/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 In Exercise \(4.1\) there is a g... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 23

In Exercise \(4.1\) there is a graph of the relationship between SAT score and college GPA. SAT score was the predictor and college GPA was the response variable. If you reverse the variables so that college GPA was the predictor and SAT score was the response variable, what effect would this have on the numerical value of the correlation coefficient?

Short Answer

Expert verified
Switching the roles of the variables (SAT score and college GPA) from predictor to response or vice versa does not affect the numerical value of the correlation coefficient.

Step by step solution

01

Understanding the Concept

The correlation coefficient is a value that measures the strength and direction of a linear relationship between two variables. The coefficient returns a value which is between -1 and 1 that represents the limits of correlation from a full negative correlation to a full positive correlation. A correlation of -1 shows a perfect negative correlation, while a correlation of 1 shows a perfect positive correlation. A correlation of zero means there is no linear relationship between the variables. In our case, the predictor variable, \(X\), is SAT score and the response variable, \(Y\), is college GPA.
02

Changing the Roles of the Variables

The exercise asks what would happen if we swapped the roles of the variables. That is, if we make the SAT score the response variable and the college GPA the predictor variable. In other words, we need to assess how the correlation coefficient would be affected if instead of looking at how SAT scores predict college GPA, we now want to see how college GPA predicts SAT scores.
03

Effect on the Correlation Coefficient

The value of the correlation coefficient does not change when we switch the predictor and response variables. This means that, if the SAT score and college GPA have a correlation coefficient of \(r\), then making college GPA the predictor and SAT score the response variable will result in the same correlation coefficient of \(r\). The correlation coefficient is unaffected by the labelling of the variables. It measures the strength and directionality of the relationship, but not the 'cause' and 'effect' between the two.

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Most popular questions from this chapter

The data shows the number of calories, carbohydrates (in grams) and sugar (in grams) found in a selection of menu items at McDonald's. Scatterplots suggest the relationship between calories and both carbs and sugars is linear. The data are also available on this text's website. (Source: shapefit.com) $$ \begin{array}{|c|c|c|} \hline \text { Calories } & \text { Carbs (in grams) } & \text { Sugars (in grams) } \\ \hline 530 & 47 & 9 \\ \hline 520 & 42 & 10 \\ \hline 720 & 52 & 14 \\ \hline 610 & 47 & 10 \\ \hline 600 & 48 & 12 \\ \hline 540 & 45 & 9 \\ \hline 740 & 43 & 10 \\ \hline 240 & 32 & 6 \\ \hline 290 & 33 & 7 \\ \hline 340 & 37 & 7 \\ \hline 300 & 32 & 6 \\ \hline 430 & 35 & 7 \\ \hline 380 & 34 & 7 \\ \hline 430 & 35 & 6 \\ \hline 440 & 35 & 7 \\ \hline 430 & 34 & 7 \\ \hline 750 & 65 & 16 \\ \hline 590 & 51 & 14 \\ \hline 510 & 55 & 10 \\ \hline 350 & 42 & 8 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \text { Calories } & \text { Carbs (in grams) } & \text { Sugars (in grams) } \\ \hline 670 & 58 & 11 \\ \hline 510 & 44 & 9 \\ \hline 610 & 57 & 11 \\ \hline 450 & 43 & 9 \\ \hline 360 & 40 & 5 \\ \hline 360 & 40 & 5 \\ \hline 430 & 41 & 6 \\ \hline 480 & 43 & 6 \\ \hline 430 & 43 & 7 \\ \hline 390 & 39 & 5 \\ \hline 500 & 44 & 11 \\ \hline 670 & 68 & 12 \\ \hline 510 & 54 & 10 \\ \hline 630 & 56 & 7 \\ \hline 480 & 42 & 6 \\ \hline 610 & 56 & 8 \\ \hline 450 & 42 & 6 \\ \hline 540 & 61 & 14 \\ \hline 380 & 47 & 12 \\ \hline 340 & 37 & 8 \\ \hline 260 & 30 & 7 \\ \hline 340 & 34 & 5 \\ \hline 260 & 27 & 4 \\ \hline 360 & 32 & 3 \\ \hline 280 & 25 & 2 \\ \hline 330 & 26 & 3 \\ \hline 190 & 12 & 0 \\ \hline 750 & 65 & 16 \\ \hline \end{array} $$ a. Calculate the correlation coefficient and report the equation of the regression line using carbs as the predictor and calories as the response variable. Report the slope and interpret it in the context of this problem. Then use your regression equation to predict the number of calories in a menu item containing 55 grams of carbohydrates. b. Calculate the correlation coefficient and report the equation of the regression line using sugar as the predictor and calories as the response variable. Report the slope and interpret it in the context of this problem. Then use your regression equation to predict the number of calories in a menu item containing 10 grams of sugars. c. Based on your answers to parts (a) and (b), which is a better predictor of calories for these data: carbs or sugars? Explain your choice using appropriate statistics.

The correlation between height and armspan in a sample of adult women was found to be \(r=0.948 .\) The correlation between arm span and height in a sample of adult men was found to be \(r=0.868\). Assuming both associations are linear, which association-the association between height and arm span for women, or the association between height and arm span for men-is stronger? Explain.

The following table shows the weights and prices of some turkeys at different supermarkets. a. Make a scatterplot with weight on the \(x\) -axis and cost on the \(y\) -axis. Include the regression line on your scatterplot. b. Find the numerical value for the correlation between weight and price. Explain what the sign of the correlation shows. c. Report the equation of the best-fit straight line, using weight as the predictor \((x)\) and cost as the response \((y)\). d. Report the slope and intercept of the regression line, and explain what they show. If the intercept is not appropriate to report, explain why. e. Add a new point to your data: a 30 -pound turkey that is free. Give the new value for \(r\) and the new regression equation. Explain what the negative correlation implies. What happened? f. Find and interpret the coefficient of determination using the original data. $$ \begin{array}{|c|c|} \hline \text { Weight (pounds) } & \text { Price } \\ \hline 12.3 & \$ 17.10 \\ \hline 18.5 & \$ 23.87 \\ \hline 20.1 & \$ 26.73 \\ \hline 16.7 & \$ 19.87 \\ \hline 15.6 & \$ 23.24 \\ \hline 10.2 & \$ 9.08 \end{array} $$

Five people were asked how many female first cousins they had and how many male first cousins. The data are shown in the table. Assume the trend is linear, find the correlation. and comment on what it means. $$ \begin{array}{|c|c|} \hline \text { Female } & \text { Male } \\ \hline 2 & 4 \\ \hline 1 & 0 \\ \hline 3 & 2 \\ \hline 5 & 8 \\ \hline 2 & 2 \\ \hline \end{array} $$

USA Today College published an article with the headline "Positive Correlation Found between Gym Usage and GPA." Explain what a positive correlation means in the context of this headline.
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