Questions and Answers of Statistics For Engineers And Scientists

Let X ∼ U(a, b) a. Show that if $x \leq a$ then $P(X \leq x)=0$.b. Show that if $x>b$ then $P(X \leq x)=1$.c. Show that if \(a F(x) = 0 x-a b-a 1 x a a
The article "Gypsum Effect on the Aggregate Size and Geometry of Three Sodic Soils Under Reclamation" (I. Lebron, D. Suarez, and T. Yoshida, Journal of the Soil Science Society of America,
Repeat Exercise 3, using the $S$ chart in place of the $R$ chart.Data From Exercise 3:The thickness, in mm, of metal washers is measured on samples of size 5. The following table presents the
Repeat Exercise 8, using the $S$ chart in place of the $R$ chart.Data From Exercise 8:Samples of eight bolts are taken periodically, and their diameters (in $\mathrm{mm}$ ) are measured. The
Repeat Exercise 10, using the $S$ chart in place of the $R$ chart.Data From Exercise 10:A certain type of integrated circuit is connected to its frame by five wires. Thirty samples of five units
Repeat Exercise 12, using the SS chart in place of the RR chart.Data From Exercise 12:Copper wires are coated with a thin plastic coating. Samples of four wires are taken every hour, and the
The following table presents the average annual energy expenditures (in dollars) for housing units of various sizes (in square feet).a. Compute the least-squares line for predicting energy
The following table presents the mean cost of a hospital stay, in $\$ 1000$ s, and the number of hospital stays, in millions, in the United States for each of 15 years.a. Compute the least-squares
The following table, based on data from the credit reporting company Experian, presents the average number of late payments and the average credit score for residents of selected U.S. cities in
The following output (from $\mathrm{R}$ ) is for a multiple regression. Something went wrong with the printer, so some of the numbers are missing. Fill in the missing numbers. Coefficients:
The following output (from $\mathrm{R}$ ) is for a multiple regression. Some of the numbers got smudged and are illegible. Fill in the missing numbers. Coefficients: (Intercept) X1 X2 X3 Estimate
The article "Two Different Approaches for RDC Modelling When Simulating a Solvent Deasphalting Plant" (J. Aparicio, M. Heronimo, et al., Computers and Chemical Engineering, 2002:1369-1377) reports
The article "Experimental and Statistical Investigation of Self-Consolidating Concrete Mixture Constituents for Prestressed Bridge Girder Fabrication" (E. Torres, J. Seo, and R. Lederle, Journal of
The article "Characterization of Effects of Thermal Property of Aggregate on the Carbon Footprint of Asphalt Industries in China" (A. Jamshidi, K. Kkurumisawa, et al., Journal of Traffic and
Refer to the data set pit.csv. Consider depths of pits at all durations at $40 \%$ relative humidity. The question is whether the mean depth differs among the durations.a. Construct an ANOVA table.
Repeat Exercise 21, using the values at $75 \%$ relative humidity.Data From Exercise 21:Refer to the data set pit.csv. Consider depths of pits at all durations at 40%40% relative humidity. The
It is desired to determine whether the speed of the machine is affected by the soil type.a. Construct an ANOVA table. You may give a range for the $P$-value.b. Can you conclude that there are
It is desired to determine whether the torque of the cutting wheel is affected by the soil type.a. Construct an ANOVA table. You may give a range for the $P$-value.b. Can you conclude that there
A geologist is making repeated measurements (in grams) on the mass of a rock. It is not known whether the measurements are a random sample from an approximately normal population. Following are three
A new centrifugal pump is being considered for an application involving the pumping of ammonia. The specification is that the flow rate be more than $5 \mathrm{gal}-$ lons per minute (gpm). In an
The article "Influence of Penetration Rate on Penetrometer Resistance" (G. Gagnon and J. Doubrough, Canadian Journal of Civil Engineering, 2011: 741-750) describes a study in which 20 two-liter
Specifications call for the wall thickness of two-liter polycarbonate bottles to average 4.0 mils. A quality control engineer samples 7 two-liter polycarbonate bottles from a large batch and measures
Environmental Protection Agency standards require that the amount of lead in drinking water be less than $15 \mathrm{ppb}$. Twelve samples of water from a particular source have the following
The article "Approximate Methods for Estimating Hysteretic Energy Demand on Plan-Asymmetric Buildings" (M. Rathhore, A. Chowdhury, and S. Ghosh, Journal of Earthquake Engineering, 2011: 99-123)
Measurements were made of arsenic concentrations in fish found in Northern Canada. In a sample of 8 whitefish caught in Yellowknife Bay, the mean arsenic concentration in the liver was \(0.32
Eight specimens of Incoloy 825 steel were immersed in acidified brine for four hours. The number of corrosive pits was counted on each sample. The results wereCan you conclude that the mean number of
The article "Evaluation of Mobile Mapping Systems for Roadway Data Collection" (H. Karimi, A. Khattak, and J. Hummer, Journal of Computing in Civil Engineering, 2000:168-173) describes a system for
An inspector measured the fill volume of a simple random sample of 100 cans of juice that were labeled as containing 12 ounces. The sample had mean volume 11.98 ounces and standard deviation 0.19
In a process that manufactures tungsten-coated silicon wafers, the target resistance for a wafer is $85 \mathrm{~m} \Omega$. In a simple random sample of 50 wafers, the sample mean resistance was
An ultra-low particulate air filter is used to maintain uniform air flow in production areas in a clean room. In one such room, the mean air velocity should be at least \(40 \mathrm{~cm} /
The following output (from $\mathrm{R}$ ) presents the results of a hypothesis test for a population mean $\mu$.a. Is this a one-tailed or a two-tailed test?b. What is the null hypothesis?c. Can
The following output (from $\mathrm{R}$ ) presents the results of a hypothesis test for a population mean $\mu$.a. Is this a one-tailed or a two-tailed test?b. What is the null hypothesis?c. Can
If a record high is equally likely to occur in any year of observation, the mean year in which a record is observed would occur at the midpoint of the observation period, which is \((1871+2020) /
Because of the occurrence of hot spells and cold spells, it is common for records to be set on two or more consecutive days in the same year. For example, five consecutive record highs, from June 2
For some records, the record temperature occurred more than once. In these cases, only the most recent year is listed. Explain how this might cause the mean to be greater than the midpoint of 1945.5
Drop the dates in which the record high occurred more than once, and test the hypothesis in Exercise 22 again. Does your conclusion change?Data From Exercise 22:If a record high is equally likely to
Perform a hypothesis test on the record lows, after dropping dates on which the record low occurred more than once, in which the alternative hypothesis is that the mean year is less than 1945.5. What
The choice of January 1 as a starting date is arbitrary. The results might be different if we start on a different date. For each of the days January 2 through January 7, construct a data set
The anterior cruciate ligament (ACL) runs diagonally in the middle of the knee. The article "Return to Sport After Pediatric Anterior Cruciate Ligament Reconstruction and Its Effect on Subsequent
The article "Developing a Tool to Measure the Factors Influencing Nurses' Enjoyment of Nursing" (L. Wilkes, M. Doull, et al., Journal of Clinical Nursing, 2016:1854-1860) reports that in a sample of
A sociologist sampled 200 people who work in computer-related jobs and found that 42 of them have changed jobs in the past six months. Can you conclude that more than $15 \%$ of computer-related
The General Social Survey asked a sample of 1294 people whether they performed any volunteer work during the past year. A total of 517 people said they did. Can you conclude that more than $35 \%$
Because of the occurrence of hot spells and cold spells, it is common for records to be set on two or more consecutive days in the same year. For example, five consecutive record highs, from June 2
The data set contains observations from the years 1871 through 2020 , so the midpoint is 1945.5 . For some records, the record temperature occurred more than once. In these cases, only the most
Drop the dates in which the record high occurred more than once, and perform a test where the alternative hypothesis is that the proportion of record highs occurring after 1945 is greater than 0.5 .
Drop the dates in which the record low occurred more than once, and perform a test where the alternative hypothesis is that the proportion of record lows occurring after 1945 is less than 0.5 . What
The choice of January 1 as a starting date is arbitrary. The results might be different if we start on a different date. For each of the days January 2 through January 7, construct a data set
The article "Utility of Pretest Probability and Exercise Treadmill Test in Korean Women with Suspected Coronary Artery Disease" (Y. Kim, W. Shim, et al., Journal of Women's Health, 2015:617-622)
Do you prefer taking tests on paper or online? A college instructor gave identical tests to two randomly sampled groups of 50 students. One group took the test on paper and the other took it online.
The article "Treadmill Exercise and Resistance Training in Patients With Peripheral Arterial Disease With and Without Intermittent Claudication. A Randomized Controlled Trial" (M. McDermott, P. Ades,
To determine the effect of fuel grade on fuel efficiency, 80 new cars of the same make, with identical engines, were each driven for 1000 miles. Forty of the cars ran on regular fuel and the other 40
King Tut was an ancient Egyptian ruler whose tomb was discovered and opened in 1923. Legend has it that the archaeologists who opened the tomb were subject to a "mummy's curse," which would shorten
The article "Predicting Depressed Patients with Suicidal Ideation from ECG Recordings" (A. Khandoker, V. Luthra, et al., Med Biol Eng Comput, 2017: 793-805) reports a study in which systolic blood
In a study to compare the effectiveness of distance learning with traditional classroom instruction, 12 students took a business course online while 14 students took it in a classroom. The final exam
The following output (from $\mathrm{R}$ ) presents the results of a hypothesis test for the difference between two means:a. Is this a one-tailed or a two-tailed test?b. What is the null
The following output (from $\mathrm{R}$ ) presents the results of a hypothesis test for the difference between two means:a. Is this a one-tailed or a two-tailed test?b. What is the null
The 90th percentile of ozone is 43.535. Can you conclude that people whose exposure to ozone is in the top $10 \%$ have lower mean FEV1 than those in the lower $90 \%$?
The 90th percentile of PM10 is 63.43. Can you conclude that people whose exposure to PM10 is in the top $10 \%$ have lower mean FEV1 than those in the lower $90 \%$?
The 90th percentile of $\mathrm{NO}_{2}$ is 44.93. Can you conclude that people whose exposure to $\mathrm{NO}_{2}$ is in the top $10 \%$ have lower mean FEV1 than those in the lower $90 \%$?
Compute the mean FEV1 for those with asthma and those without. What is counterintuitive about the result?
Perform a test of the null hypothesis that the mean FEV1 for those with asthma is less than or equal to the mean FEV1 for those without, against the alternative that the mean is greater. Taking the
The median age for people in the study is 10.82 years. What proportion of those older than 10.82 years have asthma? What proportion of those 10.82 years or younger have asthma? Do these proportions
The median age for people in the study is 10.82 years. Using only those individuals over the age of 10.82 , perform a test of the null hypothesis that the mean FEV1 for those with asthma is equal to
Refer to Exercise 38. Repeat this test, using only those 10.82 years old or younger.Data From Exercise 38:The median age for people in the study is 10.82 years. Using only those individuals over the
Using the results of Exercises 38 and 39, can you conclude that the mean FEV1 differs between those who have asthma and those who don't?Data From Exercise 39:Refer to Exercise 38. Repeat this test,
The article "Bioremediation of Soil Contaminated Crude Oil by Agaricomycetes" (Mohammad-Sichani, Assadi, et al., Journal of Environmental Health and Engineering, 2017, online) describes an experiment
The article "Weekly Personal Ozone Exposure and Respiratory Health in a Panel of Greek Schoolchildren" (A. Karakatsani, E. Samoli, et al., Environmental Health Perspectives, 2017, online) describes a
In a sample of 100 automobiles driven at high altitude, $17 \%$ exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. The same cars were driven at sea level, and
Refer to the data set lung.csv. The median ozone for all individuals in the study is 32.48. Can you conclude that people whose ozone exposure is greater than the median are more likely to have asthma
Refer to the data set weather.csv. Can you conclude that the proportion of times the record maximum has occurred after 1945 is greater than the proportion of times the record minimum has occurred
The article "Prediction of Performance and Evaluation of Flexible Pavement Rehabilitation Strategies" (K. Lee, K. Wilson, and S. Hassan, Journal of Traffic and Transportation Engineering,
In an experiment to determine whether there is a systematic difference between the weights obtained with two different scales, 10 rock specimens were weighed, in grams, on each scale. The following
The following output (from $\mathrm{R}$ ) presents the results of a hypothesis test for the difference $\mu_{X}-\mu_{Y}$ between two population means:a. Is this a one-tailed or a two-tailed
The article "Inconsistent Health Perceptions for US Women and Men with Diabetes" (M. McCollum, L. Hansen, et al., Journal of Women's Health, 2007:1421-1428) presents results of a survey of males and
Refer to the data set pit.csv. A pit is classified as "deep" if its depth is greater than $30 \mu \mathrm{m}$. Consider pits at $40 \%$ humidity. Can you conclude that the proportion of pits that
Refer to the data set tunnel.csv. Can you conclude that the proportion of times that the speed is greater than 50 mm/min50 mm/min varies with the type of soil?Tunnel Data Set:A tunnel boring
A company that manufactures steel wires guarantees that the mean breaking strength (in $\mathrm{kN}$ ) of the wires is greater than 50 . They measure the strengths for a sample of wires and test
Washers used in a certain application are supposed to have a thickness of 2 millimeters. A quality control engineer measures the thicknesses for a sample of washers and tests $H_{0}: \mu=2$ versus
Let $\mu$ denote the mean yield strength of a certain type of steel bar, in units of $\mathrm{mPa}$. A test will be made of $H_{0}: \mu \leq 250$ versus $H_{1}: \mu>250$. Three rejection
It is desired to check the calibration of a scale by weighing a standard 10 -gram weight 100 times. Let $\mu$ be the population mean reading on the scale, so that the scale is in calibration if
Scores on a certain IQ test are known to have a mean of 100 . A random sample of 60 students attend a series of coaching classes before taking the test. Let $\mu$ be the population mean IQ score
The rejection region for a $5 \%$ level test of $H_{0}: \mu \geq 10$ versus \(H_{1}: \mu
A coin has probability $p$ of landing heads when tossed. A test will be made of the hypotheses $H_{0}: p \leq 0.1$ ver- sus $H_{1}: p>0.1$, as follows. The coin will be tossed once. If it comes
The following output (from $\mathrm{R}$ ) presents the results of a power calculation for a test of the difference between two means $\mu_{1}-\mu_{2}$.a. Is the power calculated for a one-tailed
Refer to the data set pit.csv.a. Consider pits at 4 weeks duration. Perform a randomization test to determine whether the mean depth at $40 \%$ relative humidity differs from that at $70 \%$
Refer to the data set gravity.csv. Generate 1000 bootstrap samples. Let $\mu$ represent the population mean measurement of the acceleration due to gravity. Using the bootstrap data you generated,
A team of automotive engineers want to compare the lifetimes of two brands of tire. They obtain samples of seven tires of each brand. On each of seven cars, they mount one tire of each brand on each
A blood pressure measurement consists of two numbers: the systolic pressure, which is the maximum pressure taken when the heart is contracting, and the diastolic pressure, which is the minimum
Foot ulcers are a common problem for people with diabetes. Higher skin temperatures on the foot indicate an increased risk of ulcers. The article "An Intelligent Insole for Diabetic Patients with the
A blood pressure measurement consists of two numbers: the systolic pressure, which is the maximum pressure taken when the heart is contracting, and the diastolic pressure, which is the minimum
Concrete expands both horizontally and vertically over time. The article "Evaluation of the Expansion Attained to Date by Concrete Affected by AlkaliSilica Reaction. Part III: Application to Existing
Carrie and Javier have both computed the slope of the least-squares line using data for which the standard deviation of the $x$ values and the standard deviation of the $y$ values are equal.
Refer to the data set lung.csv.a. Compute the least-squares line for predicting FEV1 from ozone exposure.b. Let $\beta_{1}$ denote the change in FEV1 associated with an increase of \(1
Refer to Exercise 18. Which of the following is the best conclusion to be drawn from the results?i. Higher exposure to ozone is associated with lower values of FEV1, but the value of the correlation
How much noisier are streets where cars travel faster? The article "Modeling of Urban Area Stop-and-Go Traffic Noise" (P. Pamanikabud and C. Tharasawatpipat, Journal of Transportation Engineering,
In the following $\mathrm{R}$ output, some of the numbers have been accidentally erased. Recompute them, using the numbers still available. There are $n=25$ points in the data set. Coefficients:
In the following $R$ output, some of the numbers have been accidentally erased. Recompute them, using the numbers still available. There are $n=20$ points in the data set. Coefficients:
The distance advanced in a day by a tunnel boring machine, in meters, is uniformly distributed on the inter$\operatorname{val}(30,50)$.a. Find the mean distance.b. Find the standard deviation of
Let $T \sim \Gamma(4,0.5)$.a. Find $\mu_{T}$.b. Find $\sigma_{T}$.c. Find $P(T \leq 1)$.d. Find $P(T \geq 4)$.
Let $T \sim \Gamma(r, \lambda)$. If $\mu_{T}=8$ and $\sigma_{T}^{2}=16$, find $r$ and $\lambda$.

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