91Ó°ÊÓ

In Exercise 1.61 we described an informal experiment conducted at McNair Academic High School in Jersey City, New Jersey. Two freshman algebra classes were studied, one of which used laptop computers at school and at home, while the other class did not. In each class, students were given a survey at the beginning and end of the semester, measuring his or her technological level. The scores were recorded for the end of semester survey \((x)\) and the final examination \((y)\) for the laptop group. \({ }^{5}\) The data and the MINITAB printout are shown here. $$ \begin{array}{rrr|rrr} & & \text { Final } & & & \text { Final } \\ \text { Student } & \text { Posttest } & \text { Exam } & \text { Student } & \text { Posttest } & \text { Exam } \\ \hline 1 & 100 & 98 & 11 & 88 & 84 \\ 2 & 96 & 97 & 12 & 92 & 93 \\ 3 & 88 & 88 & 13 & 68 & 57 \\ 4 & 100 & 100 & 14 & 84 & 84 \\ 5 & 100 & 100 & 15 & 84 & 81 \\ 6 & 96 & 78 & 16 & 88 & 83 \\ 7 & 80 & 68 & 17 & 72 & 84 \\ 8 & 68 & 47 & 18 & 88 & 93 \\\ 9 & 92 & 90 & 19 & 72 & 57 \\ 10 & 96 & 94 & 20 & 88 & 83 \end{array} $$a. Construct a scatter plot for the data. Does the assumption of linearity appear to be reasonable? b. What is the equation of the regression line used for predicting final exam score as a function of the post test score? c. Do the data present sufficient evidence to indicate that final exam score is linearly related to the post test score? Use \(\alpha=.01\). d. Find a \(99 \%\) confidence interval for the slope of the regression line.

Step by step solution

01

Construct a scatter plot

Draw a scatter plot using the posttest scores on the x-axis and the final exam scores on the y-axis. After plotting the points, visually assess whether the assumption of linearity appears to be reasonable.

02

Find the equation of the regression line

(Calculations need to be performed using a software, calculator or manually) Calculate the sample covariance, denoted by \(S_{xy}\), and the sample variances of \(x\) and \(y\), denoted by \(S_x^2\) and \(S_y^2\). Compute the sample correlation coefficient \(r\) using the formula: $$r = \frac{S_{xy}}{\sqrt{S_x^2 \cdot S_y^2}}$$ Next, compute the slope \(b_1\) and the y-intercept \(b_0\) of the regression line using the formulas: $$b_1 = \frac{S_{xy}}{S_x^2}$$ $$b_0 = \overline{y} - b_1\overline{x}$$ Now, the equation of the regression line is given by: $$ \hat{y} = b_0 + b_1x$$

03

Test the linear relationship

Test the null hypothesis \(H_0\): There is no linear relationship (i.e., \(\rho = 0\)) between the final exam score and the posttest score against the alternative hypothesis \(H_1\): There is a linear relationship (i.e., \(\rho \ne 0\)) between the final exam score and the posttest score using the test statistic: $$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$ where \(n\) is the number of pairs. Compare the computed value of the test statistic with the critical value at \(\alpha = 0.01\) with \(n-2\) degrees of freedom to determine whether to reject or fail to reject the null hypothesis.

04

Find the confidence interval for the slope

Compute the standard error of the estimate of the slope \(b_1\), denoted by \(SE_b\) using the formula: $$SE_b = \frac{\sqrt{1-r^2}\cdot S_y}{\sqrt{n}\cdot S_x}$$ Find the critical value for the desired confidence level (in this case, 99%) with \(n-2\) degrees of freedom from the t-table and calculate the confidence interval for the slope: $$b_1 \pm t_{\alpha/2, n-2} \cdot SE_b$$

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Key Concepts

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

Scatter Plot

A scatter plot is a powerful visual tool used in statistics to display the relationship between two variables. In this context, it helps us visualize the connection between posttest scores (on the horizontal x-axis) and final exam scores (on the vertical y-axis) for a group of students. Each point on the plot represents a student's posttest score and their final exam score.

To construct a scatter plot, you plot each pair of corresponding values as a single point on the graph. For example, if one student scored 96 on their posttest and 97 on their final exam, you will mark a point at the coordinate (96, 97). Repeating this for all students helps to visualize the data spread and any pattern.

The main objective here is to assess whether a linear relationship appears likely between the two variables. If the points on the scatter plot roughly form a straight line (upward or downward), a linear relationship assumption is reasonable. However, if the points are widely scattered with no apparent trend, this assumption might not hold true.

Sample Correlation Coefficient

The sample correlation coefficient, often denoted as \( r \), is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. Here:

• \( r = 1 \) indicates a perfect positive linear relationship.
• \( r = -1 \) indicates a perfect negative linear relationship.
• \( r = 0 \) suggests no linear relationship.

To calculate \( r \), use the formula:\[ r = \frac{S_{xy}}{\sqrt{S_x^2 \cdot S_y^2}} \]Where:

• \( S_{xy} \) is the sample covariance, showing how much the two variables change together.
• \( S_x^2 \) and \( S_y^2 \) are the variances of the posttest scores and final exam scores, respectively.

The sample covariance is calculated based on how each pair of scores deviate from their respective means. If \( r \) is close to 1 or -1, it supports the presence of a strong linear relationship, making further analysis with linear regression more plausible. If it's near zero, the relationship is weak or non-linear.

Confidence Interval

A confidence interval provides a range of values within which the true parameter, like the slope of a regression line, is expected to lie with a certain level of confidence. Here, we focus on finding a 99% confidence interval for the slope of the regression line that predicts exam scores from posttest scores.

This process involves determining the standard error of the slope, \( SE_b \), using the formula:\[ SE_b = \frac{\sqrt{1-r^2} \cdot S_y}{\sqrt{n} \cdot S_x} \]Here:

• \( r \) is the sample correlation coefficient.
• \( S_y \) is the standard deviation of the exam scores.
• \( n \) is the number of students.
• \( S_x \) is the standard deviation of the posttest scores.

Using the calculated standard error and the critical value from the t-distribution table with \( n-2 \) degrees of freedom, we compute the confidence interval as:\[ b_1 \pm t_{\alpha/2, n-2} \cdot SE_b \]Where \( b_1 \) is the estimated slope of the regression line. This interval gives a range for the slope within which we can be 99% confident the true slope lies, if repeated samples were taken under the same condition.