Questions and Answers of Introduction To Probability Statistics

The data of Exercise 15.8 may be looked upon as evidence that the standard of \(3 \%\) unusable bearings is being exceeded.(a) Use the data from Exercise 15.8 to construct new control limits for the
The specifications for a certain mass-produced valve prescribe a testing procedure according to which each valve can be classified as satisfactory or unsatisfactory (defective). Past experience has
The standard for a process producing tin plate in a continuous strip is 5 defects in the form of pinholes or visual blemishes per 100 feet. Based on the following set of 25 observations, giving the
A process for the manufacturer of 4-by-8-foot woodgrained panels has performed in the past with an average of 2.7 imperfections per 100 panels. Construct a chart to be used in the inspection of the
To check the strength of carbon steel for use in chain links, the yield stress of a random sample of 25 pieces was measured, yielding a mean and a standard deviation of 52,800 psi and 4,600 psi,
In a study designed to determine the number of turns required for an artillery-shell fuse to arm, 80 fuses, rotated on a turntable, average 45.6 turns with a standard deviation of 5.5 turns.
In a random sample of 50 electrodes, the mean diameter was \(0.4 \mathrm{~cm}\), and the standard deviation was \(0.005 \mathrm{~cm}\).(a) Between what limits can it be said with \(99 \%\) confidence
Nonparametric tolerance limits can be based on the extreme values in a random sample of size \(n\) from any continuous population. The following equation relates the quantities \(n, P\), and
The specifications require that the height of shock shafts have \(\mu=1.6\) meters and \(\sigma=0.03\) meters.(a) Use the specifications to calculate a central line and three-sigma control limits for
Calculate \(\overline{\bar{x}}\) and \(\bar{R}\) for the data of part (c) of Exercise 15.17 and use these values to construct the central lines and three-sigma control limits for new \(\bar{x}\) and
Twenty-five successive samples of 200 propellers, each taken from a production line, contained, respectively, 1, 8, 4, 6, 10, 7, 9, 5, 1, 0, 4, 8, 10, 3, 12, 5, 9, 16, 13, 7, 8, 4, 2, 9 and 2
The data of Exercise 15.19 may be looked upon as evidence that the standard of \(4 \%\) defectives is being exceeded.(a) Use the data of Exercise 15.19 to construct new control limits for the
A process for the manufacture of film has performed in the past with an average of 0.8 imperfections per 10 linear feet.(a) Construct a chart to be used in the inspection of 10 -foot sections.(b)
With reference to the aluminum alloy strength data on page 29 , obtain two-sided \(95 \%\) tolerance limits on the proportion \(P=0.90\) of the population of strengths.Data From Page 29 EXAMPLE 7
With reference to the interrequest time data on page 29 , obtain \(95 \%\) tolerance limits on the proportion \(P=0.90\) of the population of interrequest times. Take logs, use the normal theory
With reference to the discussion on page 492, calculate the CUSUM using 2.25 in place of 2.00 as the centering value. Also make the CUSUM chart.Data From Page 492 5 S -5- 0 5 5 10 15 20 CUSUM chart
With reference to Example 4.(a) verify the calculation of the tolerance bound \(L\);(b) if the confidence is decreased to \(90 \%\), calculate the new tolerance bound (use \(K=2.010)\)(c) check the
Explain, from the perspective of quality improvement programs, why the \(\bar{x}, R\), and fraction defective charts should be used to listen to the process and observe its natural variability, at
An experiment is performed to compare the rotational speed of two conveyers, Conveyer \(X\) and Conveyer \(Y\). 30 belts are loaded with an optimal weight, each is put on one of the conveyers, and
A certain bon vivant, wishing to ascertain the cause of his frequent hangovers, conducted the following experiment. On the first night, he drank nothing but whiskey and water; on the second night, he
Three alternatives are suggested for electroplating to reduce dissolved metal cations so that they form a coherent metal coating on an electrode. In an experiment conducted to compare the
Using the sum of squares obtained in Exercise 12.3, test at the level of significance \(\alpha=0.01\) whether the differences among the means obtained for the 3 samples are significant.Data From
The following are the numbers of mistakes made in 5 successive days for 4 technicians working for a photographic laboratory:Test at the level of significance \(\alpha=0.01\) whether the differences
With reference to the example on page 389 , suppose one additional observation \(y_{25}=8\) is available using formula \(B\). Construct the analysis of variance table and test the equality of the
Given the following observations collected according to the one-way analysis of variance design,(a) decompose each observation \(y_{i j}\) as\[y_{i
The one-way analysis of variance is conveniently implemented using MINITAB. With reference to the example on page 389 , we first set the observations in columns: DATA: C1: 13 10 C2: 13 11 C3: 4 1 14
To find the best arrangement of equipment at the rear of a fire truck, 3 different arrangements were tested by simulating a fire condition and observing the reaction time required to extinguish the
Several different aluminum alloys are under consideration for use in heavy-duty circuit-wiring applications. Among the desired properties is low electrical resistance, and specimens of each wire are
Two tests are made of the compressive strength of each of 6 samples of poured concrete. The force required to crumble each of 12 cylindrical specimens, measured in kilograms, is as follows:Test at
Corrosion rates (percent) were measured for 4 different metals that were immersed in a highly corrosive solution:(a) Perform an the analysis of variance and test for differences due to metals using
Assume that the standard deviations of the tin-coating weights determined by any one of the 4 laboratories have the common value \(\sigma=0.012\), and that it is desired to be \(95 \%\) confident
Show that if \(\mu_{i}=\mu+\alpha_{i}\) and \(\mu\) is the mean of the \(\mu_{i}\), it follows that\[\sum_{i=1}^{k} n_{i} \alpha_{i}=0\]
Verify the alternative formulas for computing \(S S T\) and \(S S(T r)\) given on page 399 .Data From Page 399 k ni SST=-C i=1 j=1 k SS(Tr) = ni i=1 where C, called the correction term for the mean,
With reference to Exercise 12.9, determine individual \(95 \%\) confidence intervals for the differences of mean reaction times.Data From Exercise 12.9 12.9 To find the best arrangement of equipment
Samples of peanut butter produced by 2 different manufacturers are tested for aflatoxin content, with the following results:(a) Use analysis of variance to test whether the two brands differ in
A randomized-block experiment is run with three treatments and four blocks. The three treatment means are \(\bar{y}_{1}=6, \bar{y}_{2 \bullet}=7\), and \(\bar{y}_{3 \bullet}=11\).The total
Concerns about the increasing friction between some machine parts prompted an investigation of four different types of ball bearings. Five different machines were available and each type of ball
The analysis of variance for a randomized-block design is conveniently implemented using MINITAB. With reference to Example 7, first open C12Ex7.MTW in the MINITAB data bank.Use computer software to
Looking at the days (rows) as blocks, rework Exercise 12.5 by the method of Section 12.3.Data From Exercise 12.5 12.5 The following are the numbers of mistakes made in 5 successive days for 4
Four different, though supposedly equivalent, forms of a standardized reading achievement test were given to each of 5 students, and the following are the scores which they obtained:Treating students
A laboratory technician measures the breaking strength of each of 5 kinds of linen thread by means of 4 different instruments and obtains the following results (in ounces):Looking at the threads as
An industrial engineer tests 4 different shop-floor layouts by having each of 6 work crews construct a subassembly and measuring the construction times (minutes) as follows:Test at the 0.01 level of
To emphasize the importance of blocking, reanalyze the cleanness data Example 7 as a one-way classification with the 4 detergents being the different treatments.
If, in a two-way classification, the entire experiment is repeated \(r\) times, the model becomes\[Y_{i j k}=\mu+\alpha_{i}+\beta_{j}+\varepsilon_{i j k}\]Data From Theorem 12.2 for i = 1, 2,..., a,
The following are the number of defectives produced by the 4 workers operating, in turn, 3 different machines. In each case, the first figure represents the number of defectives produced on a Friday
As was pointed out on page 308 , two ways of increasing the size of a two-way classification experiment are(a) to double the number of blocks,(b) to replicate the entire experiment. Discuss and
Show that if \(\mu_{i j}=\mu+\alpha_{i}+\beta_{j}\), the mean of the \(\mu_{i j}\) (summed on \(j\) ) is equal to \(\mu+\alpha_{i}\), and the mean of \(\mu_{i j}\) (summed on \(i\) and \(j\) ) is
Verify that the computing formulas for \(S S T, S S(T r)\), \(S S(B l)\), and \(S S E\), given on page 409 , are equivalent to the corresponding terms of the identity of Theorem 12.2.Data From
Using \(q_{0.01}=5.499\) for the Tukey HSD method, examine differences among the ball bearings in Exercise 12.19.Data From Exercise 12.19 12.19 Concerns about the increasing friction between some
Referring to Example 1, use \(q_{0.05}=3.776\) for the Tukey HSD method to examine differences in mean coating weights.Data From Example 1 EXAMPLE I Conducting a one-way analysis of variance To
Referring to Exercise 12.3, use Bonferroni simultaneous confidence intervals with \(\alpha=0.06\) to compare the mean number of electrodes coated by the experiment under the 3 different
An approximation to Tukey HSD confidence intervals for mildly unequal sample sizes When the there are only small differences in the sample sizes, an approximation is available. The MSE and its
(a) Using \(q_{0.10}=3.921\) for the Tukey HSD method, compare the strength of the 5 linen threads in Exercise 12.23 .(b) Use the Bonferroni confidence interval approach on page 411, with
The Bonferroni inequality states that\[P\left(\cap_{i} C_{i}\right) \geq 1-\sum_{i} P\left(\overline{C_{i}}\right)\](a) Show that this holds for 3 events.(b) Let \(C_{i}\) be the event that the \(i\)
An experimenter wants to compare the time to failure \(y\) after rebuilding a robotic welder by three different methods but adjusting for the covariate \(x=\) age of robotic welder.Suppose the data,
MINITAB calculation of balanced analysis of covariance We illustrate the MINTAB commands for Example 11 concerning surface reflectivity.Use computer software to perform an analysis of covariance for
Four different railroad-track cross-section configurations were tested to determine which is most resistant to breakage under use conditions. Ten miles of each kind of track were laid in each of 5
To compare power consumption by a newly developed low-power USB device under three test run conditions and record the data transmission rate at the same time, a computer engineer obtained the
Use computer software to work Exercise 12.37.Data From Exercise 12.37 12.37 An experimenter wants to compare the time to failure y after rebuilding a robotic welder by three different methods but
Assume the following data obey the one-way analysis of variance model.(a) Decompose each observation \(y_{i j}\) as \[y_{i j}=\bar{y}+\left(\bar{y}_{i}-\bar{y}\right)+\left(y_{i
To determine the effect of height on power generated in a hydroelectric power plant, the following observations were made:.Use the level of significance \(\alpha=0.05\) to test whether the height of
Refer to Example 11. Ignore the covariate original reflectivity. Perform an analysis of variance take \(\alpha=0.05\)Data From Example 11 EXAMPLE II Creating an analysis of covariance table with one
Given the following data from a randomized block design,(a) Decompose each observation \(y_{i j}\) as \[\begin{aligned} y_{i j}= & \bar{y}_{\bullet}+\left(\bar{y}_{\bullet
Using \(q_{0.05}=4.339\) for the Tukey HSD method, compare the treatments in Exercise 12.45.Data From Exercise 12.45 12.45 Given the following data from a randomized block design, Blocks 1 234
Samples of groundwater were taken from 5 different toxic-waste dump sites by each of 3 different agencies: the EPA, the company that owned each site, and an independent consulting engineer. Each
Using \(q_{0.05}=4.041\) for the Tukey HSD method, compare the pollution levels of the three agencies in Exercise 12.47.Data From Exercise 12.47 12.47 Samples of groundwater were taken from 5
An experiment is conducted with \(k=5\) treatments, one covariate \(x\), and \(n=6\). Calculations result in the sums of squares error \(S S E_{x}=14.4, S S E_{t r}=4.69\), and \(S S E_{t r,
The state highway department does an experiment to compare three types of surfacing treatments and the response \(y\) is road roughness. The following table also gives average daily traffic volume
Refer to Exercise 12.50.(a) Perform an analysis of covariance. Test for a difference in treatments using level of significance 0.05 .(b) Compare your analysis in part (a) with the analysis of
Three different instrument panel configurations were tested by placing airline pilots in flight simulators and testing their reaction time to simulated flight emergencies. Eight pilots were assigned
Using the alternative calculation formula verify analysis of variance table for the paper-strength in Example 4.Data From Example 4 EXAMPLE 4 Confidence intervals quantify the amount of difference
Benjamin Franklin (1706-1790) conducted an experiment to study the effect of water depth on the amount of drag on a boat being pulled up a canal. He made a 14-foot trough and a model boat 6 inches
A chemical engineer found that by adding different amounts of an additive to gasoline, she could reduce the amount of nitrous oxides (NOx) coming from an automobile engine. A specified amount will be
A motorist found that the efficiency of her engine could be increased by adding lubricating oil to fuel. She experimented with different amounts of lubricating oil and the data are(a) Obtain the
A textile company, wanting to know the effect of temperature on the tearing strength of a fiber, obtained the data shown in the following table.(a) Draw a scatter plot to verify that a straight line
In the accompanying table, \(x\) is the tensile force applied to a steel specimen in thousands of pounds, and \(y\) is the resulting elongation in thousandths of an inch:(a) Graph the data to verify
With reference to the preceding exercise,(a) construct a \(95 \%\) confidence interval for \(\beta\), the elongation per thousand pounds of tensile stress;(b) find 95% limits of prediction for the
The following table shows how many days in December a sample of 6 students were present at their university and the number of lectures each attended on a given day.(a) Find the equation of the least
With reference to the preceding exercise, test the null hypothesis \(\beta=0.75\) against the alternative hypothesis \(\beta
With reference to Exercise 11.6, find(a) a \(90\%\) confidence interval for the average number of classes attended each day by a student present for 15 days;(b) \(90 \%\) limits of prediction for the
Scientists searching for higher performance flexible structures created a diode with organic and inorganic layers. It has excellent mechanical bending properties. Applying a bending strain to the
With reference to Exercise 11.9, construct a 95% confidence interval for \(\alpha\).Data From Exercise 11.9 11.9 Scientists searching for higher performance flexible structures created a diode with
With reference to Exercise 11.9, test the null hypothesis \(\beta=0.40\) against the alternative hypothesis \(\beta>\) 0.40 at the 0.05 level of significance.Data From Exercise 11.9 11.9
The level of pollution because of vehicular emissions in a city is not regulated. Measurements by the local government of the change in flow of vehicles and the change in the level of air pollution
With reference to the preceding exercise, find \(99 \%\) limits of prediction for the level of air pollution when the flow of vehicles is \(30 \%\). Also indicate to what extent the width of the
With reference to Exercise 11.3, express \(90 \%\) limits of prediction for the tearing strength in terms of the temperature \(x_{0}\). Choosing suitable values of \(x_{0}\), sketch graphs of the
In Exercise 11.4 it would have been entirely reasonable to impose the condition \(\alpha=0\) before fitting a straight line by the method of least squares.(a) Use the method of least squares to
Recycling concrete aggregate is an important component of green engineering. The strength of any potential material, expressed in terms of its resilient modulus, must meet standards before it is
With reference to Exercise 11.16, find a \(90 \%\) confidence interval for \(\alpha\).Data From Exercise 11.16 11.16 Recycling concrete aggregate is an important compo- nent of green engineering. The
With reference to Exercise 11.16, fit a straight line to the data by the method of least squares, using \(M_{\text {ext }}\) as the independent variable, and draw its graph on the diagram obtained in
When the sum of the \(x\) values is equal to zero, the calculation of the coefficients of the regression line of \(Y\) on \(x\) is greatly simplified; in fact, their estimates are given
Using the formulas on page 330 for \(\widehat{\alpha}\) and \(\widehat{\beta}\), show that(a) the expression for \(\widehat{\alpha}\) is linear in the \(Y_{i}\)(b) \(\widehat{\alpha}\) is an unbiased
The decomposition of the sums of squares into a contribution due to error and a contribution due to regression underlies the least squares analysis. Consider the
It is tedious to perform a least squares analysis without using a computer. We illustrate here a computer-based analysis using the MINITAB package. The observations on page 328 are entered in
Referring to Example 3, the nanopillar data on height (nm) and width ( \(\mathrm{nm})\) are(a) Fit a straight line with \(y=\) height and \(x=\) width by least squares.(b) Test, with \(\alpha=0.05\),
Nanowires, tiny wires just a few millionths of a centimeter thick, which spiral and have a pine tree-like appearance, have recently been created. \({ }^{2}\) The investigators' ultimate goal is to
The following data pertain to the growth of a colony of bacteria in a culture medium:(a) Plot \(\log y_{i}\) versus \(x_{i}\) to verify that it is reasonable to fit an exponential curve.(b) Fit an
The following data pertain to water pressure at various depths below sea level:(a) Fit an exponential curve.(b) Use the result obtained in part (a) to estimate the mean pressure at a depth of \(1,000
With reference to the preceding exercise, change the equation obtained in part (a) to the form \(\widehat{y}=a \cdot e^{-c x}\), and use the result to rework part (b).
Refer to Example 10. Two new observations are available.Add these observations to the data set in Example 10 and rework the example.Data From Example 10 Rate of discharge (A) Capacity (Ah)

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