Questions and Answers of Introduction To Probability Statistics

Fit a Gompertz curve of the form \[y=e^{e^{\alpha x+\beta}}\] to the data of Exercise 11.26.Data From Exercise 11.26 11.26 The following data pertain to water pressure at various depths below sea
Plot the curve obtained in the preceding exercise and the one obtained in Exercise 11.26 on one diagram and compare the fit of these two curves.Data From Exercise 11.26 11.26 The following data
The number of inches which a newly built structure is settling into the ground is given by\[y=3-3 e^{-\alpha x}\]where \(x\) is its age in months.Use the method of least squares to estimate
The following data pertain to the amount of hydrogen present, \(y\), in parts per million in core drillings made at 1 -foot intervals along the length of a vacuum-cast ingot, \(x\), core location in
When fitting a polynomial to a set of paired data, we usually begin by fitting a straight line and using the method on page 339 to test the null hypothesis \(\beta_{1}=0\). Then we fit a
With reference to Example 11, verify that the predicted drying time is minimum when the amount of additive used is 5.1 grams.Data From Example 11 EXAMPLE II Fitting a quadratic function by the method
Verify that the system of normal equations on page 357 corresponds to the minimization of the sum of squares. EXAMPLE 12 A multiple regression with two predictor variables The following are data on
Twelve specimens of cold-reduced sheet steel, having different copper contents and annealing temperatures, are measured for hardness with the following results:Fit an equation of the form
With reference to Exercise 11.36, estimate the hardness of a sheet of steel with a copper content of \(0.05 \%\) and an annealing temperature of 1,150 degrees Fahrenheit.Data From Exercise 11.36
A compound is produced for a coating process. It is added to an otherwise fixed recipe and the coating process is completed. Adhesion is then measured. The following data concern the amount of
The following sample data were collected to determine the relationship between processing variables and the current gain of a transistor in the integrated circuit:Fit a regression plane and use its
Multiple regression is best implemented on a computer. The following MINITAB commands fits the \(y\) values in \(\mathrm{C} 1\) to predictor values in \(\mathrm{C} 2\) and \(\mathrm{C} 3\).It
Using MINITAB we can transform the \(x\) values in \(\mathrm{C} 1\) and/or the \(y\) values in \(\mathrm{C} 2\). For instance, to obtain the logarithm to the base 10 of \(y\), selectUse the computer
To fit the quadratic regression model using MINITAB, when the \(x\) values are in C1 and the \(y\) values in \(\mathrm{C} 2\), you must selectUse the computer to repeat the analysis of Example
With reference to Exercise 11.40, in order to plot residuals, before clicking \(\mathbf{O K}\), you must selectThe additional steps, before clicking the second \(\mathbf{O K}\).will produce a plot of
With reference to Exercise 11.39, analyze the residuals from the regression plane.Data From Exercise 11.39 11.39 The following sample data were collected to determine the relationship between
The following residuals and predicted values were obtained from an experiment that related yield of a chemical process \((y)\) to the initial concentration \((x)\) of a component (the time order of
Data, collected over seven years, reveals a positive correlation between the annual starting salary of engineers and the annual sales of diet soft drinks. Will buying more diet drinks increase
Data, collected from cities of widely varying sizes, revealed a high positive correlation between the amount of beer consumed and the number of weddings in the past year. Will consuming lots of beer
Use the expressions on page 367, involving the deviations from the mean, to calculate \(r\) for the following data: x Y 8278 69 128 10
Calculate \(r\) for the air velocities and evaporation coefficients of Example 2. Also, assuming that the necessary assumptions can be met, test the null hypothesis \(ho=0\) against the alternative
The following data pertain to the processing speed (GHz) of a computer and the time (minutes) it takes to boot up:Calculate \(r\). Processing Speed Boot Time 1.2 5 1.0 1 1.3 1.6 2 1.8 324 3 1.1 8 0.8
With reference to Exercise 11.50, test \(ho=0\) against \(ho eq 0\) at \(\alpha=0.05\).Data From Exercise 11.50 11.50 The following data pertain to the processing speed (GHz) of a computer and the
Calculate \(r\) for the temperatures and tearing strengths of Exercise 11.3. Assuming that the necessary assumptions can be met, test the null hypothesis \(ho=0.60\) against the alternative
Calculate \(r\) for the changes in the flow of vehicles and the level of pollution in Exercise 11.12. Assuming that the necessary assumptions can be met, construct a 99% confidence interval for the
The following are measurements of the total dissolved salts (TDS) and hardness index of 22 samples of water.(a) Calculate \(r\).(b) Find 95% confidence limits for \(ho\). TDS (ppm) Hardness Index 200
Referring to Example 3 concerning nanopillars, calculate the correlation coefficient between height and width.Data From Example 3 EXAMPLE 3 One scatter plot but two different fitted lines Engineers
If \(r=0.83\) for one set of paired data and \(r=\) 0.60 for another, compare the strengths of the two relationships.
If data on the ages and prices of 25 pieces of equipment yielded \(r=-0.58\), test the null hypothesis \(ho=-0.40\) against the alternative hypothesis \(ho
Assuming that the necessary assumptions are met, construct a 95% confidence interval for \(ho\) when(a) \(r=0.72\) and \(n=19\);(b) \(r=0.35\) and \(n=25\);(c) \(r=0.57\) and \(n=40\).
(a) Evaluating the necessary integrals, verify the identities\[\mu_{2}=\alpha+\beta \mu_{1} \quad \text { and } \quad \sigma_{2}^{2}=\sigma^{2}+\beta^{2} \sigma_{1}^{2}\]on page 374 .(b) Substitute
Show that for the bivariate normal distribution(a) independence implies zero correlation;(b) zero correlation implies independence.
Instead of using the computing formula on page 367, we can obtain the correlation coefficient \(r\) with the formula\[r= \pm \sqrt{1-\frac{\sum(y-\widehat{y})^{2}}{\sum(y-\bar{y})^{2}}}\]which is
With reference to Exercise 11.39, use the theory of the preceding exercise to calculate the multiple correlation coefficient (which measures how strongly the current gain is related to the two
Referring to the nano twisting data in Exercise 11.24, calculate the correlation coefficient.Data From Exercise 11.24 11.24 Nanowires, tiny wires just a few millionths of a centimeter thick, which
To calculate \(r\) using MINITAB when the \(x\) values are in \(C 1\) and the \(y\) values are in \(C 2\), useAlso, you can make a scatter plot using the plot procedure in Exercise 11.22.Use the
The data below pertains to the number of hours a laptop has been charged for and the number of hours of backup provided by the battery.(a) Use the first set of expressions on page 330, involving
With reference to Exercise 11.65, construct a \(99 \%\) confidence interval for \(\alpha\).
With reference to Exercise 11.65, test the null hypothesis \(\beta=1.5\) against the alternative hypothesis \(\beta>1.5\) at the 0.01 level of significance.Data From Exercise 11.65 11.65 The data
With reference to Exercise 11.65,(a) find a \(99 \%\) confidence interval for the mean battery backup at \(x=1.25\);(b) find \(95 \%\) limits of prediction for the battery backup provided by a laptop
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 was
With reference to Exercise 11.69, find the 95% limits of prediction when the amount of additive is 4.5.Data From Exercise 11.69 11.69 A chemical engineer found that by adding different amounts of an
With reference to Exercise 11.69, find the proportion of variance in the amount of NOx explained by the amount of additive.Data From Exercise 11.69 11.69 A chemical engineer found that by adding
To determine how well existing chemical analyses can detect lead in test specimens in water, a civil engineer submits specimens spiked with known concentrations of lead to a laboratory. The chemists
With reference to the preceding exercise, construct a 95% confidence interval for \(\alpha\).
With reference to Example 15,(a) find the least squares line for predicting the chromium in the effluent from that in the influent after taking natural logarithms of each variable;(b) predict the
In an experiment designed to determine the specific heat ratio \(\gamma\) for a certain gas, measurements of the volume and corresponding pressure \(p\) produced the data:Assuming the ideal gas law
With reference to Exercise 11.75, use the method of Section 11.2 to construct a \(95 \%\) confidence interval for \(\gamma\). State what assumptions will have to be made.Data From Exercise 11.75
The rise of current in an inductive circuit having the time constant \(\tau\) is given by\[I=1-e^{-t / \tau}\]where \(t\) is the time measured from the instant the switch is closed, and \(I\) is the
The following are sample data provided by a moving company on the weights of six shipments, the distances they are moved, and the damage that was incurred:(a) Fit an equation of the form
With reference to Exercise 11.9,(a) find a 95% confidence interval for the mean current density when the strain is \(x=0.50\);(b) find 95% limits of prediction for the current density when a new
Use the expression on page 367, involving deviations from the mean, to calculate \(r\) for the following data: x Y 3 8 6 367 -205
If \(r=0.41\) for one set of paired data and \(r=0.29\) for another, compare the strengths of the two relationships.
If for certain paired data \(n=18\) and \(r=0.44\), test the null hypothesis \(ho=0.30\) against the alternative hypothesis \(ho>0.30\) at the 0.01 level of significance.
Assuming that the necessary assumptions are met, construct a 95\% confidence interval for \(ho\) when(a) \(r=0.78\) and \(n=15\);(b) \(r=-0.62\) and \(n=32\);(c) \(r=0.17\) and \(n=35\).
With reference to Exercise 11.78, use the theory of Exercise 11.61 to calculate the multiple correlation coefficient (which measures how strongly the damage is related to both weight and
Robert A. Millikan (1865-1953) produced the first accurate measurements on the charge \(e\) of an electron. He devised a method to observe a single drop of water or oil under the influence of both
Robert Boyle (1627-1691) established the law that (pressure \(\times\) volume) \(=\) constant for a gas at a constant temperature. By pouring mercury into the open top of the long side of a
In a random sample of 150 complaints filed against a construction company for mixing excess sand in their concrete mixture, 95 complaints showed that the proportion of sand in the mix exceeded 75
With reference to Exercise 10.1, what can we say with \(95 \%\) confidence about the maximum error if we use the sample proportion as an estimate of the true proportion of complaints filed against
In a random sample of 400 industrial accidents, it was found that 231 were due at least partially to unsafe working conditions. Construct a \(99 \%\) confidence interval for the corresponding true
With reference to Exercise 10.3, what can we say with \(95 \%\) confidence about the maximum error if we use the sample proportion to estimate the corresponding true proportion?Data From Exercise
In a random sample of 140 observations of workers on a site, 25 were found to be idle. Construct a \(99 \%\) confidence interval for the true proportion of workers found idle, using the large sample
In an experiment, 85 of 125 processors were observed to process data at a speed of 4,700 MIPS. If we estimate the corresponding true proportion as \(\frac{85}{125}=0.68\), what can we say with \(99
Among 100 fish caught in a large lake, 18 were inedible due to the pollution of the environment. If we use \(\frac{18}{100}=0.18\) as an estimate of the corresponding true proportion, with what
New findings suggest many persons possess symptoms of motion sickness after watching a 3D movie. One scientist administered a questionnaire to \(n=451\) adults after they watched a 3D movie of their
What is the size of the smallest sample required to estimate an unknown proportion of customers who would pay for an additional service, to within a maximum error of 0.06 with at least \(95 \%\)
With reference to Exercise 10.9, how would the required sample size be affected if it is known that the proportion to be estimated is at least 0.75 ?Data From Exercise 10.9 10.9 What is the size of
Suppose that we want to estimate what percentage of all bearings wears out due to friction within a year of installation. How large a sample will we need to be at least \(90 \%\) confident that the
Refer to Example 1. How large a sample of wind turbines is needed to ensure that, with at least \(95 \%\) confidence, the error in our estimate of the sample proportion is at most 0.06 if(a) nothing
MINITAB determination of confidence interval for \(p\)When the sample size is not large, the confidence interval for a proportion \(p\) can be obtained using the following commands. We illustrate the
Use Exercise 10.13 or other software to obtain the interval requested in Exercise 10.3.Data From Exercise 10.13 10.13 MINITAB determination of confidence interval for p When the sample size is not
Show that the inequality on page 304 leads to the following \((1-\alpha) 100 \%\) confidence limits:\[\frac{x+\frac{1}{2} z_{\alpha / 2}^{2} \pm z_{\alpha / 2} \sqrt{\frac{x(n-x)}{n}+\frac{1}{4}
Use the formula of Exercise 10.15 to rework Exercise 10.3.Data From Exercise 10.3Data From Exercise 10.15 10.3 In a random sample of 400 industrial accidents, it was found that 231 were due at least
A chemical laboratory was facing issues with the concentration of the sulfuric acid they prepared. The first step was to collect data on the magnitude of the problem. Of 5,186 recently supplied acid
An international corporation needed several millions of words, from thousands of documents and manuals, translated. The work was contracted to a company that used computer-assisted translation, along
A manufacturer of submersible pumps claims that at most \(30 \%\) of the pumps require repairs within the first 5 years of operation. If a random sample of 120 of these pumps includes 47 which
A supplier of imported vernier calipers claims that \(90 \%\) of their instruments have a precision of 0.999. Testing the null hypothesis \(p=0.90\) against the alternative hypothesis \(p eq 0.90\),
To check on an ambulance service's claim that at least \(40 \%\) of its calls are life-threatening emergencies, a random sample was taken from its files, and it was found that only 49 of 150 calls
In a random sample of 600 cars making a right turn at a certain intersection, 157 pulled into the wrong lane. Test the null hypothesis that actually \(30 \%\) of all drivers make this mistake at the
An airline claims that only \(6 \%\) of all lost luggage is never found. If, in a random sample, 17 of 200 pieces of lost luggage are not found, test the null hypothesis \(p=0.06\) against the
Suppose that 4 of 13 undergraduate engineering students are going on to graduate school. Test the dean's claim that \(60 \%\) of the undergraduate students will go on to graduate school, using the
A manufacturer of machine bearings claims that \(90 \%\) of the heavy machine bearings have a work life of more than 5 years. You doubt this claim and want to refute it on the basis of a sample of
Refer to Exercise 10.25. Suppose a sample of 650 moderate machine bearings yielded 550 bearings that had a work life of more than 5 years. Obtain a \(90 \%\) confidence interval for the difference in
Tests are made on the proportion of defective castings produced by 5 different molds. If there were 14 defectives among 100 castings made with Mold I, 33 defectives among 200 castings made with Mold
A study showed that 64 of 180 persons who saw a photocopying machine advertised during the telecast of a baseball game and 75 of 180 other persons who saw it advertised on a variety show remembered
The following data come from a study in which random samples of the employees of three government agencies were asked questions about their pension plan:Use the 0.01 level of significance to test the
A factory owner must decide which of two alternative electric generator systems should be installed in their factory. If each generator is tested 175 times and the first generator fails to work (does
With reference to the preceding exercise, verify that the square of the value obtained for \(Z\) in part (b) equals the value obtained for \(\chi^{2}\) in part (a).
Photolithography plays a central role in manufacturing integrated circuits made on thin disks of silicon. Prior to a quality-improvement program, too many rework operations were required. In a sample
With reference to Exercise 10.32, find a large sample 99% confidence interval for the true difference of the proportions.Data From Exercise 10.32 10.32 Photolithography plays a central role in
To test the null hypothesis that the difference between two population proportions equals some constant \(\delta_{0}\), not necessarily 0 , we can use the statisticwhich, for large samples, is a
With reference to part (b) of Exercise 10.34, find a large sample \(99 \%\) confidence interval for the true difference of the proportions.Data From Exercise 10.34 10.34 To test the null hypothesis
Verify that the formulas for the \(\chi^{2}\) statistic on page 311 (with \(\widehat{p}\) substituted for the \(p_{i}\) ) and on page 312 are equivalent. EXAMPLE 8 Testing the equality of three
Verify that if the expected frequencies are determined in accordance with the rule on page 312 , the sum of the expected frequencies for each row and column equals the sum of the corresponding
Verify that the square of the \(Z\) statistic on page 314 equals the \(\chi^{2}\) statistic on page 312 for \(k=2\). Statistic for test concerning difference between two proportions X1 X2 n1 - n2 Z =
Referring to Example 12 and the data on repair, use the 0.05 level of significance to test whether there is homogeneity among the shops' repair distributions.Data From Example 12 EXAMPLE 12
A large electronics firm that hires many workers with disabilities wants to determine whether their disabilities affect such workers' performance. Use the level of significance \(\alpha=0.05\) to
Tests of the fidelity and the selectivity of 190 digital radio receivers produced the results shown in the following table:Use the 0.01 level of significance to test whether there is a relationship
An engineer takes samples on a daily basis of \(n=5\) cars coming to a workshop to be checked for repairs and on 250 consecutive days the data summarized in the following table are obtained:To test

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