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Chapter 4: Problem 55

If there is a positive correlation between number of years studying math and shoe size (for children), does that prove that larger shoes cause more studying of math or vice versa? Can you think of a confounding variable that might be influencing both of the other variables?

Short Answer

Expert verified
No, the correlation between years of studying math and shoe size does not prove that one causes the other. A possible confounding variable in this scenario can be the age of the children.

Step by step solution

01

Understanding Correlation and Causation

First, it's important to understand that correlation - a statistical relationship between two variables - does not necessarily imply causation. Just because there is a positive correlation between the number of years studying math and shoe size does not mean that having larger shoes causes more studying of math or vice versa.
02

Identifying a Confounding Variable

A confounding variable is a variable that influences both the dependent variable and independent variable and can cause a spurious association. In this case, a possible confounding variable can be the age of children. As children get older, they usually study more complex subjects such as math, and at the same time, their feet grow larger leading to bigger shoe sizes.

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Most popular questions from this chapter

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.

Seth Wagerman, a former professor at California Lutheran University, went to the website RateMyProfessors.com and looked up the quality rating and also the "easiness" of the six full-time professors in one department. The ratings are 1 (lowest quality) to 5 (highest quality) and 1 (hardest) to 5 (easiest). The numbers given are averages for each professor. Assume the trend is linear, find the correlation, and comment on what it means. $$ \begin{array}{|c|c|} \hline \text { Quality } & \text { Easiness } \\ \hline 4.8 & 3.8 \\ \hline 4.6 & 3.1 \\ \hline 4.3 & 3.4 \\ \hline 4.2 & 2.6 \\ \hline 3.9 & 1.9 \\ 3.6 & 2.0 \\ \hline \end{array} $$

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. $$ \begin{array}{|c|c|} \hline \text { Female } & \text { Male } \\ \hline 2 & 4 \\ \hline 1 & 0 \\ \hline 3 & 2 \\ \hline 5 & 8 \\ \hline 2 & 2 \\ \hline \end{array} $$

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. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array}\\\ &\begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} \end{aligned} $$

Data from the National Data shown in the table are the 4 th-grade reading and math scores for a sample of states from the National Assessment of Educational Progress. The scores represent the percentage of 4 thgraders in each state who scored at or above basic level in reading and math. A scatterplot of the data suggests a linear trend. (Source: nationsreportcard.gov) $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { 4th-Grade } \\ \text { Reading } \\ \text { Scores } \end{array} & \begin{array}{c} \text { 4th-Grade } \\ \text { Math Scores } \end{array} & \begin{array}{c} \text { 4th-Grade } \\ \text { Reading } \\ \text { Scores } \end{array} & \begin{array}{c} \text { 4th-Grade } \\ \text { Math Scores } \end{array} \\ \hline 65 & 75 & 68 & 78 \\ \hline 61 & 78 & 61 & 79 \\ \hline 62 & 79 & 69 & 80 \\ \hline 65 & 79 & 68 & 77 \\ \hline 59 & 72 & 75 & 89 \\ \hline 72 & 82 & 71 & 84 \\ \hline 74 & 81 & 68 & 83 \\ \hline 70 & 82 & 75 & 84 \\ \hline 56 & 69 & 63 & 78 \\ \hline 75 & 85 & 71 & 85 \\ \hline \end{array} $$ a. Find and report the value for the correlation coefficient and the regression equation for predicting the math score from the reading score. Use the words Reading and Math in your regression equation and round off to two decimal places. Then find the predicted math score for a state with a reading score of 70 . b. Find and report the value of the correlation coefficient regression equation for predicting the reading score from the math score. Then find the predicted reading score for a state with a math score of 70 . c. Discuss the effect of changing the choice of dependent and independent variable on the value of \(r\) and on the regression equation.
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