Question

In Exercise 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. \({ }^{6}\) The data and the MINITAB printout are shown here. $$ \begin{array}{crr|ccc} & & \text { Final } & & & \text { Final } \\ \text { Student } & \text { Posttest } & \text { Exam } & \text { Student } & \text { Posttest } & \text { Exam } \\ \hline 1 & 100 & 98 & 11 & 88 & 84 \\ 2 & 96 & 97 & 12 & 92 & 93 \\ 3 & 88 & 88 & 13 & 68 & 57 \\ 4 & 100 & 100 & 14 & 84 & 84 \\ 5 & 100 & 100 & 15 & 84 & 81 \\ 6 & 96 & 78 & 16 & 88 & 83 \\ 7 & 80 & 68 & 17 & 72 & 84 \\ 8 & 68 & 47 & 18 & 88 & 93 \\ 9 & 92 & 90 & 19 & 72 & 57 \\ 10 & 96 & 94 & 20 & 88 & 83 \end{array} $$ a. Construct a scatterplot 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 posttest score? c. Do the data present sufficient evidence to indicate that final exam score is linearly related to the posttest score? Use \(\alpha=.01\) d. Find a \(99 \%\) confidence interval for the slope of the regression line.

Short answer

Answer: The main steps to analyze the relationship between posttest scores and final exam scores using linear regression are: 1. Construct a scatterplot to assess linearity. 2. Find the equation of the regression line. 3. Test for significance of the relationship using a specified alpha value. 4. Calculate a confidence interval for the slope of the regression line.

Step-by-step solution

Construct a Scatterplot

First, plot the posttest scores (x-axis) against the final exam scores (y-axis) to visually inspect if there is a linear relationship between the two sets of scores. This can be done using a graphing calculator, spreadsheet software like Excel, or statistical analysis software.

Obtain the Regression Line Equation

Using the given data, calculate the slope and the y-intercept of the regression line using the formula: \( y = bx + a\), where: \( b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^{2}) - (\sum x)^{2}}\) \( a = \bar{y} - b\bar{x} \)

Test for Significance of the Relationship

We want to test if there is enough evidence to indicate that the final exam score (y) is linearly related to the posttest score (x) at a significance level (alpha) of 0.01. State the null and alternative hypotheses: \(H_{0}: \rho = 0\) (There is no significant linear relationship between x and y) \(H_{A}: \rho \neq 0\) (There is a significant linear relationship between x and y) Calculate the test statistic (t-value) using the formula: \( t = \frac{b - 0}{s_{b}}\) Where \(s_{b}\) is the standard error of the slope which can be calculated using: \( s_{b} = \sqrt{\frac{(n-1)s_{y}^{2}}{(n-1)s_{x}^{2}}} \) Use a t-distribution table to find the critical t-value corresponding to the given alpha value (0.01) and degrees of freedom (n - 2). If the calculated t-value is greater than the critical t-value, we reject the null hypothesis, concluding that there is a significant linear relationship between the final exam score and the posttest score.

Calculate the Confidence Interval for the Slope

To find the 99% confidence interval for the slope of the regression line, use the formula: \( CI = b \pm t_{\alpha/2} * s_{b} \) Where \(t_{\alpha/2}\) is the critical t-value corresponding to the 99% confidence level and the degrees of freedom (n - 2), and \(s_{b}\) is the standard error of the slope. This will give you the range within which you can be 99% certain that the true population slope lies.

Key concepts

Scatterplot Construction

Initiating our journey into linear regression analysis with an understanding of scatterplots is a vital step. A scatterplot provides a visual representation of how two variables are possibly related. To create a scatterplot, one variable is plotted along the x-axis, while the other is plotted along the y-axis. Each point on the scatterplot corresponds to a single observation, connecting the value of one variable to the other. In the context of the McNair Academic High School experiment, students' posttest scores are placed on the x-axis with their final exam scores on the y-axis.

After plotting all the points, we can visually analyze the pattern. If the points roughly form a line or a curve, that suggests a relationship between the variables. For our educational data, if the points ascend from left to right, it indicates that higher posttest scores might be associated with higher final exam scores. This visualization is crucial for the initial assessment of whether a linear model could be appropriate before diving deeper into analytical methods.

Regression Line Equation

With the understanding of our scatterplot, we turn to crafting the equation of the regression line, also known as the 'line of best fit'. The equation encapsulates the linear relationship between our two variables. In the simplest form, the equation is expressed as \( y = bx + a \), where \( y \) is the response variable (final exam scores), \( x \) is the predictor variable (posttest scores), \( b \) is the slope, and \( a \) is the y-intercept.

The slope, \( b \), represents the average change in the response variable for each one-unit change in the predictor. In our high school experiment, it reflects how much we expect the final exam score to increase for each additional point on the posttest score. The intercept, \( a \), signifies the expected value of \( y \) when \( x \) is zero — though in practical terms for our case, the intercept may not have a sensible interpretation, as posttest scores will not be zero. Thus, the regression line equation is a key takeaway, offering a predictive tool based on our available data.

Hypothesis Testing

Hypothesis testing in linear regression is used to ascertain the significance of the relationship between variables. For our exercise, this statistical method helps us test if the final exam scores (\( y \)) are indeed linearly related to the posttest scores (\( x \)). To do this, we formulate two hypotheses: the null hypothesis (\( H_{0} \)), which posits that there is no correlation between the variables (\( \rho = 0 \)), and the alternative hypothesis (\( H_{A} \)), which suggests that there is a significant linear correlation (\( \rho eq 0 \)).

By computing the test statistic, often a t-value, and comparing it to a critical value from a t-distribution table, we can support or refute our null hypothesis. If the test statistic exceeds the critical value, it indicates a significant relationship, and we reject the null hypothesis in favor of the alternative. This step concludes whether or not the educational data presents a statistically significant linear association.

Confidence Interval for Slope

Moving beyond hypothesis testing, we address the precision of our slope estimate through the confidence interval. This interval gives us a range in which the true population slope is likely to fall. Calculating a 99% confidence interval for the slope means we can state with 99% certainty that the population's true slope is within this range.

The formula \( CI = b \pm t_{\alpha/2} * s_{b} \) involves the estimated slope (\( b \)), the critical t-value for our desired confidence level (\( t_{\alpha/2} \)), and the standard error of the slope (\( s_{b} \)). For the McNair Academic High School data, this interval helps articulate how confident we can be in the predictive power of the posttest scores on students' final exam scores, with a bounded degree of statistical certainty. By including a confidence interval in our analysis, we significantly enhance the reliability of our conclusions.