Questions and Answers of Introduction To Probability Statistics

With reference to Example 7 on page 29, find a 95% confidence interval for the mean strength of the aluminum alloy.Data From Example 7 EXAMPLE 7 A two sample t test to show a difference in strength
While performing a certain task under simulated weightlessness, the pulse rate of 32 astronaut trainees increased on the average by 26.4 beats per minute with a standard deviation of 4.28 beats per
It is desired to estimate the mean number of hours of continuous use until a printer overheats. If it can be assumed that \(\sigma=4\) hours, how large a sample is needed so that one will be able to
A sample of 15 pneumatic thermostats intended for use in a centralized heating unit has an average output pressure of \(9 \mathrm{psi}\) and a standard deviation of \(1.5 \mathrm{psi}\). Assuming the
In order to test the durability of a new paint, a highway department had test strips painted across heavily traveled roads in 15 different locations. If on the average the test strips disappeared
Referring to Exercise 7.82 and using 14,380 as an estimate of \(\sigma\), find the sample size that would have been needed to be able to assert with \(95 \%\) confidence that the sample mean is off
A laboratory technician is timed 20 times in the performance of a task, getting \(\bar{x}=7.9\) and \(s=1.2 \mathrm{~min}-\) utes. If the probability of a Type I error is to be at most 0.05 , does
In a fatigue study, the time spent working by employees in a factory was observed. The ten readings (in hours) were\[\begin{array}{llllllllll}4.8 & 3.6 & 10.8 & 5.7 & 8.2 & 6.8 & 7.5 & 7.7 & 6.3 &
An industrial engineer concerned with service at a large medical clinic recorded the duration of time from the time a patient called until a doctor or nurse returned the call. A sample of size 180
Refer to Exercise 7.87.(a) Perform a test with the intention of establishing that the mean time to return a call is greater than 1.5 hours. Use \(\alpha=0.05\).(b) In light of your conclusion in part
The compressive strength of parts made from a composite material are known to be nearly normally distributed. A scientist, using the testing device for the first time, obtains the tensile strength
Refer to Exercise 2.58, where \(n_{1}=30\) specimens of \(2 \times 4\) lumber have \(\bar{x}=1,908.8\) and \(s_{1}=327.1\) psi. A second sample of size \(n_{2}=40\) specimens of larger dimension, \(2
Refer to Exercise 8.1 and obtain a \(95 \%\) confidence interval for the difference in mean tensile strength.Data From Exercise 8.1 Data From Exercise 2.58 8.1 Refer to Exercise 2.58, where n = 30
The dynamic modulus of concrete is obtained for two different concrete mixes. For the first mix, \(n_{1}=33\), \(\bar{x}=115.1\), and \(s_{1}=0.47\) psi. For the second mix, \(n_{2}=31,
Refer to Exercise 8.3 and obtain a \(95 \%\) confidence interval for the difference in mean dynamic modulus.Data From Exercise 8.3 8.3 The dynamic modulus of concrete is obtained for two different
An investigation of two types of bulldozers showed that 50 failures of one type of bulldozer took on an average 6.8 hours to repair with a standard deviation of 0.85 hours, while 50 failures of the
Studying the flow of traffic at two busy intersections between 4 P.M. and 6 P.M. (to determine the possible need for turn signals), it was found that on 40 weekdays there were on the average 247.3
Given the \(n_{1}=3\) and \(n_{2}=2\) observations from Population 1 and Population 2, respectively,(a) Calculate the three deviations \(x-\bar{x}\) and two deviations \(y-\bar{y}\).(b) Use your
Two methods for manufacturing a product are to be compared. Given 12 units, six are manufactured using method \(M\) and six are manufactured using method \(N\).(a) How would you assign manufacturing
Measuring specimens of nylon yarn taken from two spinning machines, it was found that 8 specimens from the first machine had a mean denier of 9.67 with a standard deviation of 1.81 , while 10
We know that silk fibers are very tough but in short supply. Breakthroughs by one research group result in the summary statistics for the stress \((\mathrm{MPa})\) of synthetic silk fibers (Source:
The following are the number of hydraulic pumps which a sample of 10 industrial machines of Type \(A\) and a sample of 8 industrial machines of Type \(B\) manufactured over a certain fixed period of
With reference to Example 5 construct a 95% confidence interval for the true difference between the average resistance of the two kinds of wire.Data From Example 5 EXAMPLE 5 The multiplication rule
In each of the parts below, first decide whether or not to use the pooled estimator of variance. Assume that the populations are normal.(a) The following are the Brinell hardness values obtained for
A civil engineer wants to compare two machines for grinding cement and sand. A sample of a fixed quantity of cement and sand is taken and put in each machine. The machines are run and the fineness of
Refer to Exercise 8.14. Test with \(\alpha=0.01\), that the mean difference is 0 versus a two-sided alternative.Data From Exercise 8.14Find a 99% confidence interval for the mean difference in
The following data were obtained in an experiment designed to check whether there is a systematic difference in the weights obtained with two different scales:Use the paired \(t\) test at the 0.05
Refer to Example 14 concerning suspended solids in effluent from a treatment plant. Take the square root of each of the measurements and then take the difference.(a) Construct a \(95 \%\) confidence
Refer to Example 14 concerning suspended solids in effluent from a treatment plant. Take the natural logarithm of each of the measurements and then take the difference.(a) Construct a 95% confidence
A shoe manufacturer wants potential customers to compare two types of shoes, one made of the current PVC material \(X\) and one made of a new PVC material \(Y\). Shoes made of both are available.
Referring to Example 13, conduct a test to show that the mean change \(\mu_{D}\) is different from 0 . Take \(\alpha=0.05\).Data From Example 13 EXAMPLE 13 The chi square test of independence To
In a study of the effectiveness of physical exercise in weight reduction, a group of 16 persons engaged in a prescribed program of physical exercise for one month showed the following results:Use the
An engineer wants to compare two busy hydraulic belts by recording the number of finished goods that are successfully transferred by the belts in a day. Describe how to select 3 of the next 6 working
An electrical engineer has developed a modified circuit board for elevators. Suppose 3 modified circuit boards and 6 elevators are available for a comparative test of the old versus the modified
It takes an average of 30 classes for an instructor to teach a civil engineering student probability. The instructor introduces a new software which they feel will lead to faster calculations. The
How would you randomize, for a two sample test, if 50 cars are available for an emissions study and you want to compare a modified air pollution device with that used in current production?
With reference to Exercise 2.64, test that the mean charge of the electron is the same for both tubes. Use \(\alpha=0.05\).Data From Exercise 2.64 2.64 J. J. Thomson (1856-1940) discovered the
Two adhesives for pasting plywood boards are to be compared. 10 tubes are prepared using Adhesive I and 8 tubes are prepared using Adhesive II. Then 18 different pairs of plywood boards are pasted
With reference to Example 2, Chapter 2, test that the mean copper content is the same for both heats.Data From Example 2Data From Figure 3.2 EXAMPLE 2 Relation of regions in Venn diagrams to events
Random samples are taken from two normal populations with \(\sigma_{1}=9.6\) and \(\sigma_{2}=13.2\) to test the null hypothesis \(\mu_{1}-\mu_{2}=41.2\) against the alternative hypothesis
With reference to Example 8, find a \(90 \%\) confidence interval for the difference of mean strengths of the alloys(a) using the pooled procedure;(b) using the large samples procedure.Data From
How would you randomize, for a two sample test, in each of the following cases?(a) Forty combustion engines are available for a speed test and you want to compare a modified exhaust valve with the
With reference to part(a) of Exercise 8.33, how would you pair and then randomize for a paired test?Data From Exercise 8.33 8.33 How would you randomize, for a two sample test, in each of the
Two samples in \(\mathrm{C} 1\) and \(\mathrm{C} 2\) can be analyzed using the MINITAB commandsIf you do not click Assume equal variances, the Smith-Satterthwaite test is performed.The output
Refer to Example 13 concerning an array of sites that smell toxic chemicals. When exposed to the common manufacturing chemical Arsine, a product of arsenic and acid, the change in the red component
Refer to Example 12 concerning the improvement in lost worker-hours. Obtain a \(90 \%\) confidence interval for the mean of this paired difference.Data From Example 12 EXAMPLE 12 Calculating a
Verify that the function of Example 1 is, in fact, a probability density.Data From Example 1 EXAMPLE I Combining events by union, intersection, and complement With reference to the sample space of
If the probability density of a random variable is given by\[f(x)= \begin{cases}(k+2) x^{3} & 0
If the probability density of a random variable is given by\[f(x)= \begin{cases}x & \text { for } 0
Given the probability density \(f(x)=\frac{k}{1+x^{2}}\) for \(-\infty
If the distribution function of a random variable is given by\[F(x)= \begin{cases}1-\frac{4}{x^{2}} & \text { for } x>2 \\ 0 & \text { for } x \leq 2\end{cases}\]find the probabilities that this
Find the probability density that corresponds to the distribution function of Exercise 5.7. Are there any points at which it is undefined? Also sketch the graphs of the distribution function and the
Let the phase error in a tracking device have probability density\[f(x)= \begin{cases}\cos x & 0
The length of satisfactory service (years) provided by a certain model of laptop computer is a random variable having the probability density\[f(x)= \begin{cases}\frac{1}{4.5} e^{-x / 4.5} & \text {
At a certain construction site, the daily requirement of gneiss (in metric tons) is a random variable having the probability density \[f(x)= \begin{cases}\frac{4}{81}(x+2)^{-(x+2) / 9} & \text { for
Prove that the identity \(\sigma^{2}=\mu_{2}^{\prime}-\mu^{2}\) holds for any probability density for which these moments exist.
Find \(\mu\) and \(\sigma^{2}\) for the probability density of Exercise 5.2.Data From Exercise 5.2 5.2 If the probability density of a random variable is given by (k+2)x f(x) = 0 0 < x < 1 elsewhere
Find \(\mu\) and \(\sigma^{2}\) for the probability density of Exercise 5.4.Data From Exercise 5.4 5.4 If the probability density of a random variable is given by X for 0 < x < 1 f(x) = 2-x for 1 < x
Find \(\mu\) and \(\sigma\) for the probability density obtained in Exercise 5.8.Data From Exercise 5.8Data From Exercise 5.7 5.8 Find the probability density that corresponds to the distribution
Find \(\mu\) and \(\sigma\) for the distribution of the phase error of Exercise 5.9.Data From Exercise 5.9 5.9 Let the phase error in a tracking device have probabil- ity density f(x) = (9) = { o COS
Find \(\mu\) for the distribution of the satisfactory service of Exercise 5.10.Data From Exercise 5.10 5.10 The length of satisfactory service (years) provided by a certain model of laptop computer
Show that \(\mu_{2}^{\prime}\) and, hence, \(\sigma^{2}\) do not exist for the probability density of Exercise 5.6.Data From Exercise 5.6 5.6 Given the probability density f(x) = -
If a random variable has the standard normal distribution, find the probability that it will take on a value(a) less than 1.75 ;(b) less than -1.25 ;(c) greater than 2.06 ;(d) greater than -1.82 .
If a random variable has the standard normal distribution, find the probability that it will take on a value(a) between 0 and 2.3 ;(b) between 1.22 and 2.43 ;(c) between -1.45 and -0.45 ;(d) between
The nozzle of a mixing vibrator is tested for its number of vibrations. The vibration frequency, for each nozzle sample, can be modeled by a normal distribution with mean 128 and standard deviation
If a random variable has a normal distribution, what are the probabilities that it will take on a value within(a) 1 standard deviation of the mean;(b) 2 standard deviations of the mean;(c) 3 standard
Verify that(a) \(z_{0.005}=2.575\);(b) \(z_{0.025}=1.96\).
Given a random variable having the normal distribution with \(\mu=16.2\) and \(\sigma^{2}=1.5625\), find the probabilities that it will take on a value(a) greater than 16.8 ;(b) less than 14.9;(c)
The time for oil to percolate to all parts of an engine can be treated as a random variable having a normal distribution with mean 20 seconds. Find its standard deviation if the probability is 0.25
Butterfly-style valves used in heating and ventilating industries have a high flow coefficient. Flow coefficient can be modeled by a normal distribution with mean \(496 C_{V}\) and standard deviation
Refer to Exercise 5.26 but suppose that a large potential contract contains the specification that at most \(7.5 \%\) can have a flow coefficient less than \(420 \mathrm{C}_{\mathrm{v}}\). If the
Find the quartiles\[-z_{0.25} \quad z_{0.50} \quad z_{0.25}\]of the standard normal distribution.
The daily high temperature in a computer server room at the university can be modeled by a normal distribution with mean \(68.7^{\circ} \mathrm{F}\) and standard deviation \(1.2^{\circ} \mathrm{F}\).
A machine produces soap bars with a weight of \(80 \pm\) \(0.10 \mathrm{~g}\). If the weight of the soap bars manufactured by the machine may be looked upon as a random variable having normal
The number of teeth of a \(12 \%\) tooth gear produced by a machine follows a normal distribution. Verify that if \(\sigma=1.5\) and the mean number of teeth is \(13,74 \%\) of the gears contain at
The quantity of aerated water that a machine puts in a bottle of a carbonated beverage follows a normal distribution with a standard deviation of \(0.25 \mathrm{~g}\). At what "normal" (mean) weight
An automatic machine fills distilled water in \(500-\mathrm{ml}\) bottles. Actual volumes are normally distributed about a mean of \(500 \mathrm{ml}\) and their standard deviation is \(20
If a random variable has the binomial distribution with \(n=25\) and \(p=0.65\), use the normal approximation to determine the probabilities that it will take on(a) the value 15 ;(b) a value less
From past experience, a company knows that, on average, \(5 \%\) of their concrete does not meet standards. Use the normal approximation of the binomial distribution to determine the probability that
The probability that an electronic component will fail in less than 1,000 hours of continuous use is 0.25 . Use the normal approximation to find the probability that among 200 such components fewer
Workers in silicon factories are prone to a lung disease called silicosis. In a recent survey in a factory, about \(11 \%\) of the workers have been infected by it. Assume the same rate of infection
Refer to Example 11 concerning the experiment that confirms electron antineutrinos change type. Suppose instead that there are 400 electron antineutrinos leaving the reactor. Repeat parts (a)-(c) of
To illustrate the law of large numbers mentioned on Page 116, find the probabilities that the proportion of drawing a club from a fair deck of cards will be anywhere from 0.24 to 0.26 when a card is
Verify the identity \(F(-z)=1-F(z)\) given on page 141. F(z) 0 N
Verify that the parameter \(\mu\) in the expression for the normal density on page 140 , is, in fact, its mean. 5.2 The Normal Distribution Among the special probability densities we study in this
Verify that the parameter \(\sigma^{2}\) in the expression for the normal density on page 140 is, in fact, its variance. 5.2 The Normal Distribution Among the special probability densities we study
Normal probabilities can be calculated using MINITAB. Let \(X\) have a normal distribution with mean 11.3 and standard deviation 5.7. The following steps yield the cumulative probability of 9.31 or
Find the distribution function of a random variable having a uniform distribution on \((0,1)\).
In a manufacturing process, the error made in determining the composition of an alloy is a random variable having the uniform density with \(\alpha=-0.075\) and \(\beta=0.010\). What are the
From experience Mr. Harris has found that the low bid on a construction job can be regarded as a random variable having the uniform density\[f(x)= \begin{cases}\frac{3}{4 C} & \text { for } \frac{2
Verify the expression given on page 154 for the mean of the log-normal distribution. Mean of log-normal distribution M = ea+B 12 Variance of log-normal Similar, but more lengthy, calculations yield
With reference to the Example 12, find the probability that \(I_{o} / I_{i}\) will take on a value between 7.0 and 7.5.Data From Example 12 EXAMPLE 12 Maximum likelihood estimator: normal
If a random variable has the log-normal distribution with \(\alpha=-3\) and \(\beta=3\), find its mean and its standard deviation.
If a random variable has the gamma distribution with \(\alpha=2\) and \(\beta=3\), find the mean and the standard deviation of this distribution.
With reference to Exercise 5.52, find the probability that the random variable will take on a value less than 5.Data From Exercise 5.52 5.52 If a random variable has the gamma distribution with = 2
At a construction site, the daily requirement of gneiss (in metric tons) is a random variable having a gamma distribution with \(\alpha=2\) and \(\beta=5\). If their supplier's daily supply capacity
With reference to the Example 14, suppose the expert opinion is in error. Calculate the probability that the supports will survive if(a) \(\mu=3.0\) and \(\sigma^{2}=0.09\);(b) \(\mu=4.0\) and
Verify the expression for the variance of the gamma distribution given on page 156 . Mean of log-normal distribution M = ea+B 12 Variance of log-normal Similar, but more lengthy, calculations yield
Show that when \(\alpha>1\), the graph of the gamma density has a relative maximum at \(x=\beta(\alpha-1)\). What happens when \(0
The server of a multinational corporate network can run for an amount of time without having to be rebooted and this amount of time is a random variable having the exponential distribution

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