Questions and Answers of Statistics For Engineers And Scientists

Let $T \sim \Gamma(r, \lambda)$. If $\mu_{T}=8$ and $r=16$, find $\lambda$ and $\sigma^{2}$.
Let $T \sim \operatorname{Weibull}(0.5,3)$.a. Find $\mu_{T}$.b. Find $\sigma_{T}$.c. Find $P(T5)$.e. Find \(P(2
Let $X_{1}, \ldots, X_{n}$ be a random sample from a population with the Poisson $(\lambda)$ distribution. Find the MLE of $\lambda$.
Let $X_{1}, \ldots, X_{n}$ be a random sample from a $N(\mu, 1)$ population. Find the MLE of $\mu$.
Consider the following data set:a. Construct a normal probability plot for these data. Do the data appear to come from an approximately normal distribution?b. Transform the data by taking the square
Consider the following data set:a. Construct a normal probability plot for these data. Do the data appear to come from an approximately normal distribution?b. Transform the data by taking the
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
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
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
Refer to Exercise 13. Assume the distribution from which the sample was drawn is unknown. We want to estimate the population mean μμμμ with the sample mean ¯X¯XX¯¯X. This exercise shows how
Medication used to treat a certain condition is administered by syringe. The target dose in a particular application is $\mu$. Because of the variations in the syringe, in reading the scale, and in
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
A sample of 114 patients were given a drug to lower cholesterol. A 95\% confidence interval for the mean reduction in cholesterol (in $\mathrm{mmol} / \mathrm{L}$ ) was $(0.88,1.02)$.a. What was
In a sample of 80 light bulbs, the mean lifetime was 1217 hours. Assume the population standard deviation is 52 hours.a. Find a 95\% confidence interval for the mean lifetime of this type of light
Refer to Exercise 7.a. Find a $90 \%$ upper confidence bound for the mean lifetime.b. Someone says that the mean lifetime is less than 1220 hours. With what level of confidence can this statement
Refer to Exercise 8.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
A chemist made six independent measurements of the sublimation point of carbon dioxide (the temperature at which it changes to dry ice). The sample mean was $196.64 \mathrm{~K}$ with a standard
In a sample of 100 steel canisters, the mean wall thickness was $8.1 \mathrm{~mm}$ with a standard deviation of $0.5 \mathrm{~mm}$.a. Find a 95\% confidence interval for the mean wall thickness
A sample of 75 concrete blocks had a mean mass of $38.3 \mathrm{~kg}$ with a standard deviation of $0.6 \mathrm{~kg}$.a. Find a 95\% confidence interval for the mean mass of this type of concrete
One step in the manufacture of a certain metal clamp involves the drilling of four holes. In a sample of 150 clamps, the average time needed to complete this step was 72 seconds and the standard
Refer to Exercise 14.a. Find a $95 \%$ upper confidence bound for the mean improvement.b. Find a $98 \%$ lower confidence bound for the mean improvement.Data From Exercise 14:The article
Refer to Exercise 15.a. Find a $95 \%$ lower confidence bound for the mean wall thickness.b. Find a $99 \%$ upper confidence bound for the mean wall thickness.Data From Exercise 15:In a sample of
Refer to Exercise 16.a. Find a $95 \%$ lower confidence bound for the mean time.b. Find a $90 \%$ upper confidence bound for the mean time.Data From Exercise 16:The article "Application of
Refer to Exercise 17.a. Find a 95%95% upper confidence bound for the mean mass.b. Find a 99.5%99.5% lower confidence bound for the mean mass.Data From Exercise 17:A sample of 75 concrete blocks had a
Refer to Exercise 18.a. Find a 95%95% upper confidence bound for the mean time.b. Find a 98%98% lower confidence bound for the mean time.Data From Exercise 18:One step in the manufacture of a certain
Refer to Exercise 19.a. Find a 95%95% lower confidence bound for the mean weight.b. Find a 90%90% upper confidence bound for the mean weight.Data From Exercise 19:In a sample of 80 ten-penny nails,
An investigator computes a $95 \%$ confidence interval for a population mean on the basis of a sample of size 70. Approximately how large a sample is needed to produce a 95\% confidence interval
The concentration of carbon monoxide (CO) in a gas sample is measured by a spectrophotometer and found to be $85 \mathrm{ppm}$. Through long experience with this instrument, it is believed that its
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
Refer to the data set pit.csv. Find a confidence interval of the specified level for the mean depth of pits for the following durations and humidity levels. When the sample size is 30 or less,
Refer to Exercise 32. For each duration, the confidence interval at 40%40% humidity is narrower than the one at 75%75% humidity. Which of the following is the best explanation for this?i. There are
Refer to the data set tunnel.csv. Find a confidence interval of the specified level for the mean speed in each of the following types of soil.a. GSG soil, 95%95% confidenceb. CCS soil, 98%98%
Find a confidence interval of the specified level for the mean torque in each of the following types of soil.a. GSG soil, $95 \%$ confidenceb. CCS soil, $98 \%$ confidencec. GSF soil, $99 \%$
Refer to the data set gravity.csv.a. How many outliers are in the data?b. Explain why it is appropriate to use the methods of this section to find a confidence interval for the mean, even though 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.
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
Refer to Exercise 4. Find a $99 \%$ lower confidence bound for the proportion of smokers who tried to quit smoking in the past year.Data From Exercise 4:The National Center for Health Statistics
Sleep apnea is a disorder in which there are pauses in breathing during sleep. People with this condition must wake up frequently to breathe. The article "Postoperative Complications in Patients With
Refer to the data set pit.csv. Pits with depths greater than $30 \mu \mathrm{m}$ are considered "deep." Find a confidence interval of the specified level for the proportion of pits that are deep
Which of the following statements best summarizes the results of Exercise 17?i. The number of pits is less at $75 \%$ humidity than at $40 \%$, but the proportion that are deep is about the same
Refer to the data set gravity.csv. The measurements are expressed in deviations from a base value of about $9.8 \mathrm{~m} / \mathrm{s}^{2}$, in units of $10^{-8} \mathrm{~m} / \mathrm{s}^{2}$.
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)
The article "Effects of Diets with Whole PlantOrigin Proteins Added with Different Ratios of Taurine:Methionine on the Growth, Macrophage Activity and Antioxidant Capacity of Rainbow Trout
In a sample of 90 large grade A eggs, the mean number of calories was 70 with a standard deviation of 2.5. In a sample of 85 large grade AA eggs, the mean number of calories was 80 with a standard
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,
In an agricultural experiment, the effects of two fertilizers on the production of oranges were measured. Sixteen randomly selected plots of land were treated with fertilizer A. The average yield, in
In a test to compare yield strengths of two grades of carbon steel bolts, strengths of eight grade 2 bolts and six grade 5 bolts were measured. The results, in MPa, wereFind a $98 \%$ confidence
For each of the following types of soil, find the median torque and median speed for all observations in that soil.a. GSG soilb. CCS soilc. GSF soild. GSGu soil
Which statement best describes the results of Exercise 30 ?i. For soils where the median torque is greater, the median speed is greater.ii. For soils where the median torque is greater, the median
For each of the following types of soil, find the median value of the torque for all observations in that soil. Then let $\mu_{L}$ be the mean speed when the torque is below the median, let
Which statement best describes the results of Exercise 32 ?i. For each of the four soils, we are $95 \%$ confident that the mean speed is greater when the torque is lower.ii. For each of the four
Refer to Exercises 31 and 33. The relationship between median torques and median speeds is called the relationship between soils. The relationships between torque and speed restricted to a certain
Compute the median torque for all observations in all soils. Let $\mu_{L}$ be the mean speed when the torque is below the median, and let $\mu_{U}$ be the mean speed when the torque is above the
Which statement best describes the results of Exercise 35 ?i. For observations over all soils, we are 95\% confident that the mean speed is greater when the torque is lower.ii. For observations over
Is the relationship between torque and speed computed from all soils combined more like the withinsoils relationship or the between-soils relationship?
For each of the following durations, find a confidence interval of the specified level for the difference between the mean depth at $40 \%$ relative humidity and the mean depth at $75 \%$
Find a confidence interval of the specified level for the difference in mean pit depth for each of the following values for humidity and pairs of durations.a. $40 \%$ humidity: Difference between
In an accelerated failure test, components are operated under extreme conditions so that a substantial number will fail in a rather short time. In such a test involving two types of microchips, 600
The article "Soft Drinks Consumption Is Associated with Nonalcoholic Fatty Liver Disease Independent of Metabolic Syndrome in Chinese Population" (G. Meng, B. Zhang, et al., European Journal of
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
A pit is considered "deep" if its depth is greater than $30 \mu \mathrm{m}$. For each of the following durations, find a confidence interval of the specified level for the difference between the
Find a confidence interval of the specified level for the difference in proportion of deep pits for each of the following values for humidity and pairs of durations.a. $40 \%$ humidity: Difference
To construct a confidence interval for each of the following quantities, say whether it would be better to use paired samples or independent samples.a. The mean difference in height between identical
To construct a confidence interval for each of the following quantities, say whether it would be better to use paired samples or independent samples.a. The mean difference in weight loss between
Find a $95 \%$ confidence interval for the mean difference between the year in which the maximum temperature was recorded and the year in which the minimum temperature was recorded.
Refer to Exercise 13. Are the results of the confidence interval consistent with the hypothesis that temperatures have been rising over time? Explain.Data From Exercise 13:Find a $95 \%$ confidence
Find the following values.a. $\chi_{12,025}^{2}$b. $\chi_{12,975}^{2}$c. $\chi_{5,005}^{2}$d. $\chi_{5,995}^{2}$e. $\chi_{22,1}^{2}$f. $\chi_{22,9}^{2}$
Following are interest rates (annual percentage rates) for a 30-year fixed-rate mortgage from a sample of lenders in Colorado on November 9, 2021. Assume that the population is normally
The pressure of air (in $\mathrm{MPa}$ ) entering a compressor is measured to be $X=8.5 \pm 0.2$, and the pressure of the air leaving the compressor is measured to be $Y=21.2 \pm 0.3$. The
The mass (in $\mathrm{kg}$ ) of a soil specimen is measured to be $X=1.18 \pm 0.02$. After the sample is dried in an oven, the mass of the dried soil is measured to be $Y=0.85 \pm 0.02$. Assume
A student measures the acceleration $A$ of a cart moving down an inclined plane by measuring the time $T$ that it takes the cart to travel $1 \mathrm{~m}$ and using the formula $A=2 / T^{2}$.
The initial temperature of a certain container is measured to be $T_{0}=20^{\circ} \mathrm{C}$. The ambient temperature is measured to be $T_{a}=4^{\circ} \mathrm{C}$. An engineer uses Newton's
Refer to the data set gravity.csv.a. Generate 1000 bootstrap samples from these data. Find the 2.5 and 97.5 percentiles.b. Compute a $95 \%$ bootstrap confidence interval for the mean, using method
Refer to the data set pit.csv.a. Generate 1000 bootstrap samples from the pit depths at four weeks and 75\% humidity. Find the 2.5 and 97.5 percentiles.b. Compute a $95 \%$ bootstrap confidence
A sample of size 64 has standard deviation $s=3.6$. Approximately how large a sample is needed so that a $99 \%$ confidence interval will specify the mean to within $\pm 1.0$ ?
Refer to Exercise 26.a. Generate 10,000 bootstrap samples from the data in Exercise 26. Find the bootstrap sample mean percentiles that are used to compute a $99 \%$ confidence interval.b. Compute
A sample of 50 copper wires had a mean resistance of $1.03 \mathrm{~m} \Omega$ with a standard deviation of $0.1 \mathrm{~m} \Omega$. Let $\mu$ represent the mean resistance of copper wires of
A sample of 65 electric motors had a mean efficiency of 0.595 with a standard deviation of 0.05 . Let $\mu$ represent the mean efficiency of electric motors of this type.a. Find the $P$-value for
In a test of corrosion resistance, a sample of $60 \mathrm{In}$ coloy steel specimens were immersed in acidified brine for four hours, after which each specimen had developed a number of corrosive
A process that manufactures steel bolts is supposed to be calibrated to produce bolts with a mean length of $5 \mathrm{~cm}$. A sample of 100 bolts has a mean length of $5.02 \mathrm{~cm}$. The
Fill in the blank: In a test of $H_{0}: \mu \geq 10$ versus \(H_{1}: \mu
An automotive engineer subjects a large number of brake pads to a stress test and measures the wear on each. The values obtained are $\bar{X}=7.4 \mathrm{~mm}$ and \(\sigma_{\bar{X}}=0.2
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. What
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. What
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 calibration of a scale is checked by weighing a standard $10 \mathrm{~g}$ weight 100 times. Let $\mu$ be the population mean reading on the scale, so that the scale is in calibration if
A sample of size $n=100$ is used to test $H_{0}: \mu \leq 20$ versus $H_{1}: \mu>20$. The value of $\mu$ will not have practical significance unless $\mu>25$. The population standard
A new method of postoperative treatment was evaluated for patients undergoing a certain surgical procedure. Under the old method, the mean length of hospital stay was 6.3 days. The sample mean for
Erin computes a $95 \%$ confidence interval for $\mu$ and obtains (94.6, 98.3). Jamal performs a test of the hypotheses $H_{0}: \mu=100$ versus $H_{1}: \mu eq 100$ and obtains a $P$-value
A $99 \%$ confidence interval for $\mu$ is $(5.1,5.8)$. Someone wants to use the data from which this confidence interval was constructed to test $H_{0}: \mu=6$ versus $H_{1}: \mu eq 6$.
A shipment of fibers is not acceptable if the mean breaking strength of the fibers is less than $50 \mathrm{~N}$. A large sample of fibers from this shipment was tested, and a $98 \%$ lower
Refer to Exercise 23. It is discovered that the mean of the sample used to compute the confidence bound is $\bar{X}=3.40$. Is it possible to determine whether \(P
Refer to Exercise 24. It is discovered that the standard deviation of the sample used to compute the confidence interval is 5 N5 N. Is it possible to determine whether P
The following output presents the results of a hypothesis test for a population mean $\mu$.a. Can $H_{0}$ be rejected at the $5 \%$ level? How can you tell?b. Someone asks you whether the null
The General Social Survey asked a sample of adults how many siblings (brothers and sisters) they had $(X)$ and also how many children they had $(Y)$. We show results for those who had no more
Refer to Exercise 9.a. Find $\mu_{X+Y}$.b. Find $\sigma_{X+Y}$.c. Find $P(X+Y=5)$.Data From Exercise 9:The General Social Survey asked a sample of adults how many siblings (brothers and
Refer to Exercise 9.a. Find the conditional probability mass function $p_{Y \mid X}(y \mid 4)$.b. Find the conditional probability mass function $p_{X \mid Y}(x \mid 3)$.c. Find the conditional
Let $a,b, c, d$ be any numbers with $aIn other words, \(f(x, y)$ is constant on the rectangle $aa. Show that \(k=\frac{1}{(b-a)(d-c)}$.b. Show that the marginal density of $X$ is
A calibration laboratory has received a weight that is labeled as $1 \mathrm{~kg}$. It is weighed five times. The measurements are as follows, in units of micrograms above $1 \mathrm{~kg}$ :
A measurement of the diameter of a disk has an uncertainty of $1.5 \mathrm{~mm}$. How many measurements must be made so that the diameter can be estimated with an uncertainty of only \(0.5

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