similarities and differences in the boxplots.
=+4.34 ● The article “Compression of Single-Wall Corrugated Shipping Containers Using Fixed and Floating Test Platens” (Journal of Testing and Evaluation [1992]: 318–320) described an experiment in which several different types of boxes were compared with respect to compressive strength.
=+b. Construct a comparative boxplot for blood lead level for the two samples. Write a few sentences comparing the blood lead level distributions for the two samples.
=+a. Compute the values of the mean and the median for blood lead level for the sample of African Americans.Which of the mean or the median is larger? What characteristic of the data set explains the relative values of the mean and the median?
=+of African Americans. Data consistent with the given summary quantities follow:Whites 8.3 0.9 2.9 5.6 5.8 5.4 1.2 1.0 1.4 2.1 1.3 5.3 8.8 6.6 5.2 3.0 2.9 2.7 6.7 3.2 African 4.8 1.4 0.9 10.8 2.4 0.4 5.0 Americans 5.4 6.1 2.9 5.0 2.1 7.5 3.4 13.8 1.4 3.5 3.3 14.8 3.7
=+4.33 ● ▼ The paper “Relationship Between Blood Lead and Blood Pressure Among Whites and African Americans” (a technical report published by Tulane University School of Public Health and Tropical Medicine, 2000)gave summary quantities for blood lead level (in micrograms per deciliter)
=+d. If you were asked by an insurance company to decide which, if any, occupations should be offered a professional discount on auto insurance, which occupations would you recommend? Explain.
=+c. Draw a modified boxplot for this data set.
=+b. Are there outliers in this data set? If so, which observations are mild outliers? Which are extreme outliers?
A lab technician wishes to determine the mean time to perform a lab experiment. What sample size will be needed to be 96% confident that the sample mean will be within 10 min of the true mean?Previous studies have indicated that σ = 25 min.
A laboratory believes that its equipment needs adjustment when the standard deviation of five samples of a known concentration exceeds 4 mg/L. During one test, the following sample measurements were recorded: 50.7, 58.2, 46.3, 53.0, and 62.4 mg/L. Does a confidence interval suggest that the
The corrosion depth of four samples taken from a steel I beam are 72, 217, 145, and 259 mils.Assuming an interest in being 90% certain, use a confidence interval to decide whether or not the true variance could be as small as 2500 (mils)2.
Using the sample standard deviation of the data of Problem 11.6, compute a one-sided upper 90%confidence interval on the variance. Does the interval cover the true variance of 0.25(%)2?
Using the first sample of 5 in Table 8.3, compute a two-sided, 95% confidence interval for the variance for the first sample (S2 = 1.764). Does the interval cover the true variance in this case?
Using the second sample of 5 in Table 8.3, compute a one-sided, upper 90% confidence interval on the mean assuming the population variance is not known.
A laboratory testing of five samples on the content of a certain water-quality indicator results in the following values: 3.41, 3.06, 2.94, 3.27, and 3.32 mg/L. (a) Find the 95% confidence interval for the mean content of this indicator. (b) What assumptions are made?
A random sample of test scores of the GRE exam by 25 students produces a mean of 575 and standard deviation of 60. (a) Determine the 99% confidence interval on the mean score for all students at the university. (b) What assertion can be made about the magnitude of the error, if we state that the
Twenty-five engineering students independently measured the melting point of lead. The mean and standard deviation of the 25 measurements were 329.6 and 5.1°C, respectively. Using a level of significance of 10%, what is the error that can be expected if the sample mean is used as an estimate of
For the conditions of Example 11.2, show the variation of the confidence interval with change in the confidence coefficient. Assume v = 9.
Using the first sample of 5 in Table 8.3, compute a two-sided, 95% confidence interval for the mean using the sample mean and the known population variance.
A wetland ecologist measures the trap efficiency of sediment from seven similar wetlands, with the results as follows: {62, 47, 54, 43, 66, 58, 50}%. Other studies have suggested a mean and standard deviation of 70 and 9%, respectively. Using a 90% confidence interval on the mean, is the assumption
An oceanographer is studying the relationship between the slope of beaches and the mean grain size(D, mm) of the sand. For slopes of about 6°, the expected mean is 0.45 mm, with a variance of 0.012 mm2. What is the confidence interval on the mean if the oceanographer collects the following mean
A chemical laboratory is given the following samples to test for a pollutant known to have a standard deviation of 1.1 mg/L: {48.3, 49.9, 48.7, 50.8, 48.1, 50.3} mg/L. Compute the 90% confidence interval on the mean.
Five cores are removed from a new section of an asphalt highway. The following percentages of air voids were obtained from laboratory analyses: 3.7, 4.5, 4.1, 4.7, 3.9. Past studies have suggested that the standard deviation of the air void content is 0.5%. Compute the 95%, two-sided confidence
The mean life of a random sample of 20 tires of a certain brand is 27,500 mi. (a) If the standard deviation of the population is known to be 3500 mi, what is the two-sided 90% confidence on the mean?(b) If the variance of the population is not known and the standard deviation of the random sample
An engineering firm manufactures a missile component that will have a life of use that is approximately normally distributed with a population standard deviation of 3 h. If a random sample of ten components has mean life of use of 13.4 h, find a two-sided 99.5% confidence interval for the
Identify the four factors that influence the width of a confidence interval on the mean. Provide general statements about the effect of each factor on the width of the interval.
A confidence on a statistic depends on a theorem. What must a theorem specify in order for it to be used in defining a confidence interval?
The skewness is the third statistical moment, which is measured about the mean. If a confidence interval was needed on the skew and assuming Equation 11.1 applied, discuss the nature of each element of Equation 11.1 as it would apply to a confidence interval on the skewness.
Note the similarities and differences between confidence intervals and hypothesis tests.
Compare the six steps of hypothesis testing with the six steps used to compute a confidence interval.
Discuss each element of a confidence interval (Equation 11.1) and how randomness plays a role in each of the three elements.
Provide a definition of a confidence interval and discuss how the confidence interval is used in decision making.
A 240-acre field is divided into 24 10-acre plots. The plots are randomly assigned to 12 groups, with the two fields in each group receiving the same treatment of fertilizer and irrigation. Three levels of fertilizer are applied (low, medium, and high), and four levels of water application are used
Given the data below, perform a two-way ANOVA. Use a level of significance of 1% to test (a) the hypothesis of interaction; (b) the null hypothesis of equal column means; and (c) the null hypothesis of equal row means.Factor Level A Level B Level C Level 1 27 29 18 24 13 15 Level 2 17 21 21 21 12
Given the data below, perform a two-way ANOVA. Use a level of significance of 5% to test (a) the hypothesis of interaction, (b) the null hypothesis of equal column means, and (c) the null hypothesis of equal row means.Factor Level 1 Level 2 Level A 14 16 16 18 12 14 14 16 Level B 10 12 14 16 10 12
Using the data below, perform a two-way ANOVA. Test the hypothesis of interaction at the 1% level of significance. Also, use a 1% level of significance to test the null hypotheses of equal column and equal row means.Factor Level 1 Level 2 Level A 14 16 18 12 16 16 Level B 10 12 16 12 12 14
A series of experiments is conducted, and the coefficient of friction μ between a shaft and a bearing is computed. Tests are made using four different lubricants and three levels of machining of the bearings (highly polished, little polishing, no polishing). Using a 1% level of significance,
For the given data set, which consists of four treatments and four blocks, use a 1% level of significance to test the significance of the treatment and block variables.Treatment Block 1 2 3 4 1 15 18 12 10 2 13 19 11 13 3 17 22 8 10 4 16 17 14 8
Measurements of the percentage soil moisture are made on the top of a sloping field (X1), the side of the slope (X2), and at the bottom of the slope (X3). Using a 1% level of significance, determine whether or not the soil moisture characteristics are constant over the field.X1 9.6 10.3 8.8 8.5
For the data of Problem 10.14, determine whether or not the noise intensity is uniform throughout the office. Use a 5% level of significance.
Measurements in lumens per square foot are made for testing the uniformity of lighting using four lighting systems (X1 to X4). Using a 5% level of significance and the following data, determine whether or not the four lighting systems provide equal uniformity in lighting intensity:X1 56 63 49 58 65
Test the null hypothesis of equal variances in the population for the database given below. Use a 5%level of significance.Group 1 2 3 4 5 Group 2 –1 0 7 8 Group 3 –4 –3 10 11
Test the null hypothesis of equal variances in the population for the coefficients of thermal expansion measured at four laboratories (A, B, C, D). At approximately what level of significance would the variances differ?LAB Measurements A 12.2 12.7 12.3 12.8 13.0 B 11.6 12.1 12.1 11.5 11.4 C 12.4
A 20-acre field is divided into l-acre plots, with the plots randomly assigned to one of four groups.For each plot fertilizer is either applied (F) or not applied (NF). For each plot, water is either applied(W) or not applied (NW). Thus, there are five plots for each of the four treatments: (F, W),
Use the Scheffé test with the data of Problem 10.16 to determine why the hypothesis of equal means of hardness was rejected.
Using the Scheffé test and the database for Example 10.2, determine why the null hypothesis of equal means was rejected.
Use the Duncan test with the data of Problem 10.11 to help decide which classes differed in mean grade.
Use the Duncan test with the data of Problem 10.16 to determine why the hypothesis of equal means of hardness was rejected.10-20. A 20-acre field is divided into l-acre plots, with the plots randomly assigned to one of four groups.For each plot fertilizer is either applied (F) or not applied (NF).
Use the Duncan test to determine which days of the week show different noise levels (see Problem 10.17).
An acoustical engineer is studying noise levels in center cities with day of the work week. She takes measurements each day of the week for four weeks at the same intersection and at the same time(5 pm). The following measurements (decibels) were taken:M Tu W Th F 66 76 66 73 70 71 74 73 67 66 70
Five brands of hardness testing machines are used to test the hardness of a new polymer, with the following results for three specimens:Brand A B C D E 13.5 13.3 13.5 12.4 13.4 13.6 12.9 12.8 12.6 13.6 13.4 13.0 13.1 12.5 13.0 Do the hardness machines provide different estimates of hardness?
For the six measurements on the two variables X1 and X2, conduct both a two-sample t test (assumeσ1 and σ2 are unknown but equal) and an ANOVA test for comparing means. Perform the tests for(a) Y1 = Y2 = 10, and (b) Y1 = Y2 = 15. Use a level of significance of 5%. Explain the difference in the
Three different acoustical systems are proposed for controlling noise intensity in an office.Measurements of noise intensity (decibels) that simulate heavy street traffic are made for all three acoustical systems at various locations in the office. Using a 5% level of significance, test whether or
Four electric resistance furnaces are tested to determine the temperature produced in each one.The four furnaces differ only in the composition of the coil of wire that is wound on the refractory material. A number of measurements of temperature (°C) are made for each furnace and are given below.
A series of experiments is conducted, and the coefficient of friction μ between a shaft and bearing is computed. Tests are run using six different lubricants, with four measurements of μ for each lubricant.At the 1% level of significance, can we conclude that any of the lubricants are more
Four sections of the same statistics course are taught by four teachers. Using the data given below and a level of significance of 5%, are the average grades of the four classes significantly different?Class 1 78 83 65 74 91 83 — —Class 2 92 81 87 76 94 85 90 —Class 3 63 71 65 68 83 — —
For the five measurements on the two variables R1, and R2, conduct both a two-sample t test (assumeσ1 = σ2, both unknown) and an ANOVA test on means. Perform two-tailed tests for (a) Y1 = 8, Y2 = 8, and (b) Y1 = 8, Y2 = 14 use a level of significance of 5%. Explain any difference in the results
Discuss the meaning of each term in Equation 10.5.
Assuming normal populations sketch the distributions of each of the k groups when the null hypothesis is incorrect and should be rejected.
When using the ANOVA1 test, discuss the meaning of the region of rejection?
When might it be appropriate to use a level of significance other than 5% when conducting an ANOVA1 test?
Discuss the rationale of Equation 10.2.
Why is it incorrect to apply the two-sample t test to each pair of samples rather than performing an ANOVA1 test of the k groups?
Compare the steps of the ANOVA1 hypothesis test with those of the two-sample t test.
A heavy drinker conducts an experiment. On four consecutive nights he drinks 20 oz. of rum mixed with the same brand of a cola drink. Each night a different brand of rum is used. Each following morning he has a hangover, and so he concludes that the hangover was the result of the common element,
An automotive engineer wants to see if the brand of tire makes a difference in stopping distance.She has new sets of Brand A tires placed on each of four cars of different makes and measures the stopping distances after the cars had reached a speed of 55 miles/h. Then she repeats the experiment
Simulate the critical values for the lower tail of the t statistic for the normal, uniform, and exponential distributions following the procedure of Example 9.12. Use a sample size of 10.
Write a pseudocode that could be used to simulate the critical values for the normal, uniform, and exponential distributions presented in Example 9.12.
For what situations would you use the Kolmogorov-Smirnov test instead of using chi-square test?
Using the Kolmogorov–Smirnov test, decide whether or not it is safe to assume that the 87-octane data of Problem 9.34 were obtained from a lognormal distribution, with the parameters defined by the method of moments.
Can we conclude that the ten taxicab times of Problem 9.29 are from a normal population with the parameters obtained by the method of moments. Use the Kolmogorov–Smirnov test.
Is it reasonable to conclude that the 93-octane data of Problem 9.34 were obtained from a uniform population with parameters estimated using the method of moments? Use the Kolmogorov–Smirnov test.
Use the Kolmogorov–Smirnov test to test the retention time data for the without-island condition(Problem 9.32) to determine if the data can be represented by a uniform distribution with parameters obtained from the method of moments.
The time for collecting tolls at a tollbooth on an interstate was measured for a short period, with the following results: 17.2, 23.4, 16.7, 19.0, 21.2, 20.8, 18.7, 20.4, 18.0, 22.1, 17.9, and 19.3 s. Can the traffic authority responsible for staffing the tollbooth legitimately claim that the
Nine measurements of the compressive strength of a Boston blue clay are 41.6, 48.7, 45.4, 44.0, 46.1, 44.8, 47.7, 45.5, and 42.9 lb/in.2. Using a 5% level of significance and the Kolmogorov–Smirnov onesample test, decide whether or not the values are normally distributed.
Use the data in Table 9.9 for the Piscataquis River with the sample log-mean and log-standard deviation(i.e., parameters) of 3.8894 and 0.2031, respectively, to test for the goodness of fit of the lognormal distribution. Use the Kolmogorov–Smirnov test.
Use the data in Table 9.9 for the Piscataquis River with the sample mean and standard deviation of 8620 and 4128 ft3/s, respectively, to test for the goodness of fit of the normal distribution. Use the Kolmogorov–Smirnov test.
The histogram of annual maximum discharges for the Piscataquis River is shown in Table 9.9 and Figure 9.9. Test whether or not it can be assumed that this random variable has a normal distribution with μ = 8620 and σ = 4128. Assume a 10% level of significance and use the chi-square test.
Using the sediment yield data presented in the histogram of Figure 9.6, test whether or not the sediment yield can be represented by a uniform distribution with a = 0.0 and b = 2.5. Use a level of significance of 5% and the chi-square test.
The histogram of annual maximum discharges for the Piscataquis River is shown in Table 9.9 and Figure 9.9. Test whether or not it can be assumed that this random variable has a lognormal distribution with (i.e., parameters) μY = 3.8 and σY = 0.25. Assume a 1% level of significance and use the
Use the data in Table 9.9 for the Piscataquis River with the sample log-mean (to the base 10) and log-standard deviation of 3.8894 and 0.2031 ft3/s, respectively, to test for the goodness of fit of the lognormal distribution. Use the chi-square test. Note that sample log-mean is the mean of the
Use the data in Table 9.9 for the Piscataquis River with the sample mean and standard deviation (i.e., parameters) of 8620 and 4128 ft3/s, respectively, to test for the goodness of fit of the normal distribution.Use the chi-square test.
Using the sample from Table 8.3 with the largest sample variance, test the null hypothesis that the variance is equal to the smallest sample variance of Table 8.3. Use a 1% level of significance.
Two irrigation systems are compared on their ability to provide uniform water distribution over a field. Twelve measurements with system X yield a standard deviation of 0.81 cm/h. Nine measurements with system Y yield a standard deviation of 0.35 cm/h. Can the conclusion be drawn that the systems
Using the data from Table 8.3, test the significance of the two sample variances for the last two of the 40 samples (0.421 vs. 1.192). Use a 10% level of significance.
Two competing lighting systems are installed in two adjacent parking lots, with system A being significantly less costly than system B. The manufacturer of system B argues that its system provides more uniform lighting. Ten random measurements in parking lots A and B yield readings with standard
Do the retention times of Problem 9.32 differ in variance?
Ideally, the catalytic reactors of Problem 9.35 should yield measurements with a variance less than 0.0025. Is it safe to conclude that reactor X produced consistent values?
Each car in Problem 9.34 was driven by a different person. Driving habits could partially account for the differences in mileage. If someone argues a standard deviation significantly greater than 0.5 mi/gal indicates driver differences, would the data for the 87-octane gasoline suggest that driver
The standard deviation is an indication of consistency. A resistance thermometer that provides readings with a standard deviation less than 5 mv is considered consistent. Do the readings on Brand A of Problem 9.33 suggest a standard deviation significantly greater than 5 mv?
Can the bus company of Problem 9.29 safely argue that the standard deviation of the trip is not greater than 5 min?
Can the taxicab company of Problem 9.29 safely argue that the standard deviation of the trip is not greater than 5 min?
Using the sample from Table 8.3 with the smallest sample variance, test the null hypothesis that the variance is equal to 1 against the one-sided lower alternative. Use a 1% level of significance. Is it reasonable for the null hypothesis to be rejected?
Using the sample from Table 8.3 with the largest sample variance, test the null hypothesis that the variance is equal to 1 against the two-sided alternative. Use a 5% level of significance. What is the approximate rejection probability?
The standard deviation of the 58 annual maximum discharges for the Piscataquis River (see Table 9.9) is 4128 ft3/s. Regional studies indicate that the standard deviation for that watershed should be at least 3500 ft3/s. Using a 1% level of significance, study whether or not the sample value of σ
The standard deviation of the 58 annual maximum discharges for the Piscataquis River (see Table 9.9) is 4128 ft3/s. Test the hypothesis that the population standard deviation is 3500 ft3/s. Make a twotailed test and use a 5% level of significance.
The yields from catalytic reactors depend on the catalyst. For one series of tests the yield was measured for two catalysts, X and Y. Nine measurements of yield were measured with catalyst X and 6 with catalyst Y X: 1.74, 1.62, 1.59, 1.70, 1.73, 1.60, 1.56, 1.66, 1.71 Y: 1.46, 1.53, 1.49, 1.45,
Two nonlinear models are fitted to a sample of data of 12 observations. Model 1 has a bias of –0.34 and a standard error of estimate of 0.62. The corresponding values for model 2 are 0.27 and 0.81, respectively. Can we conclude that the biases of the models are not significantly different?
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