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Chapter 5: Problem 49

The paper "Effects of Canine Parvovirus (CPV) on Gray Wolves in Minnesota" (Journal of Wildlife Management \([1995]: 565-570\) ) summarized a regression of \(y=\) percentage of pups in a capture on \(x=\) percentage of \(\mathrm{CPV}\) prevalence among adults and pups. The equation of the least-squares line, based on \(n=10\) observations, was \(\hat{y}=62.9476-0.54975 x\), with \(r^{2}=.57\) a. One observation was \((25,70)\). What is the corresponding residual? b. What is the value of the sample correlation coefficient? c. Suppose that SSTo \(=2520.0\) (this value was not given in the paper). What is the value of \(s_{e} ?\)

Short Answer

Expert verified
a. The residual for the observation (25,70) can be calculated using the given regression equation. \n b. The sample correlation coefficient is the square root of the provided coefficient of determination, taking on the sign of the regression slope. \n c. The value of \(s_{e}\) is found by taking the square root of the ratio of the sum of squares due to error (residuals) and the degrees of freedom (n-2).

Step by step solution

01

Calculate the Residual

The residual is calculated by subtracting the predicted value \(\hat{y}\) from the actual value \(y\). We know the actual value from the provided observation \((25,70)\), where \(x=25\) and \(y=70\). To find the predicted value, we substitute \(x=25\) into the regression equation \(\hat{y}=62.9476-0.54975 x\). The residual is then calculated as \(residual = y - \hat{y}\).
02

Find the Sample Correlation Coefficient

The coefficient of determination, \(r^{2}=.57\), provided in the problem is the square of the sample correlation coefficient, \(r\). Therefore, to find \(r\), we take the square root of \(r^{2}\). The sign of the correlation coefficient is the same as the sign of the slope of the least-squares line, which is negative in this case. Thus \(r = -\sqrt{r^{2}}\).
03

Calculate the standard error, \(s_{e}\)

Given the total sum of squares (SSTo) as 2520 and \(r^{2}=0.57\), the variation explained by the model (SSR - Sum of Squares for Regression) can be calculated as \(r^{2}*SSTo\). The remaining unexplained variation (SSE - Sum of squares due to error or residuals) is given by \(SSTo - SSR\). The standard error, \(s_{e}\) is then the square root of \(\frac{SSE}{n-2}\), where \(n\) is the number of observations (10 in this case).

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Key Concepts

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

Least-Squares Method
Regression analysis is a powerful tool, and at the heart of it lies the least-squares method. This method is all about finding the line of best fit for a set of data points. This "line of best fit" is the one that minimizes the sum of the squares of the vertical distances (residuals) between the observed values and the values predicted by the line. The least-squares method applies a simple formula to determine these distances and hence the best fit line. When we talk about minimizing these distances, we mean that the line should have the smallest possible sum of squared differences between the actual data points and the predicted points on the line. For the equation given in the problem, \( \hat{y}=62.9476-0.54975x \), it tells us that for every unit increase in \(x\), the value of \(y\) decreases by 0.54975 units. This step-by-step approach allows the model to adjust its slope and intercept in a way that conforms closest to our data observations.
Correlation Coefficient
The correlation coefficient, denoted as \(r\), is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1. A value of 1 means a perfect positive linear relationship, -1 means a perfect negative linear relationship, and 0 indicates no linear relationship at all.In regression analysis, the square of the correlation coefficient (\(r^2\)), known as the coefficient of determination, gives the proportion of the variability in the dependent variable that is predictable from the independent variable. In this problem, \(r^2\) is 0.57, indicating that 57% of the variation in the percentage of pups can be explained by the CPV prevalence. This suggests a moderate correlation.To find \(r\), we take the square root of \(r^2\). Because our regression slope is negative, \(r\) will also be negative. Thus, \(r = -\sqrt{0.57}\), indicating a negative correlation between the variables.
Standard Error
The standard error, \(s_{e}\), in the context of regression analysis, is an estimate of the standard deviation of the residuals. It provides insight into the accuracy of the predictions made by the regression model. Standard error helps us understand how well the model fits the data. If the standard error is small, the data points are close to the regression line, indicating a good fit. Conversely, a large standard error suggests that the data points are more spread out around the line, indicating a poor fit.In this problem, to calculate \(s_{e}\), we use the total sum of squares \(SSTo\), which is 2520. We know from \(r^2 = 0.57\) that 57% of this variation is explained by the model, giving us the sum of squares due to regression (\(SSR\)). The unexplained variation, or sum of squares due to error (\(SSE\)), is \(SSTo - SSR\). The formula for the standard error is \(s_{e} = \sqrt{\frac{SSE}{n-2}}\), where \(n\) is the number of observations, which in this case is 10.

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Most popular questions from this chapter

The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003): $$ \begin{array}{cc} & \begin{array}{l} \text { Number of } \\ \text { Transplants } \\ \text { Year } \end{array} & \text { (in thousands) } \\ \hline 1(1990) & 15.0 \\ 2 & 15.7 \\ 3 & 16.1 \\ 4 & 17.6 \\ 5 & 18.3 \\ 6 & 19.4 \\ 7 & 20.0 \\ 8 & 20.3 \\ 9 & 21.4 \\ 10 \text { (1999) } & 21.8 \\ \hline \end{array} $$ a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between \(y=\) number of transplants performed and \(x=\) year. Describe how the number of transplants performed has changed over time from 1990 to 1999 . b. Compute the 10 residuals, and construct a residual plot. Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.

The article "Characterization of Highway Runoff in Austin, Texas, Area" (Journal of Environmental Engineering \([1998]: 131-137\) ) gave a scatterplot, along with the least-squares line for \(x=\) rainfall volume (in cubic meters) and \(y=\) runoff volume (in cubic meters), for a particular location. The following data were read from the plot in the paper: $$ \begin{array}{rrrrrrrrr} x & 5 & 12 & 14 & 17 & 23 & 30 & 40 & 47 \\ y & 4 & 10 & 13 & 15 & 15 & 25 & 27 & 46 \\ x & 55 & 67 & 72 & 81 & 96 & 112 & 127 & \\ y & 38 & 46 & 53 & 70 & 82 & 99 & 100 & \end{array} $$ a. Does a scatterplot of the data suggest a linear relationship between \(x\) and \(y\) ? b. Calculate the slope and intercept of the least-squares line. c. Compute an estimate of the average runoff volume when rainfall volume is 80 . d. Compute the residuals, and construct a residual plot. Are there any features of the plot that indicate that a line is not an appropriate description of the relationship between \(x\) and \(y\) ? Explain.

The article "Reduction in Soluble Protein and Chlorophyll Contents in a Few Plants as Indicators of Automobile Exhaust Pollution" (International Journal of Environmental Studies [1983]: 239-244) reported the following data on \(x=\) distance from a highway (in meters) and \(y=\) lead content of soil at that distance (in parts per million): $$ \begin{array}{rrrrrrr} x & 0.3 & 1 & 5 & 10 & 15 & 20 \\ y & 62.75 & 37.51 & 29.70 & 20.71 & 17.65 & 15.41 \\ x & 25 & 30 & 40 & 50 & 75 & 100 \\ y & 14.15 & 13.50 & 12.11 & 11.40 & 10.85 & 10.85 \end{array} $$ a. Use a statistical computer package to construct scatterplots of \(y\) versus \(x, y\) versus \(\log (x), \log (y)\) versus \(\log (x)\) and \(\frac{1}{y}\) versus \(\frac{1}{x}\). b. Which transformation considered in Part (a) does the best job of producing an approximately linear relationship? Use the selected transformation to predict lead content when distance is \(25 \mathrm{~m}\).

The paper "Crop Improvement for Tropical and Subtropical Australia: Designing Plants for Difficult Climates" (Field Crops Research [1991]: 113-139) gave the following data on \(x=\) crop duration (in days) for soybeans and \(y=\) crop yield (in tons per hectare): $$ \begin{array}{rrrrrr} x & 92 & 92 & 96 & 100 & 102 \\ y & 1.7 & 2.3 & 1.9 & 2.0 & 1.5 \\ x & 102 & 106 & 106 & 121 & 143 \\ y & 1.7 & 1.6 & 1.8 & 1.0 & 0.3 \end{array} $$ $$ \begin{gathered} \sum x=1060 \quad \sum y=15.8 \quad \sum x y=1601.1 \\ a=5.20683380 \quad b=-0.3421541 \end{gathered} $$ a. Construct a scatterplot of the data. Do you think the least-squares line will give accurate predictions? Explain. b. Delete the observation with the largest \(x\) value from the sample and recalculate the equation of the least-squares line. Does this observation greatly affect the equation of the line? c. What effect does the deletion in Part (b) have on the value of \(r^{2}\) ? Can you explain why this is so?

Cost-to-charge ratios (the percentage of the amount billed that represents the actual cost) for 11 Oregon hospitals of similar size were reported separately for inpatient and outpatient services. The data are $$ \begin{array}{lcc} \text { Hospital } & \text { Inpatient } & \text { Outpatient } \\ \hline \text { Blue Mountain } & 80 & 62 \\ \text { Curry General } & 76 & 66 \\ \text { Good Shepherd } & 75 & 63 \\ \text { Grande Ronde } & 62 & 51 \\ \text { Harney District } & 100 & 54 \\ \text { Lake District } & 100 & 75 \\ \text { Pioneer } & 88 & 65 \\ \text { St. Anthony } & 64 & 56 \\ \text { St. Elizabeth } & 50 & 45 \\ \text { Tillamook } & 54 & 48 \\ \text { Wallowa Memorial } & 83 & 71 \\ \hline \end{array} $$ a. Does there appear to be a strong linear relationship between the cost-to- charge ratio for inpatient and outpatient services? Justify your answer based on the value of the correlation coefficient and examination of a scatterplot of the data. b. Are any unusual features of the data evident in the scatterplot? c. Suppose that the observation for Harney District was removed from the data set. Would the correlation coefficient for the new data set be greater than or less than the one computed in Part (a)? Explain.
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