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Chapter 4: Problem 53

Answer the questions using complete sentences. a. What is an influential point? How should influential points be treated when doing a regression analysis? b. What is the coefficient of determination and what does it measure? c. What is extrapolation? Should extrapolation ever be used?

Short Answer

Expert verified
An influential point is a point that significantly influences a regression analysis, and as such, it should be appropriately identified and handled to prevent skewing the results. The coefficient of determination, or \( R^2 \), measures the part of the variance in the dependent variable that can be predicted from the independent variable(s). Extrapolation is a means of predicting a value beyond the known data range, and though it can be used, it should be approached with caution due to the declining certainty of predictions as one moves further from the known data.

Step by step solution

01

Identification and Treatment of an Influential Point during Regression Analysis

An influential point refers to any point in a data set that heavily affects the outcome of a regression analysis because of its value or position. When doing a regression analysis, an influential point should be identified and handled appropriately so as not to skew the regression line unduly. Therefore, these points are often removed before the analysis or treated separately to prevent outliers from disproportionately influencing the outcome of the model.
02

Understanding of the Coefficient of Determination

The coefficient of determination, also known as \( R^2 \), is a statistical measure that shows the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It ranges between 0.0 and 1.0, where 0.0 indicates that the model explains none of the variability of the response data around its mean, and 1.0 indicates that the model explains all the variability of the response data around its mean.
03

Explanation of Extrapolation

Extrapolation involves predicting a value outside the range of the known data. While it can be used, it should be done with caution since the certainty of predictions declines the further you move from the known data. This is because extrapolation involves making assumptions that the current trend will continue, but this is not always accurate.

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Most popular questions from this chapter

Answer the questions using complete sentences. a. An economist noted the correlation between consumer confidence and monthly personal savings was negative. As consumer confidence increases, would we expect monthly personal savings to increase, decrease, or remain constant? b. A study found a correlation between higher education and lower death rates. Does this mean that one can live longer by going to college? Why or why not?

The following table shows the fat content (in grams) and calories for a sample of granola bars. (Source: calorielab. com ) $$ \begin{array}{|c|l|} \hline \text { Fat (in grams) } & \text { Calories } \\ \hline 7.6 & 370 \\ \hline 3.3 & 106.1 \\ \hline 18.7 & 312.4 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Fat (in grams) } & \text { Calories } \\ \hline 3.8 & 113.1 \\ \hline 5 & 117.8 \\ \hline 5.5 & 131.9 \\ \hline 7.2 & 140.6 \\ \hline 6.1 & 118.8 \\ \hline 4.6 & 124.4 \\ \hline 3.9 & 105.1 \\ \hline 6.1 & 136 \\ \hline 4.8 & 124 \\ \hline 4.4 & 119.3 \\ \hline 7.7 & 192.6 \\ \hline \end{array} $$ a. Use technology to make a scatterplot of the data. Use fat as the independent variable \((x)\) and calories as the dependent variable \((y)\). Does there seem to be a linear trend to the data? b. Compute the correlation coefficient and the regression equation, using fat as the independent variable and calories as the dependent variable. c. What is the slope of the regression equation? Interpret the slope in the context of this problem. d. What is the \(y\) -intercept of the regression equation? Interpret the \(y\) -intercept in the context of this problem or explain why it would be inappropriate to do so. e. Find and interpret the coefficient of determination. f. Use the regression equation to predict the calories in a granola bar containing 7 grams of fat. g. Would it be appropriate to use the regression equation to predict the calories in a granola bar containing 25 grams of fat? If so, predict the number of calories in such a bar. If not, explain why it would be inappropriate to do so. h. Looking at the scatterplot there is a granola bar in the sample that has an extremely high number of calories given the moderate amount of fat it contains. Remove its data from the sample and recalculate the correlation coefficient and regression equation. How did removing this unusual point change the value of \(r\) and the regression equation?

The following table shows the number of text messages sent and received by some people in one day. (Source: StatCrunch: Responses to survey How often do you text? Owner: Webster West. A subset was used.) a. Make a scatterplot of the data, and state the sign of the slope from the scatterplot. Use the number sent as the independent variable. b. Use linear regression to find the equation of the best-fit line. Graph the line with technology or by hand. c. Interpret the slope. d. Interpret the intercept. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array}\\\ &\begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} \end{aligned} $$

Some investors use a technique called the "Dogs of the Dow" to invest. They pick several stocks that are performing poorly from the Dow Jones group (which is a composite of 30 wellknown stocks) and invest in these. Explain why these stocks will probably do better than they have done before.

Suppose that the growth rate of children looks like a straight line if the height of a child is observed at the ages of 24 months, 28 months, 32 months, and 36 months. If you use the regression obtained from these ages and predict the height of the child at 21 years, you might find that the predicted height is 20 feet. What is wrong with the prediction and the process used?
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