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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

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)
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 \(\alpha\).

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