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Chapter 14: Problem 17

A researcher wants to determine a model that can be used to predict the 28 -day strength of a concrete mixture. The following data represent the 28 -day and 7 -day strength (in pounds per square inch) of a certain type of concrete along with the concrete's slump. Slump is a measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture. $$ \begin{array}{ccc} \text { Slump (inches) } & \text { 7-Day psi } & \text { 28-Day psi } \\ \hline 4.5 & 2330 & 4025 \\ \hline 4.25 & 2640 & 4535 \\\ \hline 3 & 3360 & 4985 \\ \hline 4 & 1770 & 3890 \\ \hline 3.75 & 2590 & 3810 \\ \hline 2.5 & 3080 & 4685 \\ \hline 4 & 2050 & 3765 \\ \hline 5 & 2220 & 3350 \\ \hline 4.5 & 2240 & 3610 \\ \hline 5 & 2510 & 3875 \\ \hline 2.5 & 2250 & 4475 \end{array} $$ (a) Construct a correlation matrix between slump, 7 -day psi, and 28 -day psi. Is there any reason to be concerned with multicollinearity based on the correlation matrix? (b) Find the least-squares regression equation \(\hat{y}=b_{0}+b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is slump, \(x_{2}\) is 7 -day strength, and \(y\) is the response variable, 28 -day strength. (c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model. (d) Interpret the regression coefficients for the least-squares regression equation. (e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1}:\) at least one of the \(\beta_{1} \neq 0\) at the \(\alpha=0.05\) level of significance. (g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\) level of significance. (h) Predict the mean 28 -day strength of all concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (i) Predict the 28 -day strength of a specific sample of concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (j) Construct \(95 \%\) confidence and prediction intervals for concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. Interpret the results.

Short Answer

Expert verified
Construct a correlation matrix, build a regression equation, check residuals, calculate and interpret regression coefficients, and predict 28-day strength using given Slump and 7-Day strength values.

Step by step solution

01

Construct the Correlation Matrix (Part (a))

Calculate the correlation coefficients between Slump, 7-Day Strength, and 28-Day Strength using the given data. This matrix helps identify multicollinearity issues.
02

Calculate the Correlation Coefficients

Use the formula \(\rho_{xy} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}\) to find the correlation between each pair of variables.
03

Check for Multicollinearity

Interpret the correlation coefficients. If any coefficient is close to \(\rho = 1\) or \(\rho = -1\), it indicates potential multicollinearity.
04

Calculate the Least-Squares Regression Equation (Part (b))

Use multiple linear regression to find \(\hat{y} = b_{0} + b_{1} x_{1} + b_{2} x_{2}\), where \(x_{1}\) is Slump and \(x_{2}\) is 7-Day Strength.
05

Find Regression Coefficients

Compute \(b_{0}, b_{1},\) and \(b_{2}\) using the least-squares method. Formulas for \(b_{1}\) and \(b_{2}\) typically involve normal equations.
06

Draw Residual Plots and Boxplot (Part (c))

Plot residuals against fitted values and predictor variables to check for patterns. Create a boxplot of residuals to assess the model鈥檚 adequacy.
07

Analyze Residual Plots

Look for randomness in residual plots. Patterns suggest model inadequacy. Use the boxplot to check for outliers and extreme values.
08

Interpret Regression Coefficients (Part (d))

Understand the significance and meaning of \(b_{0}, b_{1},\) and \(b_{2}\) in the regression equation. \(b_{0}\) is the intercept; \(b_{1}\) and \(b_{2}\) show change in 28-Day strength per unit change in Slump and 7-Day strength respectively.
09

Determine and Interpret \(R^{2}\) and Adjusted \(R^{2}\) (Part (e))

Calculate \(R^{2}\) using the formula \(\frac{SS_{reg}}{SS_{tot}}\) and adjusted \(R^{2} = 1 - (1 - R^{2})(\frac{n-1}{n-p-1})\). Interpret them to understand the proportion of variance explained by the model.
10

Test Overall Significance (Part (f))

Conduct an F-test to determine if at least one of the regression coefficients is not equal to zero. Use an \(\text{ANOVA table}\) for this purpose.
11

Null Hypotheses \(H_{0}: \beta_{1}=0\) vs \(H_{1}: \beta_{1} eq 0\) (Part (g))

Use a t-test to assess the significance of each individual coefficient. Compute t-statistics to test \(H_{0}: \beta_{1} = 0\) and \(H_{0}: \beta_{2} = 0\).
12

Predict Mean 28-Day Strength (Part (h))

Substitute \(x_{1}=3.5\) and \(x_{2}=2450\) into the regression equation to predict the mean 28-Day Strength.
13

Predict 28-Day Strength for Specific Sample (Part (i))

Use the regression equation with \(x_{1}=3.5\) and \(x_{2}=2450\) to make a specific prediction for 28-Day Strength.
14

Construct Confidence and Prediction Intervals (Part (j))

Calculate the 95% Confidence Interval for the mean and the Prediction Interval for a new observation. Use the relevant formulas for both intervals and interpret the results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Matrix
A correlation matrix is a table that shows the correlation coefficients between multiple variables. In this exercise, we are comparing slump, 7-day psi, and 28-day psi. Each cell in the matrix shows the correlation (a value between -1 and 1) between two variables. A correlation close to 1 indicates a strong positive relationship, while a value close to -1 means a strong negative relationship. By examining the correlation matrix, we can detect multicollinearity 鈥� a condition where two or more predictors are highly correlated, making it difficult to isolate their individual effects on the response variable.
Residual Plots
Residual plots are graphs that display the residuals (the differences between observed and predicted values) on the vertical axis and the fitted values or predictors on the horizontal axis. These plots help us assess the adequacy of our regression model. Ideally, residuals should randomly scatter around zero, indicating a good fit. Patterns or systematic structures in the residuals can suggest problems like non-linearity, heteroscedasticity (variable spread of residuals), or outliers. A boxplot of residuals provides a visual summary of the residual distribution, highlighting potential outliers and the spread of the data.
Regression Coefficients
In multiple linear regression, regression coefficients (denoted as \(b_0\), \(b_1\), \(b_2\), etc.) quantify the relationship between each predictor and the response variable. The equation \(\hat{y} = b_0 + b_1 x_1 + b_2 x_2\) represents the predicted response. Here, \(b_0\) is the intercept, indicating the expected value of \(y\) when all predictors are zero. \(b_1\) shows how much the 28-day strength changes with a one-inch change in slump, while \(b_2\) indicates the change in 28-day strength with a one-psi change in 7-day strength. These coefficients help us understand and quantify the impact of predictors on the response.
R-squared
The \(R^2\) value, or coefficient of determination, measures how well the regression model explains the variability of the response variable. It ranges from 0 to 1. An \(R^2\) of 1 means that the model perfectly explains the variability, while a lower \(R^2\) indicates that the model does not explain much of the variance. Adjusted \(R^2\) is a modified version of \(R^2\) that adjusts for the number of predictors in the model, providing a more accurate measure when multiple predictors are involved. It is generally lower than \(R^2\) to account for the possibility of overfitting.
T-test
A t-test in the context of regression assesses whether a particular regression coefficient is significantly different from zero, indicating a meaningful relationship between the predictor and the response variable. The null hypothesis \(H_0: \beta = 0\) suggests no effect, while the alternative \(H_1: \beta \oteq 0\) suggests a significant effect. We calculate the t-statistic and compare it against a critical value from the t-distribution. If the t-statistic exceeds the critical value, we reject the null hypothesis, indicating that the predictor significantly contributes to the model.
F-test
An F-test evaluates the overall significance of a regression model. Specifically, it tests whether at least one of the regression coefficients is non-zero 鈥� meaning at least one predictor has a significant effect on the response variable. The null hypothesis \(H_0: \beta_1 = \beta_2 = 0\) implies no predictors are useful. The F-statistic is calculated using the ratio of the model's explained variance to the unexplained variance (mean squared regression divided by mean squared error). A significant F-statistic (p-value < 0.05) leads to rejection of the null hypothesis, suggesting the model is meaningful overall.
Confidence Intervals
Confidence intervals provide a range within which we expect the true value of a regression coefficient to fall, with a certain level of confidence (e.g., 95%). For a regression model, they can also estimate the range for the mean response variable given specific predictor values. A 95% confidence interval means there is a 95% chance that the interval contains the true population parameter. This helps quantify the uncertainty around our estimates, offering insight into the precision and reliability of our predictions.
Prediction Intervals
Prediction intervals are used to predict the value of a new observation given specific predictor values, also with a certain level of confidence (e.g., 95%). Unlike confidence intervals, which focus on the mean response, prediction intervals account for individual variability and thus are wider. For example, in predicting a specific 28-day strength of concrete given slump and 7-day strength, a 95% prediction interval provides a range where the new measurement is expected to fall 95% of the time. This interval incorporates both the uncertainty in estimating the mean and the natural variability of the data.

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Most popular questions from this chapter

Researchers developed a model to explain the age gap between husbands and wives at first marriage. The model is below: $$ \hat{y}=0.0321 x_{1}+0.9848 x_{2}+0.5391 x_{3}-0.000145 x_{4}^{2}+3.8483 $$ where y: Age gap at first marriage (male - female) \(x_{1}:\) Percent of children aged 10 to 14 involved in child labor \(x_{2}:\) Indicator variable where 1 is an African country, 0 otherwise \(x_{3}:\) Percent of the population that is Muslim \(x_{4}:\) Percent of the population that is literate Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton "The Husband- Wife Age Gap at First Marriage: A Cross-Country Analysis" (a) Use the model to predict the age gap at first marriage for an African country where the percent of children aged 10 to 14 who are involved in child labor is \(12,\) the percent of the population that is Muslim is \(30,\) and the percent of the population that is literate is \(75 .\) (b) What would be the mean difference in age gap between an African country and a non-African country? (c) Interpret the coefficient of "percent of children aged 10 to 14 involved in child labor." (d) The coefficient of determination for this model is 0.593 . Interpret this result. (e) The \(P\) -value for the test \(H_{1}: \beta_{1} \neq 0\) versus \(H_{1}: \beta_{1} \neq 0\) is \(0.008 .\) What would you conclude about this test?

Kepler's Law of Planetary Motion The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. Johann Kepler studied the relation between the sidereal year of a planet and its distance from the sun in 1618 . The following data show the distances that the planets are from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto* } & 3675 & 248.0 \end{array} $$ (a) Determine the least-squares regression equation, treating distance from the sun as the explanatory variable. (b) A normal probability plot of the residuals indicates that the residuals are approximately normally distributed. Test whether a linear relation exists between distance from the sun and sidereal year. (c) Draw a scatter diagram, treating distance from the sun as the explanatory variable. (d) Plot the residuals against the explanatory variable, distance from the sun. (e) Does a linear model seem appropriate based on the scatter diagram and residual plot? (Hint: See Section 4.3.) (f) What is the moral?

Putting It Together: Purchasing Diamonds The value of a diamond is determined by the four C's: carat weight, color, clarity, and cut. Carat weight is the standard measure for the size of a diamond. Generally, the more a diamond weighs, the more valuable it will be. The Gemological Institute of America (GIA) determines the color of diamonds using a 22 -grade scale from D (almost clear white) to \(Z\) (light yellow). Colorless diamonds are generally considered the most desirable. The clarity of a diamond refers to how "free" the diamond is of imperfections and is determined using an 11 -grade scale: flawless (FL), internally flawless (IF), very, very slightly imperfect (VVS1, VVS2), very slightly imperfect (VS1, VS2), slightly imperfect (SI1,SI2), and imperfect (I1, I2, I3). The cut of a diamond refers to the diamond's proportions and finish. Put simply, the better the diamond's cut is, the better it reflects and refracts light, which makes it more beautiful and thus more valuable. The cut of a diamond is rated using a five-grade scale: Excellent, Very Good, Good, Fair, and Poor. Finally, the shape of a diamond (which is not one of the four C's) refers to its basic form: round, oval, pear-shaped, marquis, and so on. A novice might confuse shape with cut, so be careful not to confuse the two. Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(14_{-} 6_{-} 8\) using the file format of your choice for the version of the text you are using. The data represent a random sample of 40 unmounted, round-shaped diamonds. Use the data to answer the questions that follow: (a) Determine the level of measurement for each variable. (i) Carat weight (iv) Cut (ii) Color (v) Price (iii) Clarity (vi) Shape (b) Construct a correlation matrix. To do so, first convert the variables color, clarity, and cut to numeric values as follows: Color: \(\mathrm{D}=1, \mathrm{E}=2, \mathrm{~F}=3, \mathrm{G}=4, \mathrm{H}=5, \mathrm{I}=6, \mathrm{~J}=7\) Clarity: \(\mathrm{FL}=1, \mathrm{IF}=2, \mathrm{VVS} 1=3, \mathrm{VVS} 2=4, \mathrm{VS} 1=5\) \(\mathrm{VS} 2=6, \mathrm{SI} 1=7, \mathrm{SI} 2=8\) Cut: Excellent \(=1,\) Very Good \(=2,\) Good \(=3\) If price is to be the response variable in our model, is there reason to be concerned about multicollinearity? Explain. (c) Find the "best" model for predicting the price of a diamond. (d) Draw residual plots, a boxplot of the residuals, and a normal probability plot of the residuals to assess the adequacy of the "best" model. (e) For the "best" model, interpret each regression coefficient. (f) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (g) Predict the mean price of a round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1, Excellent. (h) Construct a \(95 \%\) confidence interval for the mean price found in part (g). (i) Predict the price of an individual round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1 Excellent. (j) Construct a \(95 \%\) prediction interval for the price found in \(\operatorname{part}(\mathrm{i})\) (k) Explain why the predictions in parts \((\mathrm{g})\) and (i) are the same, yet the intervals in parts \((\mathrm{h})\) and \((\mathrm{j})\) are different.

For the data set $$ \begin{array}{ccccc} \boldsymbol{x}_{1} & \boldsymbol{x}_{2} & \boldsymbol{x}_{3} & \boldsymbol{x}_{4} & \boldsymbol{y} \\ \hline 47.3 & 0.9 & 4 & 76 & 105.5 \\ \hline 53.1 & 0.8 & 6 & 55 & 113.8 \\ \hline 56.7 & 0.8 & 4 & 65 & 115.2 \\ \hline 48.8 & 0.5 & 7 & 67 & 118.9 \\ \hline 42.7 & 1.1 & 7 & 74 & 148.9 \\ \hline 44.3 & 1.1 & 6 & 76 & 120.2 \\ \hline 44.5 & 0.7 & 8 & 68 & 121.6 \\ \hline 37.7 & 0.7 & 7 & 79 & 140.0 \\ \hline 36.9 & 1.0 & 5 & 73 & 141.5 \\ \hline 28.1 & 1.8 & 6 & 68 & 141.9 \\ \hline 32.0 & 0.8 & 8 & 81 & 152.8 \\ \hline 34.7 & 0.8 & 10 & 68 & 156.5 \\\\\hline \end{array} $$ (a) Construct a correlation matrix between \(x_{1}, x_{2}, x_{3}, x_{4},\) and \(y\). Is there any evidence that multicollinearity may be a problem? (b) Determine the multiple regression line using all the explanatory variables listed. Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0 ?\) Which explanatory variables have slope coefficients that are not significantly different from zero? (c) Remove the explanatory variable with the highest \(P\) -value from the model and recompute the regression model. Does the \(F\) -test still indicate that the model is significant? Remove any additional explanatory variables on the basis of the \(P\) -value of the slope coefficient. Then compute the model with the variable removed. (d) Draw residual plots and a box plot of the residuals to assess the adequacy of the model. (e) Use the final model constructed in part (c) to predict the value of \(y\) if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\) (f) Draw a normal probability plot of the residuals. Is it reasonable to construct confidence and prediction intervals? (g) Construct \(95 \%\) confidence and prediction intervals if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\)
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