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statistical sampling to auditing

Principles Of Statistics For Engineers And Scientists 2nd International Edition William Navidi - Solutions

Five measurements are taken of the octane rating for a particular type of gasoline. The results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3. These data also appear in Exercise 4 in Section 5.4.a. Find a 95% prediction interval for a single measurement.b. Find a tolerance interval for the pH that
Six measurements were made of the concentration (in percent) of ash in a certain variety of spinach. The sample mean was 19.35, and the sample standard deviation was 0.577. Assume that the concentrations are normally distributed.a. Find a 90% prediction interval for a single measurement.b. Find a
The article “Ozone for Removal of Acute Toxicity from Logyard Run-off” (M. Zenaitis and S.Duff, Ozone Science and Engineering, 2002:83–90)presents chemical analyses of runoff water from sawmills in British Columbia. Included were measurements of pH for six water specimens: 5.9, 5.0, 6.5, 5.6,
In a sample of 20 bolts, the average breaking torque was 89.7 J with a standard deviation of 8.2 J. Assume that the breaking torques are normally distributed.a. Find a 99% prediction interval for the breaking torque of a single bolt.b. Find a tolerance interval for the breaking torque that includes
A sample of 25 resistors, each labeled 100 Ω, had an average resistance of 101.4 Ω with a standard deviation of 2.3 Ω. Assume the resistances are normally distributed.a. Find a 95% prediction interval for the resistance of a single resistor.b. Find a tolerance interval for the resistance that
The article “Arsenic and Mercury in Lake Whitefish and Burbot Near the Abandoned Giant Mine on Great Slave Lake” (P. Cott, B. Zajdlik, et al., Journal of Great Lakes Research, 2016:223–232) presents measurements of arsenic concentrations in fish found in Northern Canada.a. In a sample of 8
The concentration of carbon monoxide (CO) in a gas sample is measured by a spectrophotometer and found to be 85 ppm. Through long experience with this instrument, it is believed that its measurements are unbiased and normally distributed, with a standard deviation of 8 ppm. Find a 95% confidence
The following MINITAB output presents a confidence interval for a population mean, but some of the numbers got smudged and are now illegible. Fill in the missing numbers. One Sample T: X Variable N X Mean 20 2.39374 St Dev SE Mean (a) 99% CI 0.52640 ((b), (c) )
The following MINITAB output presents a confidence interval for a population mean.a. How many degrees of freedom does the Student’s t distribution have?b. Use the information in the output, along with the t table, to compute a 99% confidence interval. One-Sample T: X Variable N X StDev SE Mean
Ten samples of coal from a Northern Appalachian source had an average mercury content of 0.242 ppm with a standard deviation of 0.031 ppm. Find a 95%confidence for the mean mercury content of coal from this source.
The article “Influence of Penetration Rate on Penetrometer Resistance” (J. Oliveira, M. Almeida, et al., Journal of Geotechnical and Geoenvironmental Engineering, 2011:695–703) presents measures of penetration resistance for a certain fine-grained soil.Fifteen measurements, expressed as a
The article “An Automatic Visual System for Marble Tile Classification” (L. Carrino, W. Polini, and S. Turchetta, Journal of Engineering Manufacture, 2002:1095–1108) describes a measure for the shade of marble tile in which the amount of light reflected by the tile is measured on a scale of
The following are summary statistics for a data set.Would it be appropriate to use the Student’s t distribution to construct a confidence interval from these data? Explain.N Mean Median StDev 10 8.905 6.105 9.690 Minimum Maximum Q1 Q3 0.512 39.920 1.967 8.103
The article “Wind-Uplift Capacity of Residential Wood Roof-Sheathing Panels Retrofitted with Insulating Foam Adhesive” (P. Datin, D. Prevatt, and W. Pang, Journal of Architectural Engineering, 2011:144–154) presents a study of the failure pressures of roof panels. Following are the failure
True or false: The Student’st distribution may be used to construct a confidence interval for the mean of any population, so long as the sample size is small.
The article “Ozone for Removal of Acute Toxicity from Logyard Run-off” (M. Zenaitis and S. Duff, Ozone Science and Engineering, 2002:83–90)presents chemical analyses of runoff water from sawmills in British Columbia. Included were measurements of pH for six water specimens: 5.9, 5.0, 6.5,
A chemist made six independent measurements of the sublimation point of carbon dioxide (the temperature at which it changes to dry ice). She obtained a sample mean of 196.64 K with a standard deviation of 0.68 K.a. Use the Student’s t distribution to find a 95%confidence interval for the
A model of heat transfer from a cylinder immersed in a liquid predicts that the heat transfer coefficient for the cylinder will become constant at very low flow rates of the fluid. A sample of 10 measurements is taken. The results, in W/m2 K, are 13.7 12.0 13.1 14.1 13.1 14.1 14.4 12.2 11.9 11.8
Five measurements are taken of the octane rating for a particular type of gasoline. The results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3. Find a 99% confidence interval for the mean octane rating for this type of gasoline.
Find the level of a two-sided confidence interval that is based on the given value of tn−1,????∕2 and the given sample size.a. t = 2.179, sample size 13.b. t = 3.365, sample size 6.c. t = 1.729, sample size 20.d. t = 3.707, sample size 7.e. t = 3.707, sample size 27.
Find the value of tn−1,???? needed to construct an upper or lower confidence bound in each of the situations in Exercise 1.
Find the value of tn−1,????∕2 needed to construct a twosided confidence interval of the given level with the given sample size:a. Level 90%, sample size 9.b. Level 95%, sample size 5.c. Level 99%, sample size 29.d. Level 95%, sample size 2.
A stock market analyst notices that in a certain year, the price of IBM stock increased on 131 out of 252 trading days. Can these data be used to find a 95% confidence interval for the proportion of days that IBM stock increases? Explain.
The article “A Music Key Detection Method Based on Pitch Class Distribution Theory” (J. Sun, H. Li, and L. Ma, International Journal of Knowledge-based and Intelligent Engineering Systems, 2011:165–175)describes a method of analyzing digital music files to determine the key in which the music
Stainless steels can be susceptible to stress corrosion cracking under certain conditions. A materials engineer is interested in determining the proportion of steel alloy failures that are due to stress corrosion cracking.a. In the absence of preliminary data, how large a sample must be taken so as
A sociologist is interested in surveying workers in computer-related jobs to estimate the proportion of such workers who have changed jobs within the past year.a. In the absence of preliminary data, how large a sample must be taken to ensure that a 95% confidence interval will specify the
A voltmeter is used to record 100 independent measurements of a known standard voltage. Of the 100 measurements, 85 are within 0.01 V of the true voltage.a. Find a 95% confidence interval for the probability that a measurement is within 0.01 V of the true voltage.b. Find a 98% confidence interval
When the light turns yellow, should you stop or go through it? The article “Evaluation of Driver Behavior in Type II Dilemma Zones at High-Speed Signalized Intersections” (D. Hurwitz, M. Knodler, and B. Nyquist, Journal of Transportation Engineering, 2011:277–286) defines the “indecision
Refer to Exercise 4. Find a 99% lower confidence bound for the proportion of smokers who have tried to quit smoking in the past year.
Refer to Exercise 2. Find a 98% upper confidence bound for the proportion of residences that reduced their water consumption.
Refer to Exercise 1. Find a 95% lower confidence bound for the proportion of automobiles whose emissions exceed the standard.
High levels of blood sugar are associated with an increased risk of type 2 diabetes. A level higher than normal is referred to as “impaired fasting glucose.”The article “Association of Low-Moderate Arsenic Exposure and Arsenic Metabolism with Incident Diabetes and Insulin Resistance in the
The National Center for Health Statistics interviewed 5409 adult smokers in 2015, and 2636 of them said they had tried to quit smoking during the past year.Consider this to be a random sample.a. Find a 95% confidence interval for the proportion of smokers who have tried to quit within the past
A soft-drink manufacturer purchases aluminum cans from an outside vendor. A random sample of 70 cans is selected from a large shipment, and each is tested for strength by applying an increasing load to the side of the can until it punctures. Of the 70 cans, 52 meet the specification for puncture
During a recent drought, a water utility in a certain town sampled 100 residential water bills and found that 73 of the residences had reduced their water consumption over that of the previous year.a. Find a 95% confidence interval for the proportion of residences that reduced their water
In a simple random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard.a. What proportion of the automobiles in the sample had emission levels that exceed the standard?b. Find a 95% confidence interval for the proportion
Refer to Exercise 13,a. Find a 95% upper confidence bound for the mean sugar content.b. The claim is made that the mean sugar content is greater than 11.7 g. With what level of confidence can this statement be made?
Refer to Exercise 12,a. Find a 98% lower confidence bound for the mean efficiency.b. The claim is made that the mean efficiency is greater than 84.6%. With what level of confidence can this statement be made?
Refer to Exercise 11,a. Find a 90% upper confidence interval for the mean lifetime.b. Someone says that the mean lifetime is less than 1220 hours. With what level of confidence can this statement be made?
Refer to Exercise 10.a. Find a 99% upper confidence bound for the mean temperature.b. The claim is made that the mean temperature is less than 349.5◦F. With what level of confidence can this statement be made?
Refer to Exercise 9.a. Find a 95% lower confidence interval for the mean mass.b. An engineer claims that the mean mass is greater than 38.25 kg. With what level of confidence can this statement be made?
Refer to Exercise 8.a. Find a 98% lower confidence bound for the mean time.b. Someone says that the mean time is greater than 134.3 minutes. With what level of confidence can this statement be made?
Refer to Exercise 7.a. Find a 95% lower confidence interval for the mean wall thickness.b. Someone says that the mean thickness is less than 8.2 mm. With what level of confidence can this statement be made?
The sugar content in a one-cup serving of a certain breakfast cereal was measured for a sample of 140 servings. The average was 11.9 g and the standard deviation was 1.1 g.a. Find a 95% confidence interval for the mean sugar content.b. Find a 99% confidence interval for the mean sugar content.c.
In a sample of 60 electric motors, the average efficiency (in percent) was 85 and the standard deviation was 2.a. Find a 95% confidence interval for the mean efficiency.b. Find a 99.5% confidence interval for the mean efficiency.c. What is the confidence level of the interval(84.63, 85.37)?d. How
In a sample of 80 light bulbs, the mean lifetime was 1217 hours with a standard deviation of 52 hours.a. Find a 95% confidence interval for the mean lifetime of this type of light bulb.b. Find a 98% confidence interval for the mean lifetime of this type of light bulb.c. An engineer claims that the
Oven thermostats were tested by setting them to 350◦F and measuring the actual temperature of the oven. In a sample of 67 thermostats, the average temperature was 348.2◦F and the standard deviation was 5.1◦F.a. Find a 90% confidence interval for the mean oven temperature.b. Find a 95%
A sample of 75 concrete blocks had a mean mass of 38.3 kg with a standard deviation of 0.6 kg.a. Find a 95% confidence interval for the mean mass of this type of concrete block.b. Find a 99% confidence interval for the mean mass of this type of concrete block.c. An engineer claims that the mean
The article “Application of Surgical Navigation to Total Hip Arthroplasty” (T. Ecker and S. Murphy, Journal of Engineering in Medicine, 2007: 699–712)reports that in a sample of 123 hip surgeries of a certain type, the average surgery time was 136.9 minutes with a standard deviation of 22.6
In a sample of 100 steel canisters, the mean wall thickness was 8.1 mm with a standard deviation of 0.5 mm.a. Find a 95% confidence interval for the mean wall thickness of this type of canister.b. Find a 99% confidence interval for the mean wall thickness of this type of canister.c. An engineer
The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations”(Z. Li, Journal of Transportation Engineering 2011:445–454) presents a new method for timing traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in
A sample of 114 patients were given a drug to lower cholesterol. A 95% confidence interval for the mean reduction in cholesterol (in mmol/L) was (0.88, 1.02).a. What was the sample mean reduction?b. What was the sample standard deviation of the reduction amounts?
Three confidence intervals for a population mean are constructed, all from the same random sample.The confidence levels are 95%, 98%, and 99.9%. The confidence intervals are (6.4, 12.3), (5.1, 13.6), and(6.8, 11.9). Match each confidence interval to its level.
Find the levels of the confidence intervals that have the following values of z????∕2.a. z????∕2 = 1.96b. z????∕2 = 2.81c. z????∕2 = 2.17d. z????∕2 = 2.33
Find the value of z????∕2 to use in expression (5.4) to construct a confidence interval with levela. 90%b. 83%c. 99.5%d. 75%
Let X1,… , Xn be a simple random sample from a N(????, ????2) population. For any constant k > 0, define????̂ 2 k =∑n i=1(Xi − X)2 k. Consider ????̂ 2 k as an estimator of ????2.a. Compute the bias of ????̂ 2 k in terms of k. [Hint:The sample variance s2 is unbiased, and ????̂ 2 k =(n −
Refer to Exercise 3. For what values of ???? does ????̂3 have smaller mean squared error than ????̂2?
Refer to Exercise 3. For what values of ???? does ????̂3 have smaller mean squared error than ????̂1?
Let X1 and X2 be independent, each with unknown mean ???? and known variance ????2 = 1.a. Let ????̂1 = X1 + X2 2 . Find the bias, variance, and mean squared error of ????̂1.b. Let ????̂2 = X1 + 2X2 3 . Find the bias, variance, and mean squared error of ????̂2.c. Let ????̂3 = X1 + X2 4 . Find
Choose the best answer to fill in the blank. The variance of an estimator measures i. how close the estimator is to the true value.ii. how close repeated values of the estimator are to each other.iii. how close the mean of the estimator is to the true value.iv. how close repeated values of the mean
Choose the best answer to fill in the blank. If an estimator is unbiased, then i. the estimator is equal to the true value.ii. the estimator is usually close to the true value.iii. the mean of the estimator is equal to the true value.iv. the mean of the estimator is usually close to the true value.
Someone claims that the number of hits on his website has a Poisson distribution with mean 20 per hour.Let X be the number of hits in five hours.a. If the claim is true, what is P(X ≤ 95)?b. Based on the answer to part (a), if the claim is true, is 95 hits in a five-hour time period an unusually
A cereal manufacturer claims that the gross weight(including packaging) of a box of cereal labeled as weighing 12 oz has a mean of 12.2 oz and a standard deviation of 0.1 oz. You gather 75 boxes and weigh them all together. Let S denote the total weight of the 75 boxes of cereal.a. If the claim is
The length of time to perform an oil change at a certain shop is normally distributed with mean 29.5 minutes and standard deviation 3 minutes. What is the probability that a mechanic can complete 16 oil changes in an eight-hour day?
The lifetime of a bearing (in years) follows the Weibull distribution with parameters ???? = 1.5 and ???? = 0.8.a. What is the probability that a bearing lasts more than 1 year?b. What is the probability that a bearing lasts less than 2 years?
A process that polishes a mirrored surface leaves an average of 2 small flaws per 5 m2 of surface. The number of flaws on an area of surface follows a Poisson distribution.a. What is the probability that a surface with area 3 m × 5 m will contain more than 5 flaws?b. What is the probability that a
In a certain process, the probability of producing a defective component is 0.07.a. In a sample of 250 randomly chosen components, what is the probability that fewer than 20 of them are defective?b. In a sample of 10 randomly chosen components, what is the probability that one or more of them is
The stiffness of a certain type of steel beam used in building construction has mean 30 kN/mm and standard deviation 2 kN/mm.a. Is it possible to compute the probability that the stiffness of a randomly chosen beam is greater than 32 kN/mm? If so, compute the probability. If not, explain why not.b.
The intake valve clearances on new engines of a certain type are normally distributed with mean 200 ????m and standard deviation 10 ????m.a. What is the probability that the clearance is greater than 215 ????m?b. What is the probability that the clearance is between 180 and 205 ????m?c. An engine
A certain type of plywood consists of five layers.The thicknesses of the layers are independent and normally distributed with mean 5 mm and standard deviation 0.2 mm.a. Find the mean thickness of the plywood.b. Find the standard deviation of the thickness of the plywood.c. Find the probability that
Medication used to treat a certain condition is administered by syringe. The target dose in a particular application is ????. Because of the variations in the syringe, in reading the scale, and in mixing the fluid suspension, the actual dose administered is normally distributed with mean ???? and
In the article “Occurrence and Distribution of Ammonium in Iowa Groundwater” (K. Schilling, Water Environment Research, 2002:177–186), ammonium concentrations (in mg/L) were measured at a large number of wells in Iowa. The mean concentration was 0.71, the median was 0.22, and the standard
The Environmental Protection Agency (EPA) has contracted with your company for equipment to monitor water quality for several lakes in your water district. A total of 10 devices will be used. Assume that each device has a probability of 0.01 of failure during the course of the monitoring period.a.
A simple random sample X1,…, Xn is drawn from a population, and the quantities ln X1,…, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false:a. X1,…, Xn come from a population that is approximately lognormal.b. X1,…, Xn come from a
Pea plants contain two genes for seed color, each of which may be Y (for yellow seeds) or G (for green seeds). Plants that contain one of each type of gene are called heterozygous. According to the Mendelian theory of genetics, if two heterozygous plants are crossed, each of their offspring will
The number of large cracks in a length of pavement along a certain street has a Poisson distribution with a mean of 1 crack per 100 m.a. What is the probability that there will be exactly 8 cracks in a 500 m length of pavement?b. What is the probability that there will be no cracks in a 100 m
An airplane has 100 seats for passengers. Assume that the probability that a person holding a ticket appears for the flight is 0.90. If the airline sells 105 tickets, what is the probability that everyone who appears for the flight will get a seat?
A new process has been designed to make ceramic tiles. The goal is to have no more than 5% of the tiles be nonconforming due to surface defects. A random sample of 1000 tiles is inspected. Let X be the number of nonconforming tiles in the sample.a. If 5% of the tiles produced are nonconforming,
A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and a standard deviation of 5 hours. Let X represent the mean lifetime of the batteries in a simple random sample of size 100.a. If the claim is true, what is P(X ≤ 36.7)?b. Based on the
The concentration of particles in a suspension is 50 per mL. A 5-mL volume of the suspension is withdrawn.a. What is the probability that the number of particles withdrawn will be between 235 and 265?b. What is the probability that the average number of particles per mL in the withdrawn sample is
The concentration of particles in a suspension is 30 per mL.a. What is the probability that a 2-mL sample will contain more than 50 particles?b. Ten 2-mL samples are drawn. What is the probability that at least 9 of them contain more than 50 particles?c. One hundred 2-mL samples are drawn. What is
The number of particles emitted by a radioactive mass during a given period of time follows a Poisson distribution. Radioactive mass A emits particles at a mean rate of 20 per minute, and radioactive mass B emits particles at a mean rate of 25 per minute.a. What is the probability that fewer than
Concrete blocks are produced in lots of 2000. Each block has probability 0.85 of meeting a strength specification. The blocks are independent.a. What is the probability that, in a given lot, fewer than 1690 blocks meet the specification?b. Find the 70th percentile of the number of blocks that meet
A commuter encounters four traffic lights each day on her way to work. Let X represent the number of these that are red lights. The probability mass function of X is as follows.x 01234 P(X = x) 0.1 0.3 0.3 0.2 0.1 What is the probability that in a period of 100 days, the average number of red
The time taken by an automobile mechanic to complete an oil change is random, with mean 29.5 minutes and standard deviation 3 minutes.a. What is the probability that 50 oil changes take more than 1500 minutes?b. What is the probability that a mechanic can complete 80 or more oil changes in 40
The temperature of a solution will be estimated by taking n independent readings and averaging them.Each reading is unbiased, with a standard deviation of ???? = 0.5◦C. How many readings must be taken so that the probability is 0.90 that the average is within±0.1◦C of the actual temperature?
The amount of warpage in a type of wafer used in the manufacture of integrated circuits has mean 1.3 mm and standard deviation 0.1 mm. A random sample of 200 wafers is drawn.a. What is the probability that the sample mean warpage exceeds 1.305 mm?b. Find the 25th percentile of the sample mean.c.
Bags checked for a certain airline flight have a mean weight of 15 kg with a standard deviation of 5 kg. A random sample of 60 bags is drawn.a. What is the probability that the sample mean weight is less than 14 kg?b. Find the 70th percentile of the sample mean weights.c. How many bags must be
In a process that manufactures bearings, 90% of the bearings meet a thickness specification. In a sample of 500 bearings, what is the probability that more than 440 meet the specification?
A 500-page book contains 250 sheets of paper. The thickness of the paper used to manufacture the book has mean 0.08 mm and standard deviation 0.01 mm.a. What is the probability that a randomly chosen book is more than 20.2 mm thick (not including the covers)?b. What is the 10th percentile of book
The time spent by a customer at a checkout counter has mean 4 minutes and standard deviation 2 minutes.a. What is the probability that the total time taken by a random sample of 50 customers is less than 180 minutes?b. Find the 30th percentile of the total time taken by 50 customers.
Among all monthly bills from a certain credit card company, the mean amount billed was $485 and the standard deviation was $300. In addition, for 15% of the bills, the amount billed was greater than $1000. A sample of 900 bills is drawn.a. What is the probability that the average amount billed on
Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact, the fill volume is random, with mean 12.01 oz and standard deviation 0.2 oz.a. What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz?b. If the population mean fill
Consider the following data set:0.64 0.29 −0.38 0.49 0.81 −0.83 10.04 0.79 0.16 0.20 −0.78 0.19 0.54 1.53 0.55 2.26 1.77 0.34 0.35 0.35 0.57 −0.94 0.55 0.31 0.67 0.43 0.54 3.39 0.84 −2.32 0.48 −1.03−1.17 −2.15 −0.27 0.34 0.81 −2.62 0.36 −0.55a. Construct a normal probability
Consider the following data set:150 82 15 12 27 33 98 33 75 1 30 17 3 66 55 45 7 99 58 32 44 34 2 33 17 13 10 25 0 90 66 43 42 18 93 38 20 113 35 23a. Construct a normal probability plot for these data.Do the data appear to come from an approximately normal distribution?b. Transform the data by
In the article “Assessment of Dermatopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products” (L. Pershing, J. Nelson, et al., Journal of the American Academy of Dermatology, 2003:740–751), measurements of the concentration of an anti-fungal gel, in ng
Construct a normal probability plot for the logs of the PM data in Table 1.2. Do the logs of the PM data appear to come from a normal population?
Construct a normal probability plot for the PM data in Table 1.2 (page 18). Do the PM data appear to come from a normal population?
Below are the durations (in minutes) of 40 time intervals between eruptions of the geyser Old Faithful in Yellowstone National Park.91 51 79 53 82 51 76 82 84 53 86 51 85 45 88 51 80 49 82 75 73 67 68 86 72 75 75 66 84 70 79 60 86 71 67 81 76 83 76 55 Construct a normal probability plot for these
Below are the durations (in minutes) of 40 eruptions of the geyser Old Faithful in Yellowstone National Park.4.1 1.8 3.2 1.9 4.6 2.0 4.5 3.9 4.3 2.3 3.8 1.9 4.6 1.8 4.7 1.8 4.6 1.9 3.5 4.0 3.7 3.7 4.3 3.6 3.8 3.8 3.8 2.5 4.5 4.1 3.7 3.8 3.4 4.0 2.3 4.4 4.1 4.3 3.3 2.0 Construct a normal probability

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