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Chapter 4: Problem 73

Assume that in a political science class, the teacher gives a midterm exam and a final exam. Assume that the association between midterm and final scores is linear. The summary statistics have been simplified for clarity see Guidance on page \(209 .\) Midterm: \(\quad\) Mean \(=75, \quad\) Standard deviation \(=10\) Final: Mean \(=75, \quad\) Standard deviation \(=10\) Also, \(r=0.7\) and \(n=20\). According to the regression equation, for a student who gets a 95 on the midterm, what is the predicted final exam grade? What phenomenon from the chapter does this demonstrate? Explain. See page 209 for guidance.

Short Answer

Expert verified
The predicted final exam grade for a student who scores 95 on the midterm is 89.5. This demonstrates the phenomenon of regression to the mean.

Step by step solution

01

Understand the Data and Problem

First, understand the provided data. Midterm and final exams both have means of 75 and standard deviations of 10. Given a midterm score (95 in this case) and the linear correlation coefficient (0.7), the goal is to predict the final exam score.
02

Formulate the Regression Equation

Utilizing a formula for simple linear regression, the equation for the regression line can be defined as: \(Y = a + bX\). Here, \(Y\) represents the final exam score, \(X\) the midterm exam score, \(a\) is the intercept, and \(b\) is the slope of the regression line. \nTo find the slope \(b\), use the formula \(b = r \times \frac {s_{Y}}{s_{X}}\), where \(r\) is correlation coefficient, and \(s_{Y}\) and \(s_{X}\) are standard deviations. Here, \(b = 0.7 \times \frac {10}{10} = 0.7\). As for \(a\), it is equal to Mean of Y - \(b \times\) Mean of \(X\). Hence, \(a = 75 - 0.7 \times 75 = 22.5\). Therefore, the regression equation becomes \(Y = 22.5 + 0.7X\).
03

Predict the Final Exam Score

With the regression equation and the midterm score at hand, we can now predict the final exam score. Insert 95 (the midterm score) in the equation: \(Y = 22.5 + 0.7 \times 95 = 89.5\). Therefore, the predicted final exam score based on a midterm score of 95 is 89.5.
04

Identify the Demonstrated Phenomenon

The exercise demonstrates the concept of regression to the mean. It illustrates that if a student scores high on one test (95 on the midterm), their score on another related test (final) is expected to be closer to the mean of the population.

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Key Concepts

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

Linear Regression
Linear regression is a method in statistics that allows us to predict the value of a dependent variable based on the value of an independent variable. The relationship between these two variables is assumed to be linear, which means it can be represented by a straight line. In the context of our exercise, the midterm exam score is the independent variable (\(X\)), and the final exam score is the dependent variable (\(Y\)).

By calculating the slope (\(b\)) and the intercept (\(a\)) of the regression line, we can formulate the linear regression equation, which in our case is \(Y = 22.5 + 0.7X\). This allows us to predict the final exam score for any given midterm score. For example, a student who scores 95 on the midterm would be predicted to score 89.5 on the final. This simple yet powerful tool is widely used in various fields to analyze and forecast data.
Correlation Coefficient
The correlation coefficient, often represented by the symbol \(r\), measures the strength and direction of a linear relationship between two variables. It is a value between -1 and 1 where:\

  • 1 indicates a perfect positive linear correlation.

  • 0 indicates no linear correlation.

  • -1 indicates a perfect negative linear correlation.

In our exercise, the correlation coefficient of 0.7 suggests a strong positive association between midterm and final exam scores. This means that typically, students who perform well on the midterm exam tend to also do well on the final exam. However, it's important to note that correlation does not imply causation, and other factors could influence these scores.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (average) of the set, while a high standard deviation indicates that the values are spread out over a wider range. In statistical analysis, standard deviation is key for measuring the reliability of statistical conclusions.

Both the midterm and final exam scores have a standard deviation of 10, which shows how far scores are spread around the mean of 75. If we imagine the distribution of exam scores, the standard deviation helps us understand how concentrated the scores are around the average performance. This statistic is instrumental in the regression equation, as it is used to calculate the slope and gives context to our predictions in relation to the mean score.

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Most popular questions from this chapter

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?

The table shows the heights (in inches) and weights (in pounds) of 14 college men. The scatterplot shows that the association is linear enough to proceed. $$ \begin{array}{c|c|c|c|} \hline \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} & \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 68 & 205 & 70 & 200 \\ \hline 68 & 168 & 69 & 175 \\ \hline 74 & 230 & 72 & 210 \\ \hline 68 & 190 & 72 & 205 \\ \hline 67 & 185 & 72 & 185 \\ \hline 69 & 190 & 71 & 200 \\ \hline 68 & 165 & 73 & 195 \\ \hline \end{array} $$ a. Find the equation for the regression line with weight (in pounds) as the response and height (in inches) as the predictor. Report the slope and the intercept of the regression line, and explain what they show. If the intercept is not appropriate to report, explain why. b. Find the correlation between weight (in pounds) and height (in inches). c. Find the coefficient of determination and interpret it. d. If you changed each height to centimeters by multiplying heights in inches by \(2.54\), what would the new correlation be? Explain. e. Find the equation with weight (in pounds) as the response and height (in inches) as the predictor, and interpret the slope. f. Summarize what you found: Does changing units change the correlation? Does changing units change the regression equation?

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?

Construct a set of numbers (with at least three points) with a strong negative correlation. Then add one point (an influential point) that changes the correlation to positive. Report the data and give the correlation of each set.

Coefficient of Determination If the correlation between height and weight of a large group of people is \(0.67\), find the coefficient of determination (as a percentage) and explain what it means. Assume that height is the predictor and weight is the response, and assume that the association between height and weight is linear.
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