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statistics for engineers and scientists

Questions and Answers of Statistics For Engineers And Scientists

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
Refer to Exercise 4. Assume that \(T=298.4 \pm 0.2 \mathrm{~K}\). Estimate \(V\), and find the relative uncertainty in the estimate.
A Bernoulli random variable has variance 0.21 . What are the possible values for its success probability?
Let \(X \sim \operatorname{Bin}(7,0.3)\). Finda. \(P(X=1)\)b. \(P(X=2)\)c. \(P(X4)\)e. \(\mu_{X}\)f. \(\sigma_{X}^{2}\)
Let \(X \sim \operatorname{Bin}(9,0.4)\). Finda. \(P(X>6)\)b. \(P(X \geq 2)\)c. \(P(2 \leq X
Find the following probabilities:a. \(P(X=2)\) when \(X \sim \operatorname{Bin}(4,0.6)\)b. \(P(X>2)\) when \(X \sim \operatorname{Bin}(8,0.2)\)c. \(P(X \leq 2)\) when \(X \sim
Several million lottery tickets are sold, and \(60 \%\) of the tickets are held by women. Five winning tickets will be drawn at random.a. What is the probability that three or fewer of the winners
A marketing manager samples 150 people and finds that 87 of them have made a purchase on the internet within the past month.a. Estimate the proportion of people who have made a purchase on the
A quality engineer samples 100 steel rods made on mill \(A\) and 150 rods made on mill \(B\). Of the rods from mill \(\mathrm{A}, 88\) meet specifications, and of the rods from mill B, 135 meet
A commuter must pass through three traffic lights on the way to work. For each light, the probability that it is green upon arrival is 0.6. The lights are independent.a. What is the probability that
A distributor receives a large shipment of components. The distributor would like to accept the shipment if \(10 \%\) or fewer of the components are defective and to return it if more than \(10 \%\)
Let \(X \sim \operatorname{Bin}(n, p)\), and let \(Y=n-X\). Show that \(Y \sim \operatorname{Bin}(n, 1-p)\).
Let \(X \sim\) Poisson(4). Finda. \(P(X=1)\)b. \(P(X=0)\)c. \(P(X1)\)e. \(\mu_{X}\)f. \(\sigma_{X}\)
The number of pits in a corroded steel coupon follows a Poisson distribution with a mean of 6 pits per \(\mathrm{cm}^{2}\). Let \(X\) represent the number of pits in a \(1 \mathrm{~cm}^{2}\) area.
The number of large packages delivered by a courier service follows a Poisson distribution with a rate of 5 per day. Let \(X\) be the number of large packages delivered on a given day. Finda.
To estimate the concentration of particles in a certain suspension, a chemist withdraws \(3 \mathrm{~mL}\) of the suspension and counts 48 particles. Estimate the concentration in particles per
To estimate the concentration of a certain type of bacterium in a wastewater sample, a microbiologist puts a \(0.5 \mathrm{~mL}\) sample of the wastewater on a microscope slide and counts 39
Grandpa is trying out a new recipe for raisin bread. Each batch of bread dough makes three loaves, and each loaf contains 20 slices of bread.a. If he puts 100 raisins into a batch of dough, what is
Dad and Grandpa are each baking chocolate chip cookies. Each gives you two cookies. One of Dad's cookies has 14 chips in it and the other has Grandpa's cookies have 6 and 8 chips.a. Estimate the mean
Twenty-five automobiles have been brought in for service. Fifteen of them need tuneups and ten of them need new brakes. Nine cars are chosen at random to be worked on. What is the probability that
In a lot of 15 truss rods, 12 meet a tensile strength specification. Four rods are chosen at random to be tested. Let \(X\) be the number of tested rods that meet the specification.a. Find
Among smartphone users, \(40 \%\) use a case but no screen protector, \(10 \%\) use a screen protector but no case, \(45 \%\) use both a case and a screen protector, and \(5 \%\) use neither a case
Let \(X \sim \operatorname{Geom}(p)\), let \(n\) be a non-negative integer, and let \(Y \sim \operatorname{Bin}(n, p)\). Show that \(P(X=n)=\) \((1 / n) P(Y=1)\).
Find the area under the normal curvea. To the left of \(z=0.56\).b. Between \(z=-2.93\) and \(z=-2.06\).c. Between \(z=-1.08\) and \(z=0.70\).d. Outside \(z=0.96\) to \(z=1.62\).
If \(X \sim N(2,9)\), computea. \(P(X \geq 2)\)b. \(P(1 \leq X
A process manufactures ball bearings with diameters that are normally distributed with mean \(25.1 \mathrm{~mm}\) and standard deviation \(0.08 \mathrm{~mm}\).a. What proportion of the diameters are
Depths of pits on a corroded steel surface are normally distributed with mean \(822 \mu \mathrm{m}\) and standard deviation \(29 \mu \mathrm{m}\).a. Find the 10th percentile of pit depths.b. A
In a recent study, the Centers for Disease Control reported that diastolic blood pressures (in \(\mathrm{mmHg}\) ) of adult women in the United States are approximately normally distributed with mean
The lifetime of a light bulb in a certain application is normally distributed with mean \(\mu=1400\) hours and standard deviation \(\sigma=200\) hours.a. What is the probability that a light bulb
Speeds of automobiles on a certain stretch of freeway at 11:00 PM are normally distributed with mean \(65 \mathrm{mph}\). Twenty percent of the cars are traveling at speeds between 55 and \(65
Scores on an exam were normally distributed. Ten percent of the scores were below 64 and \(80 \%\) were below 81 . Find the mean and standard deviation of the scores.
Let \(X \sim N\left(\mu, \sigma^{2}ight)\), and let \(Z=(X-\mu) / \sigma\). Use Equation (4.25) to show that \(Z \sim N(0,1)\).
Two resistors, with resistances \(R_{1}\) and \(R_{2}\), are connected in series. \(R_{1}\) is normally distributed with mean \(100 \Omega\) and standard deviation \(5 \Omega\), and \(R_{2}\) is
The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution \(\left(1ight.\) mole \(=6.02 \times 10^{23}\) molecules \()\). If \(X\) is the molarity of a
The period \(T\) of a simple pendulum is given by \(T=\) \(2 \pi \sqrt{L / g}\) where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. Assume that \(g=\) \(9.80
The volume of a cylinder is given by \(V=\pi r^{2} h\), where \(r\) is the radius of the cylinder and \(h\) is the height. Assume the radius, in \(\mathrm{cm}\), is lognormal with parameters
Refer to Exercise 5. Suppose 10 pendulums are constructed. Find the probability that 4 or more have periods greater than 3 seconds.Data From Exercise 5:The period \(T\) of a simple pendulum is given
Refer to Exercise 6. Suppose 8 cylinders are constructed. Find the probability that fewer than 5 of them have volumes between 500 and \(800 \mathrm{~cm}^{3}\).Data From Exercise 6:The volume of a
A manufacturer claims that the tensile strength of a certain composite (in MPa) has the lognormal distribution with \(\mu=5\) and \(\sigma=0.5\). Let \(X\) be the strength of a randomly sampled
The distance between flaws on a long cable is exponentially distributed with mean \(12 \mathrm{~m}\).a. What is the value of the parameter \(\lambda\) ?b. Find the median distance.c. Find the
Refer to Exercise 2.a. Find the probability that there will be exactly 5 requests in a 2 -second time interval.b. Find the probability that there will be more than 1 request in a 1.5 -second time
Refer to Exercise 4.a. Find the probability that there are exactly 5 flaws in a \(50 \mathrm{~m}\) length of cable.b. Find the probability that there are more than two flaws in a \(20 \mathrm{~m}\)
A radioactive mass emits particles according to a Poisson process. The probability that no particles will be emitted in a two-second period is 0.5 .a. What is the probability that no particles are
The lifetime of a transistor is exponentially distributed. The probability that the lifetime is greater than five years is 0.8 .a. What is the probability that the lifetime is greater than 15
A sample of 100 cars driving on a freeway during a morning commute was drawn, and the number of occupants in each car was recorded. The results were as follows:a. Find the sample mean number of
Five basketball players and 11 football players are working out in a gym. The mean height of the basketball players is \(77.6 \mathrm{in}\)., and the mean height of all 16 athletes is \(74.5
An insurance company examines the driving records of 100 policy holders. They find that 80 of them had no accidents during the past year, 15 had one accident, 4 had two, and 1 had three.a. Find the
Match each histogram to the boxplot that represents the same data set. 144 (a) e (b) X (2) (3) I +
In general a histogram is skewed to the left when the median is greater than the mean and to the right when the median is less than the mean. There are exceptions, however. Consider the following
Refer to the file lung.csv.a. Construct a histogram for the values of FEV1.b. Is the histogram symmetric, skewed to the left, or skewed to the right?c. Construct a boxplot for the values of FEV1.d.
Refer to the file gravity.csv.a. Construct a histogram for the measurements of the acceleration due to gravity.b. Apart from outliers, is the histogram unimodal or bimodal?c. Construct a boxplot for
Refer to the file pit.csv.a. Consider the pits at 52 weeks duration. Construct comparative boxplots for the depths at \(40 \%\) and \(75 \%\) relative humidity.b. Using the boxplots, what differences
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29 .a. Is it possible to determine by how much the mean changes? If so, by how much does it change?b. Is it possible
The probability that a bolt meets a strength specification is 0.87 . What is the probability that the bolt does not meet the specification?
According to a report by the Agency for Healthcare Research and Quality, the age distribution for people admitted to a hospital for an asthma-related illness was as follows.a. What is the probability
A company audit showed that of all bills that were sent out, \(71 \%\) of them were paid on time, \(18 \%\) were paid up to 30 days late, \(9 \%\) were paid between 31 and 90 days late, and \(2 \%\)
In a certain community, \(28 \%\) of the houses have fireplaces and \(51 \%\) have garages. Is it possible to compute the probability that a randomly chosen house has either a fireplace or a garage?
According to the National Health Statistics Reports, \(16 \%\) of American families have one child, and \(21 \%\) have two children. Is it possible to compute the probability that a randomly chosen
Resistors manufactured by a certain process are labeled as having a resistance of \(5 \Omega\). A sample of 100 resistors is drawn, and 87 of them have resistances between 4.9 and \(5.1 \Omega\).
License plates in a certain state consist of three letters followed by three digits.a. How many different license plates can be made?b. How many license plates are there that contain neither the
Joe, Megan, and Santana are salespeople. Their sales manager has 18 accounts and must assign six accounts to each of them. In how many ways can this be done?
Suppose that \(90 \%\) of bolts and \(85 \%\) of nails meet specifications. One bolt and one nail are chosen independently.a. What is the probability that both meet specifications?b. What is the

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