Solved: Determine the confidence level for each of the

Problem 1E Consider a normal population distribution with the value of s known.

Problem 2E Each of the following is a confidence interval for ? = true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type: (114.4, 115.6) (114.1, 115.9) a. What is the value of the sample mean resonance frequency? b. Both intervals were calculated from the same sample data. The confidence level for one of these intervals is 90% and for the other is 99%. Which of the intervals has the 90% confidence level, and why?

Problem 3E Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let ? denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.8, 9.4). a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. b. Consider the following statement: There is a 95% chance that ? is between 7.8 and 9.4. Is this statement correct? Why or why not? c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4. Is this statement correct? Why or why not? d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include ?. Is this statement correct? Why or why not?

A CI is desired for the true average stray-load loss \(\mu\) (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with \(\sigma = 3.0\). a. Compute a 95% CI for \(\mu\) when n = 25 and \(\bar {x} = 58.3\). b. Compute a 95% CI for \(\mu\) when n = 100 and \(\bar {x} = 58.3\). c. Compute a 99% CI for \(\mu\) when n = 100 and \(\bar {x} = 58.3\). d. Compute an 82% CI for \(\mu\) when n = 100 and \(\bar {x} = 58.3\). e. How large must n be if the width of the 99% interval for \(\mu\) is to be 1.0?

Problem 5E Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation .75. a. Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85. b. Compute a 98% CI for true average porosity of another seam based on 16 specimens with a sample average porosity of 4.56. c. How large a sample size is necessary if the width of the 95% interval is to be .40? d. What sample size is necessary to estimate true average porosity to within .2 with 99% confidence?

On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar is known to be normally distributed with \(\sigma = 100\). The composition of bars has been slightly modified, but the modification is not believed to have affected either the normality or the value of \(\sigma\). a. Assuming this to be the case, if a sample of 25 modified bars resulted in a sample average yield point of 8439 lb, compute a 90% CI for the true average yield point of the modified bar. b. How would you modify the interval in part (a) to obtain a confidence level of 92%?

Problem 7E By how much must the sample size n be increased if the width of the CI (7.5) is to be halved? If the sample size is increased by a factor of 25, what effect will this have on the width of the interval? Justify your assertions.

Problem 8E Let Then a. Use this equation to derive a more general expression for a 100(1– ?)% CI for ? of which the interval (7.5) is a special case. b. Let ? = .05 and . Does this result in a narrower or wider interval than the interval (7.5)?

Problem 9E a.Under the same conditions as those leading to the interval (7.5), Use this to derive a one-sided interval for ? that has infinite width and provides a lower confidence bound on ?. What is this interval for the data in Exercise 5(a)? b. Generalize the result of part (a) to obtain a lower bound with confidence level 100(1 – ? )%. c. What is an analogous interval to that of part (b) that provides an upper bound on ?? Compute this 99% interval for the data of Exercise 4(a).

Problem 10E A random sample of heat pumps of n = 15a certain type yielded the following observations on lifetime (in years): a. Assume that the lifetime distribution is exponential and use an argument parallel to that of Example 7.5 to obtain a 95% CI for expected (true average) lifetime. b. How should the interval of part (a) be altered to achieve a confidence level of 99%? c. What is a 95% CI for the standard deviation of the lifetime distribution? [Hint: What is the standard deviation of an exponential random variable?]

Problem 11E Consider the next 1000 95% CIs for ? that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of ?? What is the probability that between 940 and 960 of these intervals contain the corresponding value of ?? [Hint: Let Y = the number among the 1000 intervals that contain ?. What kind of random variable is Y?]

Problem 12E The following observations are lifetimes (days) subsequent to diagnosis for individuals suffering from blood cancer ("A Goodness of Fit Approach to the Class of Life Distributions with Unknown Age," Quality and Reliability Engr. Intl., 2012: 761-766): a. Can a confidence interval for true average calculated without assuming anything lifetime be about the nature of the lifetime distribution? Explain your reasoning. [Note: A normal probability plot of the data exhibits a reasonably linear pattern.] b. Calculate and interpret a confidence interval with a 99% confidence level for true average lifetime. [Hint: : = 1191.6 and s = 506.64

Problem 13E The article “Gas Cooking, Kitchen Ventilation, and Exposure to Combustion Products” (Indoor Air, 2006: 65–73) reported that for a sample of 50 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 164.43. a. Calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level in the population of all homes from which the sample was selected. b. Suppose the investigators had made a rough guess of 175 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 50 ppm for a confidence level of 95%?

Problem 14E The negative effects of ambient air pollution on children's lung function has been well established, but less research is available about the impact of indoor air pollution. The authors of "Indoor Air Pollution and Lung Function Growth Among Children in Four Chinese Cities" (Indoor Air, 2012: 3-11) investigated the relationship between indoor air-pollution metrics and lung function growth among children ages 6-13 years living in four Chinese cities. For each subject in the study, the authors measured an important lung-capacity index known as FEV1, the forced volume (in ml) of air that is exhaled in 1 second. Higher FEV1 values are associated with greater lung capacity. Among the children in the study, 514 came from households that used coal for cooking or heating or both. Their FEV1 mean was 1427 with a standard deviation of 325. (A complex statistical procedure was used to show that burning coal had a clear negative effect on mean FEV1 levels.) a. Calculate and interpret a 95% (two-sided) confidence interval for true average FEV1 level in the population of all children from which the sample was selected. Does it appear that the parameter of interest has been accurately estimated? b. Suppose the investigators had made a rough guess of 320 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 50 ml for a confidence level of 95%?

Problem 15E Determine the confidence level for each of the following large-sample one-sided confidence bounds:

Problem 16E The alternating current (AC) breakdown voltage of an insulating liquid indicates its dielectric strength. The article “Testing Practices for the AC Breakdown Voltage Testing of Insulation Liquids” (IEEE Electrical Insulation Magazine, 1995: 21–26) gave the accompanying sample observations on breakdown voltage (kV) of a particular circuit under certain conditions a. Construct a boxplot of the data and comment on interesting features. b. Calculate and interpret a 95% CI for true average breakdown voltage ?. Does it appear that ? has been precisely estimated? Explain. c. Suppose the investigator believes that virtually all values of breakdown voltage are between 40 and 70. What sample size would be appropriate for the 95% CI to have a width of 2 kV (so that ? is estimated to within 1 kV with 95% confidence)?.

Problem 18E The U.S. Army commissioned a study to assess how deeply a bullet penetrates ceramic body armor ("Testing Body Armor Materials for Use by the U.S. Army-Phase III," 2012). In the standard test, a cylindrical clay model is layered under the armor vest. A projectile is then fired, causing an indentation i n the clay. The deepest impression in the clay is measured as an indication of survivability of someone wearing the armor. Here is data from one testing organization under particular experimental conditions; measurements (in mm) were made using a manually controlled digital caliper: 22.4 23.6 24.0 24.9 25.5 25.6 25.8 26.1 26.4 26.7 27.4 27.6 28.3 29.0 29.1 29.6 29.7 29.8 29.9 30.0 30.4 30.5 30.7 30.7 31.0 31.0 31.4 31.6 31.7 31.9. 31.9 32.0 32.1 32.4 32.5 32.5 32.6 32.9 33.1 33.3 33.5 33.5 33.5 33.5 33.6 33.6 33.8 33.9 34.1 34.2 34.6 34.6 35.0 35.2 35.2 35.4 35.4 35.4 35.5 35.7 35.8 36.0 36.0 36.0 36.1 36.1 36.2 36.4 36.6 37.0 37.4 37.5 37.5 38.0 38.7 38.8 39.8 41.0 42.0 42.1 44.6 48.3 55.0 a. Construct a boxplot of the data and comment on interesting features. b. Construct a normal probability plot. Is it plausible that impression depth is normally distributed? Is a normal distribution assumption needed in order to calculate a confidence interval or bound for the true average depth pausing the foregoing data? Explain. c. Use the accompanying Minitab output as a basis for calculating and interpreting an upper confidence boundforiswithaconfidencelevelof99%.

Problem 17E Exercise 1.13 gave a sample of ultimate tensile strength observations (ksi). Use the accompanying descriptive statistics output from Minitab to calculate a 99% lower confidence bound for true average ultimate tensile strength, and interpret the result.

The article “Limited Yield Estimation for Visual Defect Sources” (IEEE Trans. on Semiconductor Manuf., 1997: 17–23) reported that, in a study of a particular wafer inspection process, 356 dies were examined by an inspection probe and 201 of these passed the probe. Assuming a stable process, calculate a 95% (two-sided) confidence interval for the proportion of all dies that pass the probe.

Problem 20E TV advertising agencies face increasing challenges in reaching audience members because viewing TV programs via digital streaming is gaining in popularity. The Harris poll reported on November 13, 2012, that 53% of 2343 American adults surveyed said they have watched digitally streamed TV programming on some type of device a. Calculate and interpret a confidence interval at the 99% confidence level for the proportion of all adult Americans who watched streamed programming up to that point in time. b. What sample size would be required for the width of a 99% CI to be at most .05 irrespective of the value

Problem 21E In a sample of 1000 randomly selected consumers who had opportunities to send in a rebate claim form after purchasing a product, 250 of these people said they never did so (“Rebates: Get What You Deserve,” Consumer Reports, May 2009: 7). Reasons cited for their behavior included too many steps in the process, amount too small, missed deadline, fear of being placed on a mailing list, lost receipt, and doubts about receiving the money. Calculate an upper confidence bound at the 95% confidence level for the true proportion of such consumers who never apply for a rebate. Based on this bound, is there compelling evidence that the true proportion of such consumers is smaller than 1/3? Explain your reasoning.

Problem 22E The technology underlying hip replacements has changed as these operations have become more popular (over 250,000 in the United States in 2008). Starting in 2003, highly durable ceramic hips were marketed. Unfortunately, for too many patients the increased durability has been counterbalanced by an increased incidence of squeaking. The May 11, 2008, issue of the New York Times reported that in one study of 143 individuals who received ceramic hips between 2003 and 2005, 10 of the hips developed squeaking. a. Calculate a lower confidence bound at the 95% confidence level for the true proportion of such hips that develop squeaking. b. Interpret the 95% confidence level used in (a).

Problem 23E The Pew Forum on Religion and Public Life reported on Dec. 9, 2009, that in a survey of 2003 American adults, 25% said they believed in astrology. a. Calculate and interpret a confidence interval at the 99% confidence level for the proportion of all adult Americans who believe in astrology. b. What sample size would be required for the width of a 99% CI to be at most .05 irrespective of the value of

A sample of 56 research cotton samples resulted in a sample average percentage elongation of 8.17 and a sample standard deviation of 1.42 (“An Apparent Relation Between the Spiral Angle f, the Percent Elongation E1, and the Dimensions of the Cotton Fiber,” Textile Research J., 1978: 407–410). Calculate a 95% large-sample CI for the true average percentage elongation \(\mu\). What assumptions are you making about the distribution of percentage elongation?

Problem 25E A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions. a. What sample size is necessary if the 95% CI for p is to have a width of at most .10 irrespective of p? b. If the legislator has strong reason to believe that at least of the electorate know of her position, how large a 2/3 sample size would you recommend?

Problem 26E The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poissondistribution with parameter m. Use the accompanying data on absences for 50 days to obtain a large-sample CI for ?. [Hint: The mean and variance of a Poisson variable both equal ?, so has approximately a standard normal distribution. Now proceed as in the derivation of the interval for p by making a probability statement (with probability 1 – ?) and solving the resulting inequalities for ? –– see the argument just after (7.10).]

Problem 27E Reconsider the CI (7.10) for p, and focus on a confidence level of 95%. Show that the confidence limits agree quite well with those of the traditional interval (7.11) once two successes and two failures have been appended to the sample [i.e., (7.11) based on x +2 S’s in n +4 trials]. [Hint:1.96 ? 2 . Note: Agresti and Coull showed that this adjustment of the traditional interval also has an actual confidence level close to the nominal level.]

Problem 28E Determine the values of the following quantities:

Problem 29E Determine the t critical value(s) that will capture the desired t-curve area in each of the following cases: a. Central area = .95, df = 10 b. Central area = .95, df = 20 c. Central area =.99, df =20 d. Central area = .99, df = 50 e. Upper-tail area = .01, df = 25 f. Lower-tail area = .025, df = 5

Problem 30E Determine the t critical value for a two-sided confidence interval in each of the following situations: a. Confidence level = 95%, df = 10 b. Confidence level = 95%, df = 15 c. Confidence level = 99%, df = 15 d. Confidence level = 99%, n = 5 e. Confidence level =98%, df = 24 f. Confidence level = 99%, n = 38

Determine the t critical value for a lower or an upper confidence bound for each of the situations described in Exercise 30.

Problem 32E According to the article “Fatigue Testing of Condoms” (Polymer Testing, 2009: 567–571), “tests currently used for condoms are surrogates for the challenges they face in use,” including a test for holes, an inflation test, a package seal test, and tests of dimensions and lubricant quality (all fertile territory for the use of statistical methodology!). The investigators developed a new test that adds cyclic strain to a level well below breakage and determines the number of cycles to break. A sample of 20 condoms of one particular type resulted in a sample mean number of 1584 and a sample standard deviation of 607. Calculate and interpret a confidence interval at the 99% confidence level for the true average number of cycles to break. [Note: The article presented the results of hypothesis tests based on the t distribution; the validity of these depends on assuming normal population distributions.]

Problem 33E The article “Measuring and Understanding the Aging of Kraft Insulating Paper in Power Transformers” (IEEE Electrical Insul. Mag., 1996: 28–34) contained the following observations on degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range: a. Construct a boxplot of the data and comment on any interesting features. b. Is it plausible that the given sample observations were selected from a normal distribution? c. Calculate a two-sided 95% confidence interval for true average degree of polymerization (as did the authors of the article). Does the interval suggest that 440 is a plausible value for true average degree of polymerization? What about 450?

Problem 34E A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.48 MPa and a sample standard deviation of .79 MPa (“Characterization of Bearing Strength Factors in Pegged Timber Connections,” J. of Structural Engr., 1997: 326–332). a. Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress? b. Calculate and interpret a 95% lower prediction bound for the proportional limit stress of a single joint of this type .

Problem 35E The observations on escape time given in Exercise 36 of Chapter 1 give a sample mean and sample standard deviation of 370.69and 24.36, respectively. a. Calculate an upper confidence bound for population mean escape time using a confidence level of 95%. b. Calculate an upper prediction bound for the escape time of a single additional worker using a prediction level of 95%. How does this bound compare with the confidence bound of part (a)? c. Suppose that two additional workers will be chosen to participate in the simulated escape exercise. Denote their escape times by X27 and X28, and let , and calculate a 95% two-sided interval based on the given escape data.

Problem 36E A normal probability plot of the n = 26 observations on escape time given in Exercise 36 of Chapter 1 shows a substantial linear pattern; the sample mean and sample standard deviation are 370.69 and 24.36, respectively. a. Calculate an upper confidence bound for population mean escape time using a confidence level of 95%. b. Calculate an upper prediction bound for the escape time of a single additional worker using a prediction level of 95%. How does this bound compare with the confidence bound of part (a)? c. Suppose that two additional workers will be chosen to participate in the simulated escape exercise. Denote their escape times by X27 and X28, and let new denote the average of these two values. Modify the formula for a PI for a single x value to obtain a PI for new, and calculate a 95% two-sided interval based on the given escape data.

Problem 37E A study of the ability of individuals to walk in a straight line (“Can We Really Walk Straight?” Amer. J. of Physical Anthro., 1992: 19–27) reported the accompanying data on cadence (strides per second) for a sample of n = 20 randomly selected healthy men. a. Calculate and interpret a 95% confidence interval for population mean cadence. b. Calculate and interpret a 95% prediction interval for the cadence of a single individual randomly selected from this population. c. Calculate an interval that includes at least 99% of the cadences in the population distribution using a confidence level of 95%.

Problem 38E Ultra high performance concrete (UHPC) is a relatively new construction material that is characterized by strong adhesive properties with other materials. The article "Adhesive Power of Ultra High Performance Concrete from a Thermodynamic Point of View" (J. of Materials in Civil Engr., 2012: 1050-1058) described an investigation of the intermolecular forces for UHPC connected to various substrates. The following work of adhesion measurements (in mJ/m2) for UHPC specimens adhered to steel appeared in the article: 107.1 109.5 107.4 106.8 108.1 a. Is it plausible that the given sample observations were selected from a normal distribution? b. Calculate a two-sided 95% confidence interval for the true average work of adhesion for UHPC adhered to steel. Does the interval suggest that 107 is a plausible value for the true average work of adhesion for UHPC adhered to steel? What about 110? c. Predict the resulting work of adhesion value resulting from a single future replication of the experiment by calculating a 95% prediction interval, and compare the width of this interval to the width of the CI from (b). d. Calculate an interval for which you can have a high degree of confidence that at least 95% of all UHPC specimens adhered to steel will have work of adhesion values between the limits of the interval

Problem 39E Exercise 72 of Chapter 1 gave the following observations on a receptor binding measure (adjusted distribution volume) for a sample of 13 healthy individuals: 23, 39, 40, 41, 43, 47, 51, 58, 63, 66, 67, 69, 72. a. Is it plausible that the population distribution from which this sample was selected is normal? b. Calculate an interval for which you can be 95% confident that at least 95% of all healthy individuals in the population have adjusted distribution volumes lying between the limits of the interval. c. Predict the adjusted distribution volume of a single healthy individual by calculating a 95% prediction interval. How does this interval’s width compare to the width of the interval calculated in part (b)? Reference Exercise 72 of Chapter 1 Anxiety disorders and symptoms can often be effectively treated with benzodiazepine medications. It is known that animals exposed to stress exhibit a decrease in benzodiazepine receptor binding in the frontal cortex. The paper “Decreased Benzodiazepine Receptor Binding in Prefrontal Cortex in Combat-Related Posttraumatic Stress Disorder” (Amer. J. of Psychiatry, 2000: 1120–1126) described the first study of benzodiazepine receptor binding in individuals suffering from PTSD. The accompanying data on a receptor binding measure (adjusted distribution volume) was read from a graph in the paper. Use various methods from this chapter to describe and summarize the data.

Exercise 13 of Chapter 1 presented a sample of n = 153 observations on ultimate tensile strength, and Exercise 17 of the previous section gave summary quantities and requested a large-sample confidence interval. Because the sample size is large, no assumptions about the population distribution are required for the validity of the CI. a. Is any assumption about the tensile-strength distribution required prior to calculating a lower prediction bound for the tensile strength of the next specimen selected using the method described in this section? Explain. b. Use a statistical software package to investigate the plausibility of a normal population distribution. c. Calculate a lower prediction bound with a prediction level of 95% for the ultimate tensile strength of the next specimen selected.

Problem 41E A more extensive tabulation of t critical values than what appears in this book shows that for the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20, and .15, respectively. What is the confidence level for each of the following three confidence intervals for the mean ? of a normal population distribution? Which of the three intervals would you recommend be used, and why?

Problem 42E Determine the values of the following quantities:

Problem 43E Determine the following: a. The 95th percentile of the chi-squared distribution with v= 10 b. The 5th percentile of the chi-squared distribution with v = 10

Problem 44E The amount of lateral expansion (mils) was determined for a sample of n = 9 pulsed-power gas metal arc welds used in LNG ship containment tanks. The resulting sample standard deviation was s= 2.81 mils. Assuming normality, derive a 95% CI for ? 2 and for ?.

Problem 45E Wire electrical-discharge machining (WEDM) is a process used to manufacture conductive hard metal components. It uses a continuously moving wire that serves as an electrode. Coating on the wire electrode allows for cooling of the wire electrode core and provides an improved cutting performance. The article "High-Performance Wire Electrodes for Wire Electrical-Discharge Machining—A Review" (J. of Engr. Manuf., 2012: 1757-1773) gave the following sample observations on total coating layer thickness (in gm) of eight wire electrodes used for WEDM: 21 16 29 35 42 24 24 25 Calculate a 99% CI for the standard deviation of the coating layer thickness distribution. Is this interval valid whatever the nature of the distribution? Explain.

The article “Concrete Pressure on Formwork” (Mag. of Concrete Res., 2009: 407–417) gave the following observations on maximum concrete pressure \((kN/m^2 )\): 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. Is it plausible that this sample was selected from a normal population distribution? b. Calculate an upper confidence bound with confidence level 95% for the population standard deviation of maximum pressure.

Problem 47E Example 1.11 introduced the accompanying observations on bond strength. a. Estimate true average bond strength in a way that conveys information about precision and reliability. [Hint: b. Calculate a 95% CI for the proportion of all such bonds whose strength values would exceed 10.

Problem 48E The article "Distributions of Compressive Strength Obtained from Various Diameter Cores" (ACI Materials J., 2012: 597-606) described a study in which compressive strengths were determined for concrete specimens of various types, core diameters, and length-to-diameter ratios. For one particular type, diameter, and lid ratio, the 18 tested specimens resulted in a sample mean compressive strength of 64.41 MPa and a sample standard deviation of 10.32 MPa. Normality of the compressive strength distribution was judged to be quite plausible. a. Calculate a confidence interval with confidence level 98% for the true average compressive strength under these circumstances. b. Calculate a 98% lower prediction bound for the compressive strength of a single future specimen tested under the given circumstances. [Hint: t02,17 = 2.224.]

Problem 49E For those of you who don't already know, dragon boat racing is a competitive water sport that involves 20 paddlers propelling a boat across various race distances. It has become increasingly popular over the last few years. The article 'Physiological and Physical Characteristics of Elite Dragon Boat Paddlers" (J. of Strength and Conditioning, 2013: 137-145) summarized an extensive statistical analysis of data obtained from a sample of 11 paddlers. It reported that a 95% confidence interval for true average force (N) during a simulated 200-m race was (60.2, 70.6). Obtain a 95% prediction interval for the force of a single randomly selected dragon boat paddler undergoing the simulated race.

Problem 50E A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was (229.764, 233.504). You decide that a confidence level of 99% is more appropriate than the 95% level used. What are the limits of the 99% interval? [Hint: Use the center of the interval and its width to determine and s.]

Problem 52E High concentration of the toxic element arsenic is all too common in groundwater. The article “Evaluation of Treatment Systems for the Removal of Arsenic from Groundwater” (Practice Periodical of Hazardous, Toxic, and Radioactive Waste Mgmt., 2005: 152–157) reported that for a sample of n = 5water specimens selected for treatment by coagulation, the sample mean arsenic concentration was 24.3 ?g/L, and the sample standard deviation was 4.1. The authors of the cited article used t-based methods to analyze their data, so hopefully had reason to believe that the distribution of arsenic concentration was normal. a. Calculate and interpret a 95% CI for true average arsenic concentration in all such water specimens. b. Calculate a 90% upper confidence bound for the standard deviation of the arsenic concentration distribution. c. Predict the arsenic concentration for a single water specimen in a way that conveys information about precision and reliability.

Problem 51E Unexplained respiratory symptoms reported by athletes are often incorrectly considered secondary to exercise-induced asthma. The article "High Prevalence of Exercise-Induced Laryngeal Obstruction in Athletes" (Medicine and Science in Sports and Exercise, 2013: 2030-2035) suggested that many such cases could instead be explained by obstruction of the larynx. In a sample of 88 athletes referred for an asthma workup, 31 were found to have the SILO condition. a. Calculate and interpret a confidence interval using a 95% confidence level for the true proportion of all athletes found to have the SILO condition under these circumstances. b. What sample size is required if the desired width of the 95% CI is to be at most .04, irrespective of the sample results? c. Does the upper limit of the interval in (a) specify a 95% upper confidence bound for the proportion being estimated? Explain.

Problem 53E Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees are selected for experimentation. The first three groves are sprayed with pesticides 1, 2, and 3, respectively, and the fourth is treated with ladybugs, with the following results on yield: Let ?i = the true average yield (bushels/tree) after receiving the ith treatment. Then measures the difference in true average yields between treatment with pesticides and treatment with ladybugs. When n1, n2, n3, nd n4 are all large, the estimator obtained by replacing each ?i by is approximately normal. Use this to derive a large-sample 100(1– ?)% CI for u, and compute the 95% interval for the given data.

Problem 54E It is important that face masks used by firefighters be able to withstand high temperatures because firefighters commonly work in temperatures of 200–500°F. In a test of one type of mask, 11 of 55 masks had lenses pop out at 250°. Construct a 90% CI for the true proportion of masks of this type whose lenses would pop out at 250°.

Problem 55E A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate the average force required to break the binding to within .1 lb with 95% confidence? Assume that ? is known to be .8.

Problem 56E The accompanying data on crack initiation depth (m) was read from a lognormal probability plot that appeared in the article "Incorporating Small Fatigue Crack Growth in Probabilistic Life Prediction: Effect of Stress Ratio in Ti-6A1-2Sn-6Mo" (Intl. J. of Fatigue, 2013: 83-95). Although the pattern in the plot was quite straight, a normal probability plot of the data also shows a reasonably linear pattern. And a boxplot indicates that the distribution is quite symmetric in the middle 50% of the data and only mildly skewed overall. It is therefore reasonable to estimate and predict using t intervals. 4.7 5.1 5.2 53 5.6 5.8 63 6.7 7.2 7.4 7.7 8.5 8.9 9.3 10.1 11.2 a. Estimate the true average crack initiation depth with a 99% CI and interpret the resulting interval. b. Predict the value of a single crack initiation depth by constructing a 99% PI. c. Interpret in context the meaning of 99% in (b).

Problem 57E In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and the experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter ?. Let Y1 denote the time at which the first failure occurs, Y2 the time at which the second failure occurs, and so on, so that is the total accumulated lifetime at termination. Then it can be shown that 2 ?Tr has a chi-squared distribution with 2r df. Use this fact to develop a 100(1–?) CI formula for true average lifetime 1/ ?. Compute a 95% CI from the data in Example 6.8.

Let \(X_1, X_2,\ldots ,X_n\) be a random sample from a continuous probability distribution having median \(\tilde \mu\), (so that \(P(X_i \leq \tilde \mu)= P(X_i \geq \tilde \mu) = .5\)). a. Show that \(P\left(\min \left(X_{i}\right)<\tilde{\mu}<\max \left(X_{i}\right)\right)=1-\left(\frac{1}{2}\right)^{n-1}\) so that \(\left(\min \left(x_{\mathrm{i}}\right), \max \left(x_{i}\right)\right)\) is a \(100(1-\alpha)\)% confidence interval for \(\tilde \mu\) with \(\alpha=\left(\frac{1}{2}\right)^{n-1}\). [Hint: The complement of the event {min \((X_i)\) < \(\tilde \mu\) < max \((X_i)\)} is {max \((X_i) \leq \tilde \mu\)} \(\cup\) {min \((X_i) \geq \tilde \mu\)}. But max \((X_i ) \leq \mu\) iff \(X_i \leq \tilde \mu\) for all i.] b. For each of six normal male infants, the amount of the amino acid alanine (mg/100 mL) was determined while the infants were on an isoleucine-free diet, resulting in the following data: 2.84 3.54 2.80 1.44 2.94 2.70 Compute a 97% CI for the true median amount of alanine for infants on such a diet (“The Essential Amino Acid Requirements of Infants,” Amer. J. of Nutrition, 1964: 322–330). c. Let \(x_{(2)}\) denote the second smallest of the \(x_i\)’s and \(x_{(n-1)}\) denote the second largest of the \(x_i\)’s. What is the confidence level of the interval \((x_{(2)}, x_{(n-1)})\) for \(\tilde \mu\)?

Problem 59E Let X1, X2, ….., Xn be a random sample from a uniform distribution on the interval [0, ?], so that Then if , it can be shown that the rv U = Y/? has density function a. Use fU(u) to verify that and use this to derive a 100(1 –?)% and CI for ? b. Verify that , and derive a 100(1 –?)% CI for ? % CI for u based on this probability statement. c. Which of the two intervals derived previously is shorter? If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1 = 4.2,, x2 = 3.5, x3 = 1.7, x4 = 1.2 and x = 2.4, derive a 95% CI for ? by using the shorter of the two intervals.

Suppose \(x_1,x_2, \ldots, x_n\) are observed values resulting from a random sample from a symmetric but possibly heavy-tailed distribution. Let \(\tilde x\) and \(f_s\) denote the sample median and fourth spread, respectively. Chapter 11 of Understanding Robust and Exploratory Data Analysis (see the bibliography in Chapter 6) suggests the following robust 95% CI for the population mean (point of symmetry): \(\tilde{x} \pm\left(\frac{\text { conservative } t \text { critical value }}{1.075}\right) \cdot \frac{f_{s}}{\sqrt{n}}\) The value of the quantity in parentheses is 2.10 for n = 10, 1.94 for n = 20, and 1.91 for n = 30. Compute this CI for the data of Exercise 45, and compare to the t CI appropriate for a normal population distribution.

Problem 60E Let 0? ? ? ?. Then a 100(1 – ?) % CI for ? when n is large is

a. Use the results of Example 7.5 to obtain a 95% lower confidence bound for the parameter l of an exponential distribution, and calculate the bound based on the data given in the example. b. If lifetime X has an exponential distribution, the probability that lifetime exceeds t is \(P(X>t)=e^{-\lambda t}\). Use the result of part (a) to obtain a 95% lower confidence bound for the probability that breakdown time exceeds 100 min.

Example 1.11 introduced the accompanying observations on 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. Estimate true average bond strength in a way that conveys information about precision and reliability. [Hint: oxi 5 387.8 and oxi 2 5 4247.08.] b. Calculate a 95% CI for the proportion of all such bonds whose strength values would exceed 10.

The article “Distributions of Compressive Strength Obtained from Various Diameter Cores” (ACI Materials J., 2012: 597–606) described a study in which compressive strengths were determined for concrete specimens of various types, core diameters, and length-to-diameter ratios. For one particular type, diameter, and l/d ratio, the 18 tested specimens resulted in a sample mean compressive strength of 64.41 MPa and a sample standard deviation of 10.32 MPa. Normality of the compressive strength distribution was judged to be quite plausible. a. Calculate a confidence interval with confidence level 98% for the true average compressive strength under these circumstances. b. Calculate a 98% lower prediction bound for the compressive strength of a single future specimen tested under the given circumstances. [Hint: \(t_{.02,17} = 2.224.\)]

For those of you who dont already know, dragon boat racing is a competitive water sport that involves 20 paddlers propelling a boat across various race distances. It has become increasingly popular over the last few years. The article Physiological and Physical Characteristics of Elite Dragon Boat Paddlers (J. of Strength and Conditioning, 2013: 137145) summarized an extensive statistical analysis of data obtained from a sample of 11 paddlers. It reported that a 95% confidence interval for true average force (N) during a simulated 200-m race was (60.2, 70.6). Obtain a 95% prediction interval for the force of a single randomly selected dragon boat paddler undergoing the simulated race.

A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was (229.764, 233.504). You decide that a confidence level of 99% is more appropriate than the 95% level used. What are the limits of the 99% interval? [Hint: Use the center of the interval and its width to determine x and s.]

Unexplained respiratory symptoms reported by athletes are often incorrectly considered secondary to exercise-induced asthma. The article High Prevalence of Exercise-Induced Laryngeal Obstruction in Athletes (Medicine and Science in Sports and Exercise, 2013: 20302035) suggested that many such cases could instead be explained by obstruction of the larynx. In a sample of 88 athletes referred for an asthma workup, 31 were found to have the EILO condition. a. Calculate and interpret a confidence interval using a 95% confidence level for the true proportion of all athletes found to have the EILO condition under these circumstances. b. What sample size is required if the desired width of the 95% CI is to be at most .04, irrespective of the sample results? c. Does the upper limit of the interval in (a) specify a 95% upper confidence bound for the proportion being estimated? Explain.

High concentration of the toxic element arsenic is all too common in groundwater. The article Evaluation of Treatment Systems for the Removal of Arsenic from Groundwater (Practice Periodical of Hazardous, Toxic, and Radioactive Waste Mgmt., 2005: 152157) reported that for a sample of n 5 5 water specimens selected for treatment by coagulation, the sample mean arsenic concentration was 24.3 mg/L, and the sample standard deviation was 4.1. The authors of the cited article used t-based methods to analyze their data, so hopefully had reason to believe that the distribution of arsenic concentration was normal. a. Calculate and interpret a 95% CI for true average arsenic concentration in all such water specimens. b. Calculate a 90% upper confidence bound for the standard deviation of the arsenic concentration distribution. c. Predict the arsenic concentration for a single water specimen in a way that conveys information about precision and reliability.

Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees are selected for experimentation. The first three groves are sprayed with pesticides 1, 2, and 3, respectively, and the fourth is treated with ladybugs, with the following results on yield: Let \(\mu_i=\) the true average yield (bushels/tree) after receiving the i th treatment. Then \(\theta=\frac{1}{3}\left(\mu_1+\mu_2+\mu_3\right)-\mu_4\) measures the difference in true average yields between treatment with pesticides and treatment with ladybugs. When \(n_1, n_2, n_3\), and \(n_4\) are all large, the estimator \(\hat{\theta}\) obtained by replacing each \(\mu_i\) by \(\bar{X}_i\) is approximately normal. Use this to derive a large-sample \(100(1-\alpha) \%\) CI for \(\theta\), and compute the 95% interval for the given data.

It is important that face masks used by firefighters be able to withstand high temperatures because firefighters commonly work in temperatures of 200500F. In a test of one type of mask, 11 of 55 masks had lenses pop out at 250. Construct a 90% upper confidence bound for the true proportion of masks of this type whose lenses would pop out at 250.

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate the average force required to break the binding to within .1 lb with 95% confidence? Assume that s is known to be .8.

The accompanying data on crack initiation depth (mm) was read from a lognormal probability plot that appeared in the article Incorporating Small Fatigue Crack Growth in Probabilistic Life Prediction: Effect of Stress Ratio in Ti-6Al-2Sn-6Mo (Intl. J. of Fatigue, 2013: 8395). Although the pattern in the plot was quite straight, a normal probability plot of the data also shows a reasonably linear pattern. And a boxplot indicates that the distribution is quite symmetric in the middle 50% of the data and only mildly skewed overall. It is therefore reasonable to estimate and predict using t intervals. 4.7 5.1 5.2 5.3 5.6 5.8 6.3 6.7 7.2 7.4 7.7 8.5 8.9 9.3 10.1 11.2 a. Estimate the true average crack initiation depth with a 99% CI and interpret the resulting interval. b. Predict the value of a single crack initiation depth by constructing a 99% PI. c. Interpret in context the meaning of 99% in (b).

In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter \(\gamma\). Let Y1 denote the time at which the first failure occurs, Y2 the time at which the second failure occurs, and so on, so that \(T_{r}=Y_{1}+\cdots+Y_{r}+(n-r) Y_{r}\) is the total accumulated lifetime at termination.Then it can be shown that 2lTr has a chi-square distribution with 2r df. Use this fact to develop a100(1 - \(\alpha\))% CI formula for true average lifetime 1/\(\lambda\).Compute a 95% CI from the data in Example 6.8.

Let X1, X2,, Xn be a random sample from a continuous probability distribution having median m , (so that P(Xi # m ,) 5 P(Xi $ ,m) 5 .5). a. Show that P(min (Xi ) , m , , max (Xi )) 5 1 2 1 1 22 n21 so that (min(xi ), max(xi )) is a 100(1 2 a)% confidence interval for ,m with a 5 ( 1 2 ) n21 . [Hint: The complement of the event {min (Xi ) , m , , max (Xi )} is {max (Xi ) # m ,} {min (Xi ) $ m ,}. But max (Xi ) # m , iff Xi # ,m for all i.] b. For each of six normal male infants, the amount of the amino acid alanine (mg/100 mL) was determined while the infants were on an isoleucine-free diet, resulting in the following data: 2.84 3.54 2.80 1.44 2.94 2.70 Compute a 97% CI for the true median amount of alanine for infants on such a diet (The Essential Amino Acid Requirements of Infants, Amer. J. of Nutrition, 1964: 322330). c. Let x(2) denote the second smallest of the xi s and x(n21) denote the second largest of the xi s. What is the confidence level of the interval (x(2), x(n21)) for m ,?

Let X1, X2,, Xn be a random sample from a uniform distribution on the interval [0, u], so that f(x) 5 5 1 u 0 # x # u 0 otherwise Then if Y 5 max (Xi ), it can be shown that the rv U 5 Yyu has density function fU(u) 5 5 nun21 0 # u # 1 0 otherwise a. Use fU(u) to verify that P1(ay2)1yn , Y u # (1 2 ay2)1yn 2 5 1 2 a and use this to derive a 100(1 2 a)% CI for u. b. Verify that P(a1yn # Yyu # 1) 5 1 2 a, and derive a 100(1 2 a)% CI for u based on this probability statement. c. Which of the two intervals derived previously is shorter? If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1 5 4.2, x2 5 3.5, x3 5 1.7, x4 5 1.2, and x5 5 2.4, derive a 95%CI for u by using the shorter of the two intervals

Let 0 # g # a. Then a 100(1 2 a)% CI for m when n is large is 1x 2 zg ? s n , x 1 za2g ? s n2 The choice g 5 ay2 yields the usual interval derived in Section 7.2; if g ay2, this interval is not symmetric about x#. The width of this interval is w 5 s(zg 1 za2g)yn. Show that w is minimized for the choice g 5 ay2, so that the symmetric interval is the shortest. [Hints: (a) By definition of za, F(za) 5 1 2 a, so that za 5 F21 (1 2 a); (b) the relationship between the derivative of a function y 5 f(x) and the inverse function x 5 f 21 (y) is (dydy) f 21 (y) 5 1yf9(x).]

Suppose \(x_1, x_2, \ldots, x_n\) are observed values resulting from a random sample from a symmetric but possibly heavy-tailed distribution. Let \(\tilde{x}\) and \(f_{\mathrm{s}}\) denote the sample median and fourth spread, respectively. Chapter 11 of Understanding Robust and Exploratory Data Analysis (see the bibliography in Chapter 6) suggests the following robust 95 % CI for the population mean (point of symmetry): \(\tilde{x} \pm\left(\frac{\text { conservative } t \text { critical value }}{1.075}\right) \cdot \frac{f_s}{\sqrt{n}}\) The value of the quantity in parentheses is 2.10 for n = 10,1.94 for n = 20, and 1.91 for n=30. Compute this CI for the data of Exercise 45, and compare to the t CI appropriate for a normal population distribution.

a. Use the results of Example 7.5 to obtain a 95% lower confidence bound for the parameter \(\lambda\) of an exponential distribution, and calculate the bound based on the data given in the example. b. If lifetime X has an exponential distribution, the probability that lifetime exceeds t is \(P(X > t) = e^{- \lambda t}\). Use the result of part (a) to obtain a 95% lower confidence bound for the probability that breakdown time exceeds 100 min.