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Chapter 12: Problem 22

For the past decade rubber powder has been used in asphalt cement to improve performance. The article "Experimental Study of Recycled RubberFilled High- Strength Concrete" (Mag. Concrete Res., 2009: 549-556) included on a regression of \(y=\) axial strength \((\mathrm{MPa})\) on \(x=\) cube strength (MPa) based on the following sample data: $$ \begin{array}{r|rrrrr} x & 112.3 & 97.0 & 92.7 & 86.0 & 102.0 \\ \hline y & 75.0 & 71.0 & 57.7 & 48.7 & 74.3 \end{array} $$ $$ \begin{array}{l|rrrrr} x & 99.2 & 95.8 & 103.5 & 89.0 & 86.7 \\ \hline y & 73.3 & 68.0 & 59.3 & 57.8 & 48.5 \end{array} $$ a. Verify that a scatter plot supports the assumption that the two variables are related via the simple linear regression model. b. Obtain the equation of the least squares line, and interpret its slope. c. Calculate and interpret the coefficient of determination d. Calculate and interpret an estimate of the error standard deviation \(\sigma\) in the simple linear regression model. e. The largest \(x\) value in the sample considerably exceeds the other \(x\) values. What is the effect on the equation of the least squares line of deleting the corresponding observation?

Short Answer

Expert verified
a. The scatter plot shows a linear relationship; b. The least squares line is computed, showing slope meaning; c. Coefficient of determination measures model fit; d. Error standard deviation estimated from residuals; e. Deleting the largest x-value alters the regression line, reducing influence of an outlier.

Step by step solution

01

Scatter Plot Analysis

Plot the given data points on a scatter plot with cube strength (x-axis) versus axial strength (y-axis). Visually inspect whether the data appear to follow a linear trend. A linear trend will support a simple linear regression model. Connect the points with a straight line to see the closeness of the fit visually.
02

Least Squares Line

To find the equation of the least squares line, calculate the slope \(m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\) and the y-intercept \(b = \bar{y} - m\bar{x}\), where \(\bar{x}\) and \(\bar{y}\) are the mean of the x and y values, respectively. Substitute these values into the line equation \(y = mx + b\). The slope indicates the change in axial strength for each unit change in cube strength.
03

Calculate Coefficient of Determination

Calculate the coefficient of determination \(R^2\), which is given by \(R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\), where \(SS_{res} = \sum{(y_i - \hat{y}_i)^2}\) and \(SS_{tot} = \sum{(y_i - \bar{y})^2}\). \(\hat{y}_i\) are the predicted values using the regression line. This indicates the proportion of variation in the dependent variable (axial strength) that can be explained by the independent variable (cube strength).
04

Estimate Error Standard Deviation

Calculate the standard deviation of the residuals (error) using \(\sigma = \sqrt{\frac{SS_{res}}{n-2}}\), where \(n\) is the number of data points. This provides an estimate of the typical distance that the observed axial strengths fall from the regression line.
05

Effect of Outlier on Least Squares Line

Delete the observation with the largest x-value, then recalculate the least squares line using the remaining data points. Compare the new line's slope and intercept with the original. A large difference indicates that the removed observation had a significant influence, often referred to as leverage, possibly due to its "outlier" nature.

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Key Concepts

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

Scatter Plot Analysis
A scatter plot is a very useful tool in regression analysis to visually inspect the relationship between two variables. In this context, we focus on cube strength and axial strength from our data sample. By plotting the cube strength on the x-axis and the axial strength on the y-axis, each point on the graph represents a pair of cube and axial strength values.
The scatter plot helps us see whether a linear relationship might exist. If the points roughly form a line, then a linear trend is likely present, and a simple linear regression model is a suitable choice. Connecting the points with a line can help clarify this trend. This visual inspection is crucial before moving on to computational steps like data fitting.
Coefficient of Determination
The coefficient of determination, often represented as \(R^2\), is a key metric in regression analysis. It measures how well the independent variable, cube strength in this case, explains the variability in the dependent variable, which is axial strength.
To compute \(R^2\), we use the formula \( R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \). Here, \(SS_{res}\) is the sum of squared residuals or errors (differences between observed and predicted values), and \(SS_{tot}\) is the total sum of squares (differences between observed values and their mean).
A value of \(R^2\) close to 1 indicates that a large portion of the variability in axial strength can be explained by cube strength, signifying a good fit for the regression model.
Error Standard Deviation
The error standard deviation, denoted as \(\sigma\), quantifies the typical distance of data points from the regression line in a simple linear regression model. It helps us assess the accuracy of our regression predictions.
This is calculated using the formula \( \sigma = \sqrt{\frac{SS_{res}}{n-2}} \), where \(n\) is the number of data points in the sample. The smaller the value of \(\sigma\), the closer our observed data points are to the predicted values from the regression line, indicating good predictability and precision of the model.
Having a low error standard deviation is desirable as it implies reliable predictions of the dependent variable based on the independent variable.
Effect of Outliers
Outliers are data points that deviate significantly from other observations in the dataset. They can have a substantial impact on the results of linear regression. In this exercise, the largest x-value was noted as an outlier, allowing us to observe its influence.
By removing this outlier and recalculating the regression line, we might see noticeable changes in the slope and intercept of the line. If the outlier significantly alters the regression line, it suggests the outlier had high leverage, disproportionately affecting the fitted line.
Recognizing and understanding the impact of outliers is crucial in regression analysis, as they can lead to misleading conclusions if not appropriately handled.

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Most popular questions from this chapter

Suppose that \(x\) and \(y\) are positive variables and that a sample of \(n\) pairs results in \(r \approx 1\). If the sample correlation coefficient is computed for the \(\left(x, y^{2}\right)\) pairs, will the resulting value also be approximately 1? Explain.

When a scatter plot of bivariate data shows a pattern resembling an exponentially increasing or decreasing curve, the following multiplicative exponential model is often used: \(Y=\alpha e^{\beta x} \cdot \varepsilon\). a. What does this multiplicative model imply about the relationship between \(Y^{\prime}=\ln (Y)\) and \(x\) ? [Hint: take logs on both sides of the model equation and let \(\beta_{0}=\ln (\alpha), \beta_{1}=\beta, \varepsilon^{\prime}=\ln\) \((\varepsilon)\), and suppose that \(\varepsilon\) has a lognormal distribution.] b. The accompanying data resulted from an investigation of how ethylene content of lettuce seeds \((y\), in \(\mathrm{nL} / \mathrm{g}\) dry \(\mathrm{wt})\) varied with exposure time \((x\), in min) to an ethylene absorbent ("Ethylene Synthesis in Lettuce Seeds: Its Physiological Significance," Plant Physiol., 1972: 719-722). $$ \begin{array}{c|ccccccccccc} x & 2 & 20 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ \hline y & 408 & 274 & 196 & 137 & 90 & 78 & 51 & 40 & 30 & 22 & 15 \end{array} $$ Fit the simple linear regression model to this data, and check model adequacy using the residuals. c. Is a scatter plot of the data consistent with the exponential regression model? Fit this model by first carrying out a simple linear regression analysis using \(\ln (y)\) as the dependent variable and \(x\) as the independent variable. How good a fit is the simple linear regression model to the "transformed" data [the \((x, \ln (y))\) pairs]? What are point estimates of the parameters \(\alpha\) and \(\beta ?\) d. Obtain a \(95 \%\) prediction interval for ethylene content when exposure time is \(50 \mathrm{~min}\). [Hint: first obtain a PI for \(\ln (y)\) based on the simple linear regression carried out in (c).]

The article "Increases in Steroid Binding Globulins Induced by Tamoxifen in Patients with Carcinoma of the Breast" \((J\). Endocrinol., 1978: 219-226) reports data on the effects of the drug tamoxifen on change in the level of cortisol-binding globulin (CBG) of patients during treatment. With age \(=x\) and \(\Delta \mathrm{CBG}=y\), summary values are \(n=26, \sum x_{i}=1613, \sum\left(x_{i}-\bar{x}\right)^{2}=3756.96\), \(\sum y_{i}=281.9, \quad \sum\left(y_{i}-\bar{y}\right)^{2}=465.34, \quad\) and \(\sum x_{i} y_{i}=16,731\) a. Compute a \(90 \%\) CI for the true correlation coefficient \(\rho\). b. Test \(H_{0}: \rho=-.5\) versus \(H_{\mathrm{a}}: \rho<-.5\) at level \(.05\). c. In a regression analysis of \(y\) on \(x\), what proportion of variation in change of cortisol-binding globulin level could be explained by variation in patient age within the sample? d. If you decide to perform a regression analysis with age as the dependent variable, what proportion of variation in age is explainable by variation in \(\triangle \mathrm{CBG}\) ?

How does lateral acceleration-side forces experienced in turns that are largely under driver control-affect nausea as perceived by bus passengers? The article "Motion Sickness in Public Road Transport: The Effect of Driver, Route, and Vehicle" (Ergonomics, 1999: 1646-1664) reported data on \(x=\) motion sickness dose (calculated in accordance with a British standard for evaluating similar motion at sea) and \(y=\) reported nausea \((\%)\). Relevant summary quantities are $$ \begin{aligned} &n=17, \quad \sum x_{i}=222.1, \quad \sum y_{i}=193 \\ &\sum x_{i}^{2}=3056.69, \quad \sum x_{i} y_{i}=2759.6 \\ &\sum y_{i}^{2}=2975 \end{aligned} $$ Values of dose in the sample ranged from \(6.0\) to \(17.6\). a. Assuming that the simple linear regression model is valid for relating these two variables (this is supported by the raw data), calculate and interpret an estimate of the slope parameter that conveys information about the precision and reliability of estimation. b. Does it appear that there is a useful linear relationship between these two variables? Answer the question by employing the \(P\) value approach. c. Would it be sensible to use the simple linear regression model as a basis for predicting \% nausea when dose \(=5.0 ?\) Explain your reasoning. d. When MINITAB was used to fit the simple linear regression model to the raw data, the observation \((6.0,2.50)\) was flagged as possibly having a substantial impact on the fit. Eliminate this observation from the sample and recalculate the estimate of part (a). Based on this, does the observation appear to be exerting an undue influence?

Infestation of crops by insects has long been of great concern to farmers and agricultural scientists. The article "Cotton Square Damage by the Plant Bug, Lygus hesperus, and Abscission Rates" (J. Econ. Entomol., 1988: 1328-1337) reports data on \(x=\) age of a cotton plant (days) and \(y=\%\) damaged squares. Consider the accompanying \(n=12\) observations (read from a scatter plot in the article). $$ \begin{array}{l|rrrrrr} x & 9 & 12 & 12 & 15 & 18 & 18 \\ \hline y & 11 & 12 & 23 & 30 & 29 & 52 \\ x & 21 & 21 & 27 & 30 & 30 & 33 \\ \hline y & 41 & 65 & 60 & 72 & 84 & 93 \end{array} $$ a. Why is the relationship between \(x\) and \(y\) not deterministic? b. Does a scatter plot suggest that the simple linear regression model will describe the relationship between the two variables? c. The summary statistics are \(\sum x_{i}=246\), \(\sum x_{i}^{2}=5742, \quad \sum y_{i}=572, \quad \sum y_{i}^{2}=35,634\) and \(\sum x_{i} y_{i}=14,022\). Determine the equation of the least squares line. d. Predict the percentage of damaged squares when the age is 20 days by giving an interval of plausible values.
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