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MATH 302 Spring 2017 Homework Assignments:

Homework 1, due February 13

1) Corrosion of reinforcing steel is a serious problem in concrete structures located in environments affected by severe weather conditions. For this reason, researchers have been investigating the use of reinforcing bars made of composite materials. One study was carried out to develop guidelines for bonding glass-fiber-reinforced plastic reinforcing bars to concrete. Here are 48 observations on measured bond strength:

11.5 12.1 9.9 9.3 7.8 6.2 6.6 7.0 13.4 17.1 9.3 5.6 5.7 5.4 5.2 5.1 4.9 10.7 15.2 8.5 4.2 4.0 3.9 3.8 3.6 3.4 20.6 25.5 13.8 12.6 13.1 8.9 8.2 10.7 14.2 7.6 5.2 5.5 5.1 5.0 5.2 4.8 4.1 3.8 3.7 3.6 3.6 3.6

a) Assess whether the assumption of normality is reasonable for this data. If not, comment on whether the resulting inferences are valid or suspect.

b) Analyze the natural logarithm of the measured bond strength. Construct a 98% confidence interval for the true mean bond strength. (Construct the interval for log bond strength, and then translate into original units.)

c) Construct a 99% prediction interval for a new observation. (Construct the interval for log bond strength, and then translate into original units.)

d) Construct a 90% tolerance interval for capturing at least 95% of the log bond strength values. (Construct the interval for log bond strength, and then translate into original units.)

2) The observations below are residual flame time (sec) for strips of treated children’s nightwear. Suppose a true average flame time of at most 9.75 seconds has been mandated.

9.85 9.93 9.75 9.77 9.67 9.87 9.67 9.94 9.85 9.75 9.83 9.92 9.74 9.99 9.88 9.95 9.95 9.93 9.92 9.89

a) Does the data suggest that this condition has been met or does the data suggest otherwise? Carry out an appropriate test after first investigating the plausibility of assumptions that underlie your method of inference.

b) How many standard errors above average is a mean of 9.78? What conclusion would you come to using 𝛼 = 0.01?

c) If the decision for the test is to reject when 𝑧 ≥ 2.88, what is 𝛼?

d) Assume 𝛼 = 0.05 and using the rejection rule from part c), what is 𝛽(9.8)?

e) What sample size is necessary to ensure that 𝛽(9.8) = 0.01?

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f) Assume 𝑛 = 100, 𝛼 = 0.01. What is the power of the test when 𝜇 = 9.85?

Homework 2, due February 22 (Late homework will not be accepted for Homework 2.)

1) A study on concrete pressures on formwork gave this data:

33.2 41.8 37.3 40.2 36.7 39.1 36.2 41.8 36.0 35.2 36.7 38.9 35.8 35.2 40.1

a) Calculate an upper 95% confidence bound for the population standard deviation of maximum pressure.

b) To test 𝐻!:𝜎 = 𝜎! vs 𝐻!: 𝜎 > 𝜎! , the statistic 𝑋! = (𝑛 − 1)𝑠!/𝜎!! has a chi-squared distribution with 𝑛 − 1 degrees of freedom. Test whether the concrete pressure population has a true standard deviation of 1.5. Use 𝛼 = 0.05.

2) The following is summary data on compressive strength of concrete specimens made with a pulverized fuel-ash mix.

Age of Concrete Sample Size Sample Mean Sample SD 7 68 26.99 4.8928 74 35.76 6.43

a) Calculate and interpret a 99% confidence interval for the true difference between 7-day and 28-day concrete compressive strengths, using the t-test from Theorem 9.2.

b) Repeat using a pooled 𝑡-test.

c) Conduct a test of the equality of the two variances to see if the procedure from part (b) was warranted.

3) Construct a paired data set for which 𝑡 = ∞, so that the data is highly significant when the correct analysis is used, yet 𝑡 for the two-sample 𝑡-test is quite near zero, so the incorrect analysis yields an insignificant result.

4) Do teachers find their work rewarding and satisfying? The results of a survey of 395 elementary and 266 high school teachers showed that 224 of the elementary and 126 of the high school teachers said they were very satisfied with their jobs. Estimate the difference between the proportion of all elementary school teachers who are very satisfied and all high school teachers who are very satisfied by calculating and interpreting a 95% confidence interval.

Homework 3, due March 8

1) A study examined five different methods for teaching descriptive statistics. The five methods were traditional lecture and discussion (L/D), programmed textbook instruction (R), programmed text with lectures (R/L), computer instruction (C), and computer instruction with lectures (C/L). Forty-five students were randomly assigned, 9 to each

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method. After completing the course, the students took a 1-hour exam. In addition, a 10-minute retention test was administered 6 weeks later. Summary statistics were:

Exam Retention Test Method Average SD Average SD L/D 29.3 4.99 30.20 3.82R 28.0 5.33 28.80 5.26R/L 30.2 3.33 26.20 4.66C 32.4 2.94 31.10 4.91C/L 34.2 2.74 30.20 3.53

The grand mean for the exam was 30.82 and the grand mean for the retention test was 29.30.

a) Does the data suggest that there is a difference among the five teaching methods with respect to true mea n exam score? Use 𝛼 = 0.05.

b) Using a 0.05 significance level, test the null hypothesis of no difference among the true mean retention test scores for the five different teaching methods.

2) Four types of mortar were subjected to a compression test to measure strength (in MPa). Three strength observations for each mortar type are as follows:

OCM 32.15 35.53 34.20PIM 126.32 126.80 134.79RM 117.91 115.02 114.58PCM 29.09 30.87 29.80

a) Construct an ANOVA table.

b) Using a 0.05 significance level, determine whether the data suggests that the true mean strength is not the same for all four mortar types. If you determine that the true mean strengths are not all equal, use Tukey’s method to identify the significant differences.

3) Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated with one of the four coatings and buried in one of the three types of soil for a fixed time, after which the amount of corrosion is determined. The data are:

Soil Type (B) 1 2 3 1 64 49 50Coating (A) 2 53 51 48 3 47 45 50 4 51 43 52

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a) Assuming the validity of the additive model, carry out the ANOVA analysis using an ANOVA table to see whether the amount of corrosion depends on either the type of coating used or the type of soil. Use 𝛼 = 0.05.

b) Compute 𝜇, 𝛼!, 𝛼! 𝛼!, 𝛼!, 𝛽!, 𝛽! and 𝛽!.

4) Does yield from a chemical process depend on the formulation of one of the inputs or on mixer speed? Here is some data:

Speed 60 70 80 189.7 185.1 189.0 1 188.6 179.4 193.0 190.1 177.3 191.1Formulation 165.1 161.7 163.3 2 165.9 159.8 166.6 167.6 161.6 170.3

Summary statistics: 𝑆𝑆𝐹𝑜𝑟𝑚 = 2253.44 , 𝑆𝑆𝑆𝑝𝑒𝑒𝑑 = 230.81 , 𝑆𝑆𝐹𝑜𝑟𝑚 ∗ 𝑆𝑝𝑒𝑒𝑑 =18.58, and 𝑆𝑆𝐸 = 71.87.

a) Does there appear to be interaction between the two factors?

b) Does yield appear to depend on either formulation or speed?

c) Calculate estimates of the main effects. Use them to calculate the fitted values: 𝑥!"# = 𝜇 + 𝛼! + 𝛽! + 𝛾!".

d) Verify (using the formula) that the residuals (𝑥!"# − 𝑥!"#) are 0.23, −0.87, 0.63, 4.50, −1.20, −3.30, −2.03, 1.97, 0.07, −1.10, −0.30, 1.40, 0.67, −1.23, 0.57, −3.43, −0.13, and 3.57. Make a normal probability plot for these residuals and interpret.

Homework 4, due March 15 (Late homework will not be accepted for Homework 4.)

1) Calcium phosphate cement is gaining increasing attention for use in bone repair applications. Polypropylene fibers were used in an attempt to improve fracture behavior. Here is some data on fiber weight in percent (𝑥) and compressive strength in MPa (𝑦):

𝑥 0.00 0.00 0.00 0.00 0.00 1.25 1.25 1.25 1.25𝑦 9.94 11.67 11.00 13.44 9.20 9.92 9.79 10.99 11.32

𝑥 2.50 2.50 2.50 2.50 2.50 5.00 5.00 5.00 5.00𝑦 12.29 8.69 9.91 10.45 10.25 7.89 7.61 8.07 9.04

𝑥 7.50 7.50 7.50 7.50 10.00 10.00 10.00 10.00 𝑦 6.63 6.43 7.03 7.63 7.35 6.94 7.02 7.67

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a) Fit the simple linear regression model to this data and determine the proportion of variation explained by the model relationship. Also calculate a point estimate of the standard deviation of the error terms in the model.

b) The average strength values for the six different levels of fiber weight are 11.05,10.51, 10.32, 8.15, 6.93, and 7.24, respectively. Graph fiber weight versus these average strengths and fit an equation to these six points. Explain the difference between 𝑟! for this graph and 𝑟! for the entire data set.

2) Mist is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and work piece. Data on fluid-flow velocity in cm/sec (𝑥) and proportion of small mist droplets (𝑦) are:

𝑥 89 177 189 354 362 442 705𝑦 0.40 0.60 0.48 0.66 0.61 0.69 0.99

a) The investigators used a simple linear regression analysis. Does the graph of the data support this strategy?

b) What proportion of the variation is attributable to the model relationship? Estimate the increase in proportion of small mist droplets when fluid-flow velocity increases by 100 cm/sec, and give a 95% confidence interval for this estimate. Interpret your answer in words.

c) Calculate a 95% confidence interval and a 95% prediction interval for proportion of small mist droplets when fluid-flow velocity is 500 cm/sec. Without actually calculating the interval, indicate whether the confidence interval and the prediction interval for 900 cm/sec would be narrower or wider than the intervals for 500 cm/sec? Explain.

d) Would you recommend calculating a 95% prediction interval for a flow rate of 900 cm/sec? Explain.

e) Assuming bivariate normality, carry out a significance test at level 0.01 of 𝐻!: 𝜌 = 0. Report the 𝑝-value.

3) Failures in aircraft gas turbine engines due to high cycle fatigue are a pervasive problem. Data was collected to predict strain amplitude from cycles to failure:

Cycles Amplitude Cycles Amplitude Cycles Amplitude 1,326 0.01495 7,356 0.00576 2,672 0.008801,593 0.01470 7,904 0.00580 1,016 0.008014,414 0.01100 79 0.01212 7,532 0.008835,673 0.01190 4,175 0.00782 3,410 0.0060029,516 0.00873 34,676 0.00596 30,220 0.00676

26 0.01819 114,789 0.00600 7,101 0.00575843 0.00810

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Fit an appropriate non-linear model for predicting strain amplitude, assess the quality of the fit, and predict amplitude (using a 95% prediction interval) when cycles-to-failure is 5,000.

Homework 5, due April 7

1) The response time of a computer system to a request for a certain type of information is hypothesized to have an exponential distribution with 𝜆 = 1 sec (𝑓(𝑥) = 𝑒!!, for 𝑥 ≥ 0).

a) If you observed 𝑛 observations from this distribution, and wanted to use the chi-squared test with six class intervals having equal probability under the null hypothesis, what would be the resulting class intervals?

b) Use the following 40 data points to perform the analysis:

0.10 0.99 1.14 1.26 3.24 0.12 0.26 0.80 0.79 1.16 1.76 0.41 0.59 0.27 2.22 0.66 0.71 2.21 0.68 0.43 0.11 0.46 0.69 0.38 0.91 0.55 0.81 2.51 2.77 0.16 1.11 0.02 2.13 0.19 1.21 1.13 2.93 2.14 0.34 0.44

2) Consider a large population of 3-children families. If we assume independence of gender, then the number of male children in a randomly selected family will have a binomial distribution, based on three trials.

a) Suppose a random sample of 160 families yields the following results. Fourteen families had zero male children, 66 families had one male child, 64 families had two male children, and 16 families had three male children. Test whether the binomial is a good explanation for these data.

b) Suppose a random sample of 50 non-human families gave 15, 20, 12, and 3 cell counts, respectively. Would the degrees of freedom be the same as in part (a)? Explain.

3) Three professors are teaching large classes in introductory statistics. At the end of the semester, they compare grades to see if there are significant differences in their grading results. Here are the data:

Grade Professor A B C D F WP WF

Smith 12 45 49 6 13 18 2Jones 10 32 43 18 4 12 6White 15 19 32 20 6 9 7

a) Are these differences statistically significant? Interpret the results of your hypothesis test.

b) Are the grades assigned by Jones and White significantly different? Explain.

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4) Three different design configurations are being considered for a component. There are four possible failure modes for the component. An engineer obtained the following data on number of failures in each mode for each of the three configurations:

Failure Mode 1 2 3 4 1 20 44 17 9Configuration 2 4 17 7 12 3 10 31 14 5

Does the configuration appear to be independent of failure mode? Show the matrix of Expected Cell Counts and the matrix of the contributions to the chi-squared statistic.

5) In two separate samples, 100 couples were interviewed, and their first choices for candidates in a recent election were recorded. Here are the data for the two samples:

Sample 1 Results

Partner 2’s choice

A B Other Partner A 12 22 61’s choice B 25 21 4

Other 3 7 0

Sample 2 Results

Partner 2’s choice

A B Other Partner A 8 25 71’s choice B 23 24 3

Other 9 1 0

For both samples, compute and compare the following measures:

a) The chi-squared test statistic

b) Cramér’s Contingency Coefficient

c) Pearson’s Contingency Coefficient

Homework 6, due April 19 (Late homework will not be accepted for Homework 6.)

1) A random sample of 15 automobile mechanics was selected, and the time necessary for each one to diagnose a particular problem was determined, resulting in the following data:

30.6 30.1 15.6 26.7 27.1 25.4 35.0 30.8 31.9 53.2 12.5 23.2 8.8 24.9 30.2

Use the Wilcoxon test at significance level 0.10 to decide whether the data suggests that the true average diagnostic time is less than 30 minutes.

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2) A modification has been made to an industrial process, but due to its expense, it will only be incorporated if there is strong evidence that the modification has decreased true average time by more than one second. Assuming that the time distributions differ only with respect to location, if at all, use the Wilcoxon rank-sum test at level 0.05 on the following data to test the appropriate hypotheses.

Original Process 8.6 5.1 4.5 5.4 6.3 6.6 5.7 8.5

Modified Process 5.5 4.0 3.8 6.0 5.8 4.9 7.0 5.7

3) Revisit Problems (1) and (2) above and construct 90% confidence intervals for the true average diagnostic time and the difference in times between the original and modified processes, respectively.

4) Revisit the data from Problem (2) of Homework 3: Four types of mortar were subjected to a compression test to measure strength (in MPa). Three strength observations for each mortar type are as follows:

OCM 32.15 35.53 34.20PIM 126.32 126.80 134.79RM 117.91 115.02 114.58PCM 29.09 30.87 29.80

Reexamine this data using the Kruskal-Wallis procedure, and compare results to those from Homework 3.

5) An Olympic diver is rated on ten practice dives, with the following results:

1.7 5.3 7.6 8.9 9.0 9.1 9.3 9.6 9.9 9.9

Test the hypothesis that the distribution function of her scores is given by 𝐹(𝑥), where

𝐹 𝑥 = 0 𝑥 < 0

𝐹 𝑥 = !!

!""0 ≤ 𝑥 ≤ 10

𝐹 𝑥 = 1 10 < 𝑥

Homework 7, due May 3

1) The data below are moisture contents for some fabric specimens. Determine control limits for a chart with centerline at height 13.00 based on 𝜎 = 0.600, construct the control chart, and comment on its appearance.

Sample Observations Average SD Range 1 12.2 12.1 13.3 13.0 13.0 12.72 0.536 1.22 12.4 13.3 12.8 12.6 12.9 12.80 0.339 0.93 12.9 12.7 14.2 12.5 12.9 13.04 0.669 1.74 13.2 13.0 13.0 12.6 13.9 13.14 0.477 1.35 12.8 12.3 12.2 13.3 12.0 12.52 0.526 1.3

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6 13.9 13.4 13.1 12.4 13.2 13.20 0.543 1.57 12.2 14.4 12.4 12.4 12.5 12.78 0.912 2.28 12.6 12.8 13.5 13.9 13.1 13.18 0.526 1.39 14.6 13.4 12.2 13.7 12.5 13.28 0.963 2.410 12.8 12.3 12.6 13.2 12.8 12.74 0.329 0.911 12.6 13.1 12.7 13.2 12.3 12.78 0.370 0.912 13.5 12.3 12.8 13.1 12.9 12.92 0.438 1.213 13.4 13.3 12.0 12.9 13.1 12.94 0.559 1.414 13.5 12.4 13.0 13.6 13.4 13.18 0.492 1.215 12.3 12.8 13.0 12.8 13.5 12.88 0.432 1.216 12.6 13.4 12.1 13.2 13.3 12.92 0.554 1.317 12.1 12.7 13.4 13.0 13.9 13.02 0.683 1.818 13.0 12.8 13.0 13.3 13.1 13.04 0.182 0.519 12.4 13.2 13.0 14.0 13.1 13.14 0.573 1.620 12.7 12.4 12.4 13.9 12.8 12.84 0.619 1.521 12.6 12.8 12.7 13.4 13.0 12.90 0.316 0.822 12.7 13.4 12.1 13.2 13.3 12.94 0.541 1.3

2) A manufacturer of dustless chalk instituted a quality control program to monitor chalk density. The sample standard deviations of densities for 24 different subgroups, each consisting of 𝑛 = 8 chalk specimens, were as follows:

0.204 0.315 0.096 0.184 0.230 0.212 0.322 0.287 0.145 0.211 0.053 0.145 0.272 0.351 0.159 0.214 0.388 0.187 0.150 0.229 0.276 0.118 0.091 0.056

Calculate limits for an S chart, construct the chart, and check for out-of-control points. If there is an out-of-control point, delete it and repeat the process.

3) The target value for the diameter of a type of driveshaft is 0.75 in. The size of the shift in the average diameter considered important to detect is 0.002 in. Sample average diameters for successive groups of 𝑛 = 4 shafts are as follows (read across):

0.7507 0.7504 0.7492 0.7501 0.7503 0.7510 0.7490 0.7497 0.7488 0.7504 0.7516 0.7472 0.7489 0.7483 0.7471 0.7498 0.7460 0.7482 0.7470 0.7493 0.7462 0.7481

Use the computational form of the CUSUM procedure with ℎ = 0.003 to see whether the process mean remained on target throughout the time of observation.

4) A sample of 50 items is to be selected from a batch consisting of 5,000 items. The batch will be accepted if the sample contains at most one defective item. Calculate the probability of lot acceptance for 𝑝 = 0.01, 0.02,… , 0.10, and sketch the OC curve.

Homework 8, May 10 (Late homework will not be accepted for Homework 8.)

1) In a population with 𝑁 = 6 the values of 𝑥 are 8, 3, 1, 11, 4, and 7.

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a) Calculate the sample mean for all possible simple random samples of size 2. Verify that the sample mean is an unbiased estimate of the population mean, and that the variance of the sample mean is !

!

!!!!!

.

b) Calculate 𝑠! for all simple random samples of size 3 and verify that 𝐸 𝑠! = 𝜎!.

2) The following data show the stratification of all the farms in a county by farm size and the average acres of corn per farm in each stratum. For a sample of 100 farms, compute the same sizes in each stratum under a) proportional allocation, and b) optimum allocation.

Farm Size in acres Number of Farms Average Corn Acres SD 0 to 40 394 5.4 8.341 to 80 461 16.3 13.381 to 120 391 24.3 15.1121 to 160 334 34.5 19.8161 to 200 169 42.1 24.5201 to 240 113 50.1 26.0241 and up 148 63.8 35.2

Total 2010 26.3

3) Using a small population and a simulation, we will compare some of the various sampling schemes.

a) Simple Random Sampling: Randomly select 12 states and find the average Land Area for your states. Estimate the total for the entire US by multiplying the average of your 12 states by 50. Repeat 1,000 times, and calculate the mean and standard deviation of your 1,000 totals.

b) Stratified Sampling: From the Western states (Alaska, California, Oregon, Washington, Idaho, Montana, Wyoming, Colorado, Utah, Nevada, New Mexico, Texas, and Arizona), randomly select 4 states. From the Northeastern states (Maine, Massachusetts, New Hampshire, Vermont, Rhode Island, Connecticut, New York, New Jersey, Pennsylvania, Delaware, and Maryland), randomly select 2 states. From the remaining 26 states, randomly select 6 states. Estimate the total for the entire US by multiplying the Western average by 13, the Northeastern average by 11, and remaining average by 26, and adding all three subtotals together. Repeat 1,000 times, and calculate the mean and standard deviation of your 1,000 totals.

c) Cluster Sampling: Using the following breakdown of the states (North, Southwest, Central, Southeast, and Northeast), randomly select three regions. From each of the three regions, randomly select 4 states. Estimate the total for the entire US by multiplying the average of your 12 states by 50. Repeat 1,000 times, and calculate the mean and standard deviation of your 1,000 totals.

North: Alaska, Washington, Oregon, Idaho, Montana, Wyoming, N. Dakota, S. Dakota, Nebraska, and Minnesota.

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Southwest: Hawaii, California, Nevada, Utah, Arizona, Colorado, New Mexico, Kansas, Oklahoma, and Texas.

Central: Iowa, Missouri, Arkansas, Wisconsin, Michigan, Illinois, Indiana, Ohio, Kentucky, and Tennessee.

Southeast: Louisiana, Mississippi, Alabama, Georgia, Florida, S. Carolina, N. Carolina, Virginia, W. Virginia, and Maryland.

Northeast: Pennsylvania, Delaware, New Jersey, New York, Vermont, New Hampshire, Massachusetts, Connecticut, Rhode Island, and Maine.

State Land Area State Land Area State Land Area 1 Alabama 50,744 18 Louisiana 43,562 35 Ohio 40,948

2 Alaska 571,951 19 Maine 30,862 36 Oklahoma 68,667

3 Arizona 113,635 20 Maryland 9,774 37 Oregon 95,997

4 Arkansas 52,068 21 Massachusetts 7,840 38 Pennsylvania 44,817

5 California 155,959 22 Michigan 56,804 39 Rhode Island 1,045

6 Colorado 103,718 23 Minnesota 79,610 40 South Carolina 30,109

7 Connecticut 4,845 24 Mississippi 46,907 41 South Dakota 75,885

8 Delaware 1,954 25 Missouri 68,886 42 Tennessee 41,217

9 Florida 53,927 26 Montana 145,552 43 Texas 261,797

10 Georgia 57,906 27 Nebraska 76,872 44 Utah 82,144

11 Hawaii 6,423 28 Nevada 109,826 45 Vermont 9,250

12 Idaho 82,747 29 New Hampshire 8,968 46 Virginia 39,594

13 Illinois 55,584 30 New Jersey 7,417 47 Washington 66,544

14 Indiana 35,867 31 New Mexico 121,356 48 West Virginia 24,078

15 Iowa 55,869 32 New York 47,214 49 Wisconsin 54,310

16 Kansas 81,815 33 North Carolina 48,711 50 Wyoming 97,100

17 Kentucky 39,728 34 North Dakota 68,976