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introduction to probability statistics
Questions and Answers of
Introduction To Probability Statistics
A critical width dimension on an integrated circuit board was measured on 100 boards. The ordered measurements are:Given the specification limits \(L S L=2.496\) and \(U S L=2.516\), evaluate the
The following are the number of pounds per day shipped by a trucking company.It is suggested that the shipments be treated as a process. Because these data are not symmetrically distributed, you
An electric circuit consists of 7 transistors connected in parallel. Determine the reliability of each transistor if the overall reliability of the circuit is to be 0.99 .
A system consists of 5 identical components connected in parallel. Determine the reliability of each component if the overall reliability of the system is to be 0.96 ?
A system consists of 6 components connected as in Figure 16.3. Find the overall reliability of the system, given that the reliabilities of \(A, B, C, D, E\), and \(F\) are, respectively,
Suppose that a machine's operation is regarded as a system having three main components: \(A\) (team), \(B\) (power supply), and \(C\) (assembly). Suppose, furthermore, that component \(\mathrm{A}\)
In some reliability problems we are concerned only with initial failures, treating a component as if (for all practical purposes) it never fails, once it has survived past a certain time
As indicated in the text, one often distinguishes between initial failures, random failures during the useful life of the product, and wear-out failures. For a given product, suppose the probability
A transistor has a constant failure rate of 0.005 per ten thousand hours.(a) What is the probability that it will perform satisfactorily for at least 75,000 hours?(b) What is the 20,000-hour
After installation, the lifetime of a ball bearing is modeled as an exponential distribution with failure rate \(\alpha=0.001\) failures per day.(a) What is the probability that the ball bearing will
If the spare part of a cement mixer has the Weibull failure-time distribution with the parameters \(\alpha=\) 0.001 per hour and \(\beta=0.65\), find the probability that it will not operate
Suppose that 75 units are put on a life test, each unit that fails is immediately replaced, and the test is discontinued after 10 units have failed. If the tenth failure occurred at 850 hours,
In a non-replacement life test, 50 water turbines were put into continuous operation, and the first 3 failures occurred after 400,600, and 1,050 hours.(a) Assuming the exponential model, construct an
With reference to the data in Exercise 16.11, make a total time on test plot.Data From Exercise 16.11 16.11 In a nonreplacement life test, 50 water turbines were put into continuous operation, and
To investigate the average time to failure of a certain soldered object subject to continuous flow of current, 5 soldered objects were subjected to specified volts and amperes of current and their
In life testing, we are sometimes interested in establishing tolerance limits for the life of a component in particular, we may be interested in a one-sided tolerance limit \(t^{*}\), for which we
Using the fact that \(2 T_{r} / \mu\) is a value of a random variable having the chi square distribution with \(2 r\) degrees of freedom, derive the confidence interval for \(\mu\) given on page
One hundred devices are put on life test and the times to failure (in hours) of the first 10 that fail are\[\begin{array}{rrrrr} 7.0 & 14.1 & 18.9 & 31.6 & 52.8 \\ 80.0 & 164.5 & 355.4 & 451.0 &
A sample of 450 bulbs was placed on a life test consisting of repeated on-off cycles. The test was terminated after the fifth failure. The first five failure times were 469,724,862,995 and 1,246.
A sample of 60 diaphragm valves, used in the control system of a chemical process, are placed on life test without replacement. The first 9 failures are observed after\(\begin{array}{lllllllll}3.6 &
Using the estimates of the parameters of the Weibull model obtained in Exercise 16.18, estimate the probability that this kind of diaphragm valve will perform satisfactorily for at least 150
Show that the reliability function associated with the Weibull failure-time distribution is given by\[R(t)=e^{-\alpha t^{\beta}}\]
Derive the formula for the variance of the Weibull distribution given on page 514.Data From Page 514 Weibull failure-rate function (1) = - 1
A system consists of 7 identical components connected in parallel. Determine the reliability of each component if the overall reliability of the system is to be 0.90 ?
A transmitter has an exponential life distribution with a failure rate of \(\alpha=0.00025\) failures per hour.(a) What is the probability that the transmitter will fail during the first 400 hours it
Twenty motors are put on an accelerated life test without replacement and the test is truncated after 5 failures. If the first 5 failures occurred at 12.5, 16.8, 24.3, 27.6 and 32.1 hours, assuming
A sample of 400 high-reliability switches was placed on a life test until the first five failures occurred and the test was then terminated. The first five failures were at 268, 395, 479, 685 and 964
To investigate the performance of a logic circuit for a small electronic calculator, a laboratory puts 75 of the circuits on life test without replacement under specified environmental conditions,
Using the estimates of the parameters of the Weibull model obtained in Exercise 16.26, estimate the probability that this kind of circuit will perform satisfactorily for at least 100 hours.Data From
With reference to Exercise 16.26, make(a) a total time on test plot(b) a Weibull plotData From Exercise 16.26 16.26 To investigate the performance of a logic circuit for a small electronic
An alternative model used in reliability treats the environmental stress as a random variable \(X\), with probability density \(f(x)\), and the strength of the component to withstand this stress as
Given the two replications of a \(2 \times 3\) factorial experiment, calculate the analysis of variance table using the formulas on pages \(* * *\) and \(* * *\). Factor A Factor B Rep. I Rep. 2 1 1
MINITAB can create the analysis of variance table for Example 1 concerning recycled road materials. The three levels of \(A\) are coded 1,2, and 3, and the two levels of \(B\) are coded 1 and 2. The
To determine optimum condition for a circuit board, the effects of bond strength of FR-4 (an insulating substrate) and thickness of copper lamination on the current flow are studied in a \(2 \times
A spoilage-retarding ingredient is added in brewing beer. To determine the extent to which the taste of the beer is affected by the amount of this ingredient added to each batch, and how such taste
Suppose that in the experiment described in Example 7, Chapter 12, it is desired to determine also whether there is an interaction between the detergents and the engines; that is, whether one
An experiment was conducted to study the effects of temperature \(\left({ }^{\circ} \mathrm{C}\right)\) and quantity (g) on the solubility of a chemical in the laboratory.Analyze this experiment.
The commercial value of softwood species would be increased if the wood could be treated to meet preserver's standards. The response, \(y\), is the amount of retention \(\left(\mathrm{lb} /
A market test was performed to evaluate the impact of shelf position, and label color of a canned food product on sales.Analyze this experiment. Shelf Label Sales (dollars) Position Color Day 1 Day 2
Solve the 4 equations on page 426 for \(\mu, \alpha_{1}, \beta_{1}\), and \((\alpha \beta)_{11}\) in terms of the population means \(\mu_{i j l}\) corresponding to the 4 experimental conditions in
As a preliminary step in optimizing the coating process of iron oxide nanoparticles engineers explored the effects, of two factors each having two levels on the response \(y=\) increase in particle
Shape memory alloys can undergo a reversible phase transformation. These materials display dramatic shape memory temperature-induced deformations that are recoverable. Investigators want to evaluate
Given the following observations(a) Attach the sample means at the corner of a square. Comment on any obvious pattern.(b) Obtain the point estimates of the effects and \(99 \%\) confidence intervals.
Two factors are thought to influence the deposition rate (seconds) for a pulse laser to deposit one monolayer of material. Initially, a \(2^{2}\) design was run with two factors: spot size \(50
Tomatoes have one of highest production volumes in the world and drying is one major process for preservation. Color is an important quality index for consumers. Three factors of storage, each at two
An engineering student wanted to know which factors influence the time (in seconds) for his car to go from 0 to 30 to 0 miles per hour. Factor \(A\) was the launch, which was either no wheel spin or
The effect on engine wear of oil viscosity, temperature, and a special additive was tested using a \(2^{3}\) factorial design. Given the following results from the experiment,Interpret the effects
Two machines \(X\) and \(Y\) were used to produce two types of plastic polymers, PPET and PAMI. The polymers were produced using materials BA and PP. The production was run 3 times. The \(y\) values
The response variable \(Y_{i j}\) in a \(2^{2}\) design can also be expressed as a regression model\[Y_{i j}=\mu+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{12} x_{1} x_{2}+\varepsilon_{i j}\]where the
Refer to the Example 4. Use calculus to obtain the location of the estimated maximum yield when all terms are included in the model.Data From Example 4 EXAMPLE 4 A response surface design to maximize
MINITAB response surface analysis We illustrate the commands for the coating data in Example 4 where yield is the response. Start with the Run, Additive, Temperature, Yield in C1-C4. Dialog box:
Refer to Exercise 13.20. In Example 4, the experimenters also obtained the nine responses for adhesion. The business wants adhesion greater than 45 grams.\[\begin{array}{llllllllll} \text { Adhesion
Is there a region within the experimental region where estimated adhesion is greater than 45 grams? Construct a contour plot to show this region. Note that MINITAB does keep nonsignificant terms.
A footwear manufacturing machine manufactures each piece separately. Suppose pairs are manufactured, with the following results obtained for the range (number).Perform an appropriate analysis of
A study was conducted to measure the effect of 3 different meat tenderizers on the weight loss of steaks having the same initial (precooked) weights. The effects of cooking temperatures and cooking
An experiment was conducted to determine the effects of certain alloying elements on the ductility of a metal, and the following results were obtained:Perform an appropriate analysis of variance and
Given the two replicates of a \(2 \times 3\) factorial experiment, calculate the analysis of variance tables using the formulas on page 428 .Data From page 428 Factor A Factor B Rep. I Rep. 2 1 1 12
Given the following observations,Interpret the effects based on the confidence intervals. Factor P Factor Q Rep. I Rep. 2 1 18 12 -1 8 14 1 1 1 10 16 - 6 160 10
With reference to the example on page 443, suppose a third replicateis run. Analyze the experiment, using all 3 replicates, according to the visual procedure given in Section 13.3. Interpret the
Trouble was being experienced by a new high-tech machine for joining two pieces of sheet metal. The two factors considered first are the pressure (low/high) and temperature of the pump low/high. The
A computer engineer studied the working of a motherboard under different conditions. The response is heating effect (coded units). The factors are fan connectors (no. of pins), power connection
Given the following results from a \(2^{3}\) factorial experiment,Interpret the effects based on the confidence intervals. Factor A Factor B Factor C Rep. I Rep. 2 -1 13.8 14.6 10.8 8.4 1 9.0 9.8 1
The total sum of squares is given by \(\sum_{i=1}^{k} \sum_{j=1}^{r}\left(y_{i j}-\bar{y}\right)^{2}\) where the overall mean \(\bar{y}=\sum_{i=1}^{k} \sum_{j=1}^{r}\left(y_{i j} / n\right)\). With
With reference to the example of the \(2^{3}\) design on page \({ }^{* * *}\), express the total sum of squares as the sum of the contributions from each of the seven treatments plus the error sum of
With reference to the Example 3 concerning improvements in the safety of an ignitor, the time to reach maximum pressure was also recorded. Two replicates were run of the factorial design and the
Refer to Exercise 13.10 where the response is \(y=\) increase in particle size. Besides the first replicate, the investigators also performed the experiments that form the star part of the
In a factory, 20 observations of the factors that could heat up a conveyor belt yielded the following
The time sheet of a factory showed the following sample data (in hours) on the time spent by a worker operating a hydraulic gear lift:
With reference to Exercise 2.12, which pertained to the particle size of cement dust in a factory producing cement, use the sign test at the 0.05 level of significance to test the null hypothesis
The following are the number of classes attended by 2 students on 20 days: 3 and 5, 1 and 2, 3 and 4, 2 and 5, 5 and 3, 4 and 2, 1 and 3, 1 and 4, 1 and 2, 2 and 4, 3 and 2, 2 and 5, 5 and 5,1 and
Comparing two types of automobile engines, a consumer testing service obtained the following pickup ( \(0-100 \mathrm{kmph})\) times (rounded to the nearest tenth of a second):Use the \(U\) test at
The following are the self-reported times (hours for month), spent on homework, by random samples of juniors in two different majors.Use the \(U\) test at the 0.05 level of significance to test
The following are the data on the strength (in psi) of 2 kinds of adhesives:Use the \(U\) test at the 0.01 level of significance to test the claim that the strength of Adhesive 1 is stochastically
A company that processes health claims maintains three centers. Software was installed so they could monitor non-business internet usage by their employees. Initially, six employees were randomly
So-called Franklin tests were performed to determine the insulation properties of grain-oriented silicon steel specimens that were annealed in five different atmospheres with the following
A panel of 8 judges was asked to rate each of 3 models developed by engineering students on the likelihood that these models can be practically implemented to harness the controlled fusion energy.
The following arrangement indicates whether 60 consecutive cars which went by the toll booth of a bridge had local plates, \(L\), or out-of-state plates, \(O\) :Test at the 0.05 level of significance
The following are the graded scores (out of 20) obtained by a class of 28 students in statistics: 12,8,6,10,9,15,18,19,20,18,20,16,12,10,14,16,17,19,20,20,14,11,12,15,17,16,12, and 17. Test for
The following are 42 consecutive pizza breads baked by a newly improved oven model during 6
The \(P\)-value on page 476 was calculated using the \(R\) software command ks.test(x, "punif", 0,30 , alternative \(=\) "t")The following are 15 measurements of the boiling point of a silicon
In a vibration study, certain airplane components were subjected to severe vibrations until they showed structural failures. Given the following failure times (in minutes), test whether they can be
According to Einstein's theory of relativity, light should bend when it passes through a gravitational field. This was first tested experimentally in 1919 when photographs were taken of stars near
Referring to Exercise 12.6, use the \(U\) statistic at the 0.05 level of significance to test whether weight loss using lubricant A tends to be less than the loss using lubricant \(\mathrm{B}\).Data
To find the best order of tools on a factory workbench, two different orders were compared by simulating an operational condition and measuring the response time taken to respond to the condition
The following are the data on time taken by a computer engineer to assemble 8 computers each for 3 types of mother boards.Use the \(H\) test at the 0.05 level of significance to test whether there is
To test whether radio signals from deep space contain a message, an interval of time could be subdivided into a number of very short intervals and it could then be determined whether the signal
The total number of vehicles crossing a toll booth each day during the month of November were:Making use of the fact that the median is 328 , test at the 0.01 level of significance whether there is a
When two populations have the same probability density function, each outcome of \(n_{1}\) ranks for the first sample, out of the possible values \(1,2, \ldots, n_{1}+n_{2}\), is equally likely.(a)
With reference to Example 2, Chapter 2, use the \(U\) statistic to test the null hypothesis of equality versus the alternative that the distribution of copper content from the first heat is
The difference between the observed flux and the theoretical value was observed at 20 points within a reactor. The values wereUse a sign test at the 0.036 level to test the null hypothesis
With reference to Exercise 14.24, test for randomness with level 0.05 .Data From Exercise 14.24 14.24 The difference between the observed flux and the theo- retical value was observed at 20 points
Survival times (days) of fuel rods in a nuclear reactor are as follows:Test at the 0.01 level of significance whether these data are consistent with the assumption of a log-normal distribution of
A steel manufacturer extrudes scrap for manufacturing blades. Specifications require that the thickness of this scrap has \(\mu=0.020 \mathrm{~mm}\) and \(\sigma=0.005 \mathrm{~mm}\).(a) Use the
Calculate \(\overline{\bar{x}}\) and \(\bar{R}\) of the data of part (c) of Exercise 15.1, and use these values to construct the central lines and three-sigma control limits for new \(\bar{x}\) and
The following data give the means and ranges of 25 samples, each consisting of 4 compression test results on steel forgings, in thousands of pounds per square inch:(a) Use these data to find the
Reverse-current readings (in nanoamperes) are made at the location of a transistor on an integrated circuit. A sample of size 10 is taken every half hour. Since some of the units may prove to be
If the sample standard deviations instead of the sample ranges are used to estimate \(\sigma\), the control limits for the resulting \(\bar{x}\) chart are given by \(\overline{\bar{x}} \pm A_{1}
In order to establish control charts for an extraction job, 30 samples of five measurements of the extracted ores are taken, and the results are \(\overline{\bar{x}}=25.96\) tons and \(\bar{s}=1.5\)
Suppose that with the samples of Exercise 15.6, it is desired to establish control also over the variability of the process. Using the method of Exercise 15.5 and the values of \(\overline{\bar{x}}\)
Thirty-five consecutive samples of 100 bearings each, taken from a factory, had, respectively, 1, 2, 5, 3, 4, 2, 6, 8, 1, 2, 3, 9, 8, 0, 12, 10, 5, 4, 1, 8, 6, 7, 9, 4, 8, 1, 2, 6, 7, 5, 8, 1, 3, 4
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