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statistics

Questions and Answers of Statistics

Refer to the data in Exercise 11-5 on house selling price y and taxes paid x. y(a) Find the residuals for the least squares model.(b) Prepare a normal probability plot of the residuals and interpret
Exercise 11-6 presents data on y = steam usage and x = average monthly temperature.(a) What proportion of total variability is accounted for by the simple linear regression model?(b) Prepare a normal
Refer to the gasoline mileage data in Exercise 11-7.(a) What proportion of total variability in highway gasoline mileage performance is accounted for by engine displacement?(b) Plot the residuals
Consider the data in Exercise 11-8 on y = green liquor Na2S concentration and x = paper machine production. Suppose that a 14th sample point is added to the original data, where y14 = 59 and x14 =
Refer to Exercise 11-9, which presented data on blood pressure rise y and sound pressure level x(a) What proportion of total variability in blood pressure rise is accounted for by sound pressure
Exercise 11-10 presents data on wear volume y and oil viscosity x.(a) Calculate R2 for this model. Provide an interpretation of this quantity.(b) Plot the residuals from this model versus and versus
Refer to Exercise 11-11, which presented data on chloride concentration y and roadway area x(a) What proportion of the total variability in chloride concentration is accounted for by the regression
Consider the rocket propellant data in Exercise 11-12. (a) Calculate R2 for this model. Provide an interpretation of this quantity.(b) Plot the residuals on a normal probability scale. Do any points
Show that an equivalent way to define the test for significance of regression in simple linear regression is to base the test on R2 as follows: to test H0: B1 = 0 versus H1: B1 = 0, calculateAnd to
Suppose that a simple linear regression model has been fit to n= 25 observations and R2 = 0.90.(a) Test for significance of regression at a = 0.05. Use the results of Exercise 11-51.(b) What is the
Consider the rocket propellant data in Exercise 11-12. Calculate the standardized residuals for these data. Does this provide any helpful information about the magnitude of the residuals?
Studentized Residuals show that the variance of the ith residual isThe ithe Studentized residual is defined as(a) Explain why ri has unit standard deviation.(b) Do the standardized residuals have
The final test and exam averages for 20 randomly selected students taking a course in engineering statistics and a course in operations research follow. Assume that the final averages are jointly
The weight and systolic blood pressure of 26 randomly selected males in the age group 25 to 30 are shown in the following table. Assume that weight and blood pressure are jointly normally
Consider the NFL data introduced in Exercise 11-4.(a) Estimate the correlation coefficient between the number of games won and the yards rushing by the opponents.(b) Test the hypothesis H0: p = 0
Show that the t-statistic in Equation 11-46 for testing H0: p = 0 is identical to the t-statistic for testing H0: B1 = 0.
A random sample of 50 observations was made on the diameter of spot welds and the corresponding weld shear strength.(a) Given that r = 0.62, test the hypothesis that p = 0, using a = 0.01. What is
Suppose that a random sample of 10,000 (X, Y) pairs yielded a sample correlation coefficient of r = 0.02.(a) What is the conclusion that you would reach if you tested H0:p = 0 using a 0.05? What is
The following data gave X = the water content of snow on April 1 and Y = the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from
A random sample of n = 25 observations was made on the time to failure of an electronic component and the temperature in the application environment in which the component was used. (a) Given that r
Show that, for the simple linear regression model, the following statements are true:
An article in the IEEE Transactions on Instrumentation and Measurement (€œDirect, Fast, and Accurate Measurement of VT and K of MOS Transistor Using VT-Sift Circuit,€ Vol. 40,
The strength of paper used in the manufacture of cardboard boxes (y) is related to the percentage of hardwood concentration in the original pulp (x). Under controlled conditions, a pilot plant
The vapor pressure of water at various temperatures follows:(a) Draw a scatter diagram of these data. What type of relationship seems appropriate in relating y to x?(b) Fit a simple linear regression
An electric utility is interested in developing a model relating peak hour demand (y in kilowatts) to total monthly energy usage during the month (x, in kilowatt hours). Data for 50 residential
Consider the following data. Suppose that the relationship between Y and x is hypothesized to be Y = (B0 + B1x +έ)–1. Fit an appropriate model to the data. Does the assumed model form seem
Consider the weight and blood pressure data in Exercise 11-56. Fit a no-intercept model to the data, and compare it to the model obtained in Exercise 11-56. Which model is superior?
The following data, adapted from Montgomery, Peck, and Vining (2001), present the number of certified mental defectives per 10,000 of estimated population in the United Kingdom (y) and the number of
An article in Air and Waste (€œUpdate on Ozone Trends in California€™s South Coast Air Basin,€ Vol. 43, 1993) studied the ozone levels on the South Coast air basin of
An article in the Journal of Applied Polymer Science (Vol. 56, pp. 471€“476, 1995) studied the effect of the mole ratio of sebacic acid on the intrinsic viscosity of co-polyesters. The data
Suppose that we have n pairs of observations (xi, yi) such that the sample correlation coefficient r is unity (approximately). Now let zi = y2i and consider the sample correlation coefficient for the
The gram of solids removed from a material (y) is thought to be related to the drying time. Ten observations obtained from an experimental study follow:(a) Construct a scatter diagram for these
Two different methods can be used for measuring the temperature of the solution in a Hall cell used in aluminum smelting, a thermocouples implanted in the cell and an indirect measurement produced
Consider the simple linear regression model Y = B0 + B1 + έ, with E(έ) = 0, V(έ) = σ2, and the errors έ uncorrelated. (a) Show that cov (B0, B1) = - xo2/Sxx. (b) Show
Consider the simple linear regression model Y = B0 + B1x + έ, with E(έ) = 0, V(έ) = σ2, and the errors έ uncorrelated. (a) Show that E(σ2) = E(MSE) = σ2. (b)
Suppose that we have assumed the straight-line regression model Y = BO + B1X1 + έ but the response is affected by a second variable x2 such that the true regression function is E(Y) = BO + B1X1
Suppose that we are fitting a line and we wish to make the variance of the regression coefficient B1 as small as possible. Where should the observations xi, i = 1, 2, p, n, be taken so as to minimize
Weighted Least Squares suppose that we are fitting the line Y = Bo + B1x + , but the variance of Y depends on the level of x; that is, where the wi are constants, often called weights. Show that for
Consider a situation where both Y and X are random variables. Let sx and sy be the sample standard deviations of the observed x’s and y’s, respectively show that an alternative expression for the
Suppose that we are interested in fitting a simple linear regression model Y = B0 + B1x + έ, where the intercept, B0, is known. (a) Find the least squares estimator of B1. (b) What is the
A study was performed to investigate the shear strength of soil (y) as it related to depth in feet (x1) and moisture content (x2). Ten observations were collected, and the following summary
A regression model is to be developed for predicting the ability of soil to absorb chemical contaminants. Ten observations have been taken on a soil absorption index (y) and two regressors: x1 =
A chemical engineer is investigating how the amount of conversion of a product from a raw material (y) depends on reaction temperature (x1) and the reaction time (x2). He has developed the following
The data in Table 12-5 are the 1976 team performance statistics for the teams in the National Football League (Source: The Sporting News). (a) Fit a multiple regression model relating the number of
y: Games won (per 14 game season)x1: Rushing yards (season)x2: Passing yards (season)x3: Punting yards (yds/punt)x4: Field goal percentage (Field goals made/Field goals
The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature (x1), the number of days in the month (x2), the average product purity (x3), and
A study was performed on wear of a bearing y and its relationship to x1 = oil viscosity and x2 = load. The following data were obtained.(a) Fit a multiple linear regression model to these data.(b)
The pull strength of a wire bond is an important characteristic. The following table gives information on pull strength (y), die height (x1), post height (x2), loop height (x3), wire length (x4),
An engineer at a semiconductor company wants to model the relationship between the device HFE (y) and three parameters: Emitter-RS (x1), Base-RS (x2), and Emitter-to-Base RS (x3). The data are shown
Heat treating is often used to carburize metal parts, such as gears. The thickness of the carburized layer is considered a crucial feature of the gear and contributes to the overall reliability of
Statistics for 21 National Hockey League teams were obtained from the Hockey Encyclopedia and are shown in Table 12-8. The variables and definitions are as follows: Wins Number of games won in a
Consider the linear regression model where and where x1 = Σxi1/n and x2 = Σxi2/n. (a) Write out the least squares normal equations for this model. (b) Verify that the least squares
Consider the regression model fit to the soil shear strength data in Exercise 12-1.(a) Test for significance of regression using a = 0.05. What is the P-value for this test?(b) Construct the t-test
Consider the absorption index data in Exercise 12-2. The total sum of squares for y is SST = 742.00. (a) Test for significance of regression using a = 0.01. What is the P-value for this test?(b) Test
Consider the NFL data in Exercise 12-4.(a) Test for significance of regression using a = 0.05. What is the P-value for this test?(b) Conduct the t-test for each regression coefficient B2, B7, and B8.
Reconsider the NFL data in Exercise 12-4.(a) Find the amount by which the regressor x8 (opponents’ yards rushing) increases the regression sum of squares.(b) Use the results from part (a) above and
Consider the gasoline mileage data in Exercise 12-5.(a) Test for significance of regression using a = 0.05. What conclusions can you draw?(b) Find the t-test statistic for both regressors. Using a =
A regression model Y = B0 + B1x1 +B2x2 + B3x3 + έ has been fit to a sample of n = 25 observations. The calculated t-ratios are as follows: for B1, t0 = 4.82, for B2, t0 = 8.21 and for B3, t0 =
Consider the electric power consumption data in Exercise 12-6.(a) Test for significance of regression using a = 0.05. What is the P-value for this test?(b) Use the t-test to assess the contribution
Consider the bearing wear data in Exercise 12-7 with no interaction. (a) Test for significance of regression using a = 0.05. What is the P-value for this test? What are your conclusions?(b) Compute
Reconsider the bearing wear data from Exercises 12-7 and 12-20. (a) Refit the model with an interaction term. Test for significance of regression using a = 0.05. (b) Use the extra sum of squares
Consider the wire bond pull strength data in Exercise 12-8.(a) Test for significance of regression using a = 0.05. Find the P-value for this test. What conclusions can you draw?(b) Calculate the
Reconsider the semiconductor data in Exercise 12-9.(a) Test for significance of regression using a = 0.05. What conclusions can you draw?(b) Calculate the t-test statistic for each regression
Exercise 12-10 presents data on heat treating gears. (a) Test the regression model for significance of regression. Using a = 0.05, find the P-value for the test and draw conclusions. (b) Evaluate
Data on National Hockey League team performance was presented in Exercise 12-11.(a) Test the model from this exercise for significance of regression using a = 0.05. What conclusions can you draw?(b)
Consider the soil absorption data in Exercise 12-2.(a) Find a 95% confidence interval on the regression coefficient B1.(b) Find a 95% confidence interval on mean soil absorption index when x1 = 200
Consider the NFL data in Exercise 12-4.(a) Find a 95% confidence interval on B8.(b) What is the estimated standard error of when x2 = 2000 yards, x7 = 60%, and x8 = 1800 yards?(c) Find a 95%
Consider the gasoline mileage data in Exercise 12-5.(a) Find 99% confidence intervals on B1 and B6.(b) Find a 99% confidence interval on the mean of Y when x1 = 300 and x6 = 4.(c) Fit a new
Consider the electric power consumption data in Exercise 12-6.(a) Find 95% confidence intervals on B1, B2, B3, and B4.(b) Find a 95% confidence interval on the mean of Y when x1 = 75, x2 = 24, x3 =
Consider the wire bond pull strength data in Exercise 12-8.(a) Find 95% confidence interval on the regression coefficients. (b) Find a 95% confidence interval on mean pull strength when x2 = 20, x3 =
Consider the semiconductor data in Exercise 12-9.(a) Find 99% confidence intervals on the regression coefficients.(b) Find a 99% prediction interval on HFE when x1 = 14.5, x2 = 220, and x3 = 5.0.(c)
Consider the heat treating data from Exercise 12-10.(a) Find 95% confidence intervals on the regression coefficients.(b) Find a 95% confidence interval on mean PITCH when TEMP = 1650, SOAKTIME
Consider the bearing wear data in Exercise 12-7. (a) Find 99% confidence intervals on B1 and B2. (b) Recompute the confidence intervals in part (a) after the interaction term x1x2 is added to the
Reconsider the heat treating data in Exercises 12-10 and 12-24, where we fit a model to PITCH using regressors x1 = SOAKTIME SOAKPCT and x2 = DIFFTIME DIFFPCT.(a) Using the model with regressors
Consider the NHL data in Exercise 12-11. (a) Find a 95% confidence interval on the regression coefficient for the variable “Pts.”(b) Fit a simple linear regression model relating the response
Consider the regression model for the NFL data in Exercise 12-4.(a) What proportion of total variability is explained by this model?(b) Construct a normal probability plot of the residuals. What
Consider the gasoline mileage data in Exercise 12-5.(a) What proportion of total variability is explained by this model?(b) Construct a normal probability plot of the residuals and comment on the
Consider the electric power consumption data in Exercise 12-6.(a) Calculate R2 for this model. Interpret this quantity.(b) Plot the residuals versus y. Interpret this plot.(c) Construct a normal
Consider the wear data in Exercise 12-7.(a) Find the value of R2 when the model uses the regressors x1 and x2.(b) What happens to the value of R2 when an interaction term x1x2 is added to the model?
For the regression model for the wire bond pull strength data in Exercise 12-8.(a) Plot the residuals versus and versus the regressors used in the model. What information do these plots provide?(b)
Consider the semiconductor HFE data in Exercise 12-9.(a) Plot the residuals from this model versus y. Comment on the information in this plot.(b) What is the value of R2 for this model?(c) Refit the
Consider the regression model for the heat treating data in Exercise 12-10.(a) Calculate the percent of variability explained by this model.(b) Construct a normal probability plot for the
In Exercise 12-24 we fit a model to the response PITCH in the heat treating data of Exercise 12-10 using new regressors x1 = SOAKTIME SOAKPCT and x2 = DIFFTIME DIFFPCT(a) Calculate the R2 for this
Consider the regression model for the NHL data from Exercise 12-11.(a) Fit a model using “pts” as the only regressor.(b) How much variability is explained by this model? (c) Plot the residuals
The diagonal elements of the hat matrix are often used to denote leverage—that is, a point that is unusual in its location in the x-space and that may be influential. Generally, the ith point is
An article entitled €œA Method for Improving the Accuracy of Polynomial Regression Analysis€™€™ in the Journal of Quality Technology (1971, pp. 149€“155)
Consider the following data, which result from an experiment to determine the effect of x = test time in hours at a particular temperature on y = change in oil viscosity:(a) Fit a second-order
When fitting polynomial regression models, we often subtract x from each x value to produce a “centered’’ regressor x` = x – x. This reduces the effects of dependencies among the model terms
Suppose that we use a standardized variable x` = (x – x)/sx, where sx is the standard deviation of x, in constructing a polynomial regression model. Using the data in Exercise 12-46 and the
The following data shown were collected during an experiment to determine the change in thrust efficiency (y, in percent) as the divergence angle of a rocket nozzle (x) changes:(a) Fit a second-order
An article in the Journal of Pharmaceuticals Sciences (Vol. 80, 1991, pp. 971€“977) presents data on the observed mole fraction solubility of a solute at a constant temperature and the
Consider the gasoline mileage data in Exercise 12-5.(a) Discuss how you would model the information about the type of transmission in the car.(b) Fit a regression model to the gasoline mileage using
Consider the tool life data in Example 12-12. Test the hypothesis that two different regression models (with different slopes and intercepts) are required to adequately model the data. Use indicator
Use the National Football League Team Performance data in Exercise 12-4 to build regression models using the following techniques:(a) All possible regressions. Find the equations that minimize MSE
Use the gasoline mileage data in Exercise 12-5 to build regression models using the following techniques:(a) All possible regressions. Find the minimum Cp and minimum MSE equations.(b) Stepwise
Consider the electric power data in Exercise 12-6 Build regression models for the data using the following techniques?(a) All possible regressions.(b) Stepwise regression.(c) Forward selection.(d)
Consider the wire bond pull strength data in Exercise 12-8. Build regression models for the data using the following methods:(a) Stepwise regression.(b) Forward selection.(c) Backward elimination.(d)
Consider the NHL data in Exercise 12-11. Build regression models for these data using the following methods:(a) Stepwise regression.(b) Forward selection.(c) Backward elimination.(d) Which model
Consider the data in Exercise 12-51 use all the terms in the full quadratic model as the candidate regressors?(a) Use forward selection to identify a model.(b) Use backward elimination to identify a
Find the minimum Cp equation and the equation that maximizes the adjusted R2 statistic for the wire bond pull strength data in Exercise 12-8. Does the same equation satisfy both criteria?

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