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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

The distance (in kilometers) and price (in dollars) for one-way airline tickets from San Francisco to several cities are shown in the table. $$\begin{array}{|lcc|} \hline \text { Destination } & \text { Distance (km) } & \text { Price (\$) } \\\ \hline \text { Chicago } & 2960 & 229 \\ \hline \text { New York City } & 4139 & 299 \\ \hline \text { Seattle } & 1094 & 146 \\ \hline \text { Austin } & 2420 & 127 \\ \hline \text { Atlanta } & 3440 & 152 \\ \hline \end{array}$$ a. Find the correlation coefficient for this data using a computer or statistical calculator. Use distance as the \(x\) -variable and price as the \(y\) -variable. b. Recalculate the correlation coefficient for this data using price as the \(x\) -variable and distance as the \(y\) -variable. What effect does this have on the correlation coefficient? c. Suppose a $$\$ 50$$ security fee was added to the price of each ticket. What effect would this have on the correlation coefficient? d. Suppose the airline held an incredible sale, where travelers got a round- trip ticket for the price of a one-way ticket. This means that the distances would be doubled while the ticket price remained the same. What effect would this have on the correlation coefficient?

Seth Wagerman, a former professor at California Lutheran University, went to the website RateMyProfessors.com and looked up the quality rating and also the "easiness" of the six full-time professors in one department. The ratings are 1 (lowest quality) to 5 (highest quality) and 1 (hardest) to 5 (easiest). The numbers given are averages for each professor. Assume the trend is linear, find the correlation, and comment on what it means. $$ \begin{array}{|c|c|} \hline \text { Quality } & \text { Easiness } \\ \hline 4.8 & 3.8 \\ \hline 4.6 & 3.1 \\ \hline 4.3 & 3.4 \\ \hline 4.2 & 2.6 \\ \hline 3.9 & 1.9 \\ \hline 3.6 & 2.0 \\ \hline \end{array} $$

United Press International published an article with the headline "Study Finds Correlation between Education, Life Expectancy." Would you expect this correlation to be negative or positive? Explain your reasoning in the context of this headline.

The table shows the calories in a five-ounce serving and the \(\%\) alcohol content for a sample of wines. (Source: healthalicious.com) $$ \begin{array}{|c|c|} \hline \text { Calories } & \% \text { alcohol } \\ \hline 122 & 10.6 \\ \hline 119 & 10.1 \\ \hline 121 & 10.1 \\ \hline 123 & 8.8 \\ \hline 129 & 11.1 \\ \hline 236 & 15.5 \\ \hline \end{array} $$ a. Make a scatterplot using \(\%\) alcohol as the independent variable and calories as the dependent variable. Include the regression line on your scatterplot. Based on your scatterplot do you think there is a strong linear relationship between these variables? b. Find the numerical value of the correlation between \(\%\) alcohol and calories. Explain what the sign of the correlation means in the context of this problem. c. Report the equation of the regression line and interpret the slope of the regression line in the context of this problem. Use the words calories and \% alcohol in your equation. Round to two decimal places. d. Find and interpret the value of the coefficient of determination. e. Add a new point to your data: a wine that is \(20 \%\) alcohol that contains 0 calories. Find \(r\) and the regression equation after including this new data point. What was the effect of this one data point on the value of \(r\) and the slope of the regression equation?

The table for part (a) shows distances between selected cities and the cost of a business class train ticket for travel between these cities. a. Calculate the correlation coefficient for the data shown in the table by using a computer or statistical calculator. $$\begin{array}{|c|c|} \hline \text { Distance (in miles) } & \text { Cost (in \$) } \\ \hline 439 & 281 \\ \hline 102 & 152 \\ \hline 215 & 144 \\ \hline 310 & 293 \\ \hline 406 & 281 \\ \hline \end{array}$$ b. The table for part (b) shows the same information, except that the distance was converted to kilometers by multiplying the number of miles by \(1.609\). What happens to the correlation when the numbers are multiplied by a constant? $$ \begin{array}{|c|c|} \hline \text { Distance (in kilometers) } & \text { Cost } \\ \hline 706 & 281 \\ \hline 164 & 152 \\ \hline 346 & 144 \\ \hline 499 & 293 \\ \hline 653 & 281 \\ \hline \end{array} $$ c. Suppose a surcharge is added to every train ticket to fund track maintenance. A fee of $$\$ 20$$ is added to each ticket, no matter how long the trip is. The following table shows the new data. What happens to the correlation coefficient when a constant is added to each number? $$ \begin {array} { | c | c |} \hline \text { Distance (in miles) } & \text { Cost (in \$) } \\ \hline 439 & 301 \\ \hline 102 & 172 \\ \hline 215 & 164 \\ \hline 310 & 313 \\ \hline 406 & 301 \\ \hline \end {array} $$
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