Data from the National Data shown in the table are the 4...

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.

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.

Key Concepts

Correlation Coefficient

The correlation coefficient is a numerical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values close to -1 indicate a strong negative linear relationship, values close to 1 indicate a strong positive linear relationship, and values near 0 imply little to no linear correlation between the two variables.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between two continuous variables by fitting a linear equation to observed data. In this equation, one variable is considered the independent variable (predictor) and the other is the dependent variable (response). The regression line is characterized by a slope and an intercept, which are chosen to best fit the data in the sense of minimizing the sum of squared differences between the observed values and the values predicted by the model.

Dependent and Independent Variables

In regression analysis, the independent variable is the variable that is assumed to cause or predict changes in another variable, while the dependent variable is the outcome or response that is being predicted. The choice of which variable is independent and which is dependent is crucial because it influences the formulation of the regression equation and the interpretation of the results.

Prediction using Regression Model

Once a regression equation is established, it can be used to predict the value of the dependent variable for a given value of the independent variable. This predictive feature is key to many applications of regression analysis, allowing inferences about future or unobserved data points based on the estimated trend derived from the sample data.

Effect of Variable Choice on Regression Equation

The regression equation and its parameters are influenced by the designation of the dependent and independent variables. While the correlation coefficient remains the same regardless of which variable is chosen as the predictor, the slope and intercept of the regression line will change if the roles of the variables are reversed. This is because simple linear regression minimizes the error in the vertical direction (i.e., differences between observed and predicted values of the dependent variable), and reassigning the roles changes this error structure and the resulting best-fit line.

Transcript

00:01 Here's a solution to the verbal and math problem.

00:03 So it says that sat scores, educational researcher desired to find out if a relationship exists between the average sat verbal score and the average sat mathematical score.

00:13 So several states were randomly selected and their sat average scores are recorded below.

00:17 Is there sufficient evidence to conclude a relationship between the two scores? so on this particular example, and we're supposed to make a scatter plot and go through the whole hypothesis testing procedure.

00:28 But we're going to test that row equals zero, meaning that there is no correlation or no relationship between the two scores.

00:34 And then the alternative is that row is not equal to zero, meaning that there is some sort of relationship.

00:40 Okay, so i'm going to do most of this on the tid for calculators.

00:43 So if you go to stat edit, and then l1, l2 is where you've got your lines there, where you have a data, just make sure that your stat plot is on, and then graph.

00:52 Now, if you don't see it, that's fine.

00:53 You just do zoom and click nine, and that'll, that should should show you, you know, a pretty good graph.

00:58 So that's what your scatter plot will look like.

01:00 So that's part a.

01:02 Compute the correlation coefficient.

01:03 Okay, so correlation coefficient.

01:06 It means if we go to stat, calc, and where is it? oh, well, yeah.

01:16 So this is a linear regression.

01:18 So we're going to go to four.

01:20 Okay, so then the a and the b, so the slope is about 0 .9, and then the b, the y intercept, is 63 .47.

01:30 We'll go a couple decimal.

01:31 Places there.