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Chapter 4: Problem 10

Use the linear correlation coefficient given to determine the coefficient of determination, \(R^{2} .\) Interpret each \(R^{2}\) (a) \(r=-0.32\) (b) \(r=0.13\) (c) \(r=0.40\) (d) \(r=0.93\)

Short Answer

Expert verified
For r=-0.32, R虏=0.1024. For r=0.13, R虏=0.0169. For r=0.40, R虏=0.16. For r=0.93, R虏=0.8649.

Step by step solution

01

- Understanding the Relationship Between r and R虏

The coefficient of determination, denoted as \(R^2\), is calculated by squaring the linear correlation coefficient, denoted as \(r\). It explains the proportion of the variance in the dependent variable that is predictable from the independent variable.
02

- Calculate R虏 for r = -0.32

To find \(R^2\) for \(r = -0.32\), square the value of \(r\): \[R^2 = (-0.32)^2 = 0.1024\] Thus, \(R^2 = 0.1024\). This means approximately 10.24% of the variance in the dependent variable is predictable from the independent variable.
03

- Calculate R虏 for r = 0.13

To find \(R^2\) for \(r = 0.13\), square the value of \(r\): \[R^2 = (0.13)^2 = 0.0169\] Thus, \(R^2 = 0.0169\). This means approximately 1.69% of the variance in the dependent variable is predictable from the independent variable.
04

- Calculate R虏 for r = 0.40

To find \(R^2\) for \(r = 0.40\), square the value of \(r\): \[R^2 = (0.40)^2 = 0.16\] Thus, \(R^2 = 0.16\). This means approximately 16% of the variance in the dependent variable is predictable from the independent variable.
05

- Calculate R虏 for r = 0.93

To find \(R^2\) for \(r = 0.93\), square the value of \(r\): \[R^2 = (0.93)^2 = 0.8649\] Thus, \(R^2 = 0.8649\). This means approximately 86.49% of the variance in the dependent variable is predictable from the independent variable.
06

- Summarize Results

For \(r = -0.32\), \(R^2 = 0.1024\) which means 10.24% predictability. For \(r = 0.13\), \(R^2 = 0.0169\) which means 1.69% predictability. For \(r = 0.40\), \(R^2 = 0.16\) which means 16% predictability. For \(r = 0.93\), \(R^2 = 0.8649\) which means 86.49% predictability.

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Key Concepts

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

linear correlation coefficient
The linear correlation coefficient, also known as Pearson's correlation coefficient, is a measure of the strength and direction of a linear relationship between two variables. It's denoted by the symbol \( r \). The value of \( r \) ranges from -1 to 1. When \( r \) is close to 1, it indicates a strong positive linear relationship. Conversely, when \( r \) is close to -1, it shows a strong negative linear relationship. An \( r \) value near 0 suggests no linear correlation between the variables.
variance
Variance is a statistical concept that measures the dispersion or spread of a set of numbers. In simpler terms, it shows how much the numbers in a dataset differ from the mean of the dataset. Higher variance indicates that the numbers are more spread out from the mean, while lower variance indicates that the numbers are close to the mean.
Variance is a crucial concept in calculating the coefficient of determination because \( R^2 \) explains how much of the variance in the dependent variable can be predicted by the independent variable.
predictability
Predictability in statistics refers to the extent to which we can predict the value of one variable based on the value of another variable. The coefficient of determination, \( R^2 \), quantifies this predictability.
For example, if \( R^2 = 0.8649 \), it means 86.49% of the variance in the dependent variable is predictable from the independent variable, thus indicating a high level of predictability. Lower \( R^2 \) values indicate lower predictability. Understanding how much variance is predictable helps analysts make informed decisions based on data-driven insights.

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Most popular questions from this chapter

Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the student's age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: $$ \begin{array}{ll|ll|ll} \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } \\ \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} \\ \hline 18 & 4.2 & 19 & 5.1 & 22 & 2.1 \\ \hline 18 & 1.1 & 19 & 2.3 & 22 & 3.6 \\ \hline 18 & 4.6 & 20 & 1.7 & 24 & 5.4 \\ \hline 18 & 3.1 & 20 & 6.1 & 25 & 4.8 \\ \hline 18 & 5.3 & 20 & 3.2 & 25 & 3.9 \\ \hline 18 & 3.2 & 20 & 5.3 & 26 & 5.2 \\ \hline 19 & 2.8 & 21 & 2.5 & 26 & 4.2 \\ \hline 19 & 2.3 & 21 & 6.4 & 35 & 8.1 \\ \hline 19 & 3.2 & 21 & 4.2 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35,8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35,8.1) has on the regression line.

Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both \(x\) and \(y\) by 2 . (d) Draw a scatter diagram of the new data with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.

Attending Class The following data represent the number of days absent, \(x\), and the final grade, \(y,\) for a sample of college students in a general education course at a large state university. $$ \begin{array}{lllllllllll} \hline \begin{array}{l} \text { No. of } \\ \text { absences, } x \end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \begin{array}{l} \text { Final } \\ \text { grade, } y \end{array} & 89.2 & 86.4 & 83.5 & 81.1 & 78.2 & 73.9 & 64.3 & 71.8 & 65.5 & 66.2 \\ \hline \end{array} $$ (a) Find the least-squares regression line treating number of absences as the explanatory variable and final grade as the response variable. (b) Interpret the slope and \(y\) -intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the final grade above or below average for this number of absences? (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods? Why or why not?

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the contingency table below. 3 $$\begin{array}{lrrrrr} & 18-34 & 35-44 & 45-54 & 55+ & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & 1329 \\\\\hline \text { Less likely } & 22 & 6 & 22 & 16 & 66 \\\\\hline \begin{array}{l}\text { Neither more } \\\\\text { nor less likely }\end{array} & 282 & 201 & 164 & 118 & 765 \\\\\hline \text { Total } & 542 & 536 & 546 & 536 & 2160\end{array}$$ (a) How many adult Americans were surveyed? How many were 55 and older? (b) Construct a relative frequency marginal distribution. (c) What proportion of Americans are more likely to buy a product when the ad says "Made in America"? (d) Construct a conditional distribution of likelihood to buy "Made in America" by age. That is, construct a conditional distribution treating age as the explanatory variable. (e) Draw a bar graph of the conditional distribution found in part (d). (f) Write a couple sentences explaining any relation between likelihood to buy and age.

What is the difference between univariate data and bivariate data?
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