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Chapter 4: Problem 69

Data on the number of home runs, strikeouts, and batting averages for a sample of 50 Major League Baseball players were obtained. Regression analyses were conducted on the relationships between home runs and strikeouts and between home runs and batting averages. The StatCrunch results are shown below. (Source: mlb.com) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Strikeouts Home Runs \(=0.092770565+0.22866236\) Strikeouts Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=0.63591835\) \(\mathrm{R}-\mathrm{sq}=0.40439215\) Estimate of error standard deviation: \(8.7661607\) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Batting Average Home Runs \(=45.463921-71.232795\) Batting Average Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=-0.093683651\) \(\mathrm{R}-\mathrm{sq}=0.0087766264\) Estimate of error standard deviation: \(11.30876\) Based on this sample, is there a stronger association between home runs and strikeouts or home runs and batting average? Provide a reason for your choice based on the StatCrunch results provided.

Short Answer

Expert verified
Based on the sample, there is a stronger association between home runs and strikeouts than between home runs and batting averages. This conclusion is supported by the larger correlation coefficient and coefficient of determination for the relationship between home runs and strikeouts compared to the values for the relationship between home runs and batting averages.

Step by step solution

01

Interpreting the Results of the Linear Relationship between Home Runs and Strikeouts

We inspect the StatCrunch results for the relationship between home runs and strikeouts. The correlation coefficient, \(\mathrm{R}\), is 0.63591835, suggesting a moderately strong, positive linear relationship. The coefficient of determination, \(R^2\), is 0.40439215, indicating that about 40.44% of the variation in home runs can be explained by strikeouts.
02

Interpreting the Results of the Linear Relationship between Home Runs and Batting Average

Next, we inspect the StatCrunch results for the relationship between home runs and batting averages. The correlation coefficient, \(\mathrm{R}\), is -0.093683651, suggesting a very weak, negative linear relationship. The coefficient of determination, \(R^2\), is 0.0087766264, indicating that just about 0.88% of the variation in home runs can be explained by batting average.
03

Compare the Relationships

We compare the absolute values of the correlation coefficients and the coefficients of determination from the two analyses. Because the absolute values for the relationship between home runs and strikeouts are larger than the ones for the relationship between home runs and batting average, we can conclude that home runs are more strongly associated with the number of strikeouts than with the batting average.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient, often denoted as \( R \), is a statistical measure that describes the strength and direction of the relationship between two variables. In the context of our baseball example, it helps quantify how well the number of home runs aligns with either the number of strikeouts or the batting average.

Correlations can range from -1 to +1:
  • A correlation of +1 indicates a perfect positive linear relationship.
  • A correlation of 0 implies no linear relationship.
  • A correlation of -1 indicates a perfect negative linear relationship.
In the analysis between home runs and strikeouts, the correlation coefficient is \( 0.63591835 \). This suggests a moderately strong, positive linear relationship between these variables, meaning as strikeouts increase, home runs tend to increase as well. Alternatively, for home runs and batting average, the correlation coefficient is \( -0.093683651 \). This shows a very weak, negative linear relationship, indicating barely any linear connectivity between these two variables.

This analysis highlights that correlation not only communicates the strength but also the direction of a relationship.
Coefficient of Determination
The coefficient of determination, represented as \( R^2 \), provides insights into how well the independent variable predicts the dependent variable in a linear regression model. It is an essential metric for understanding the effectiveness of a linear model.

It is expressed as a percentage, giving the proportion of the variance in the dependent variable that is predictable from the independent variable:
  • A high \( R^2 \) value indicates a greater proportion of variance explained by the independent variable.
  • A low \( R^2 \) value suggests that the model doesn't explain much of the variance.
In our scenario, the \( R^2 \) value for the relationship between home runs and strikeouts is \( 0.40439215 \), which means approximately 40.44% of the variation in home runs can be explained by strikeouts, indicating a moderately strong association. On the other hand, the \( R^2 \) value for home runs and batting averages is \( 0.0087766264 \), showing only about 0.88% of the variance is explained, highlighting a negligible association.

This makes \( R^2 \) a powerful tool for assessing the predictiveness of our regression models.
StatCrunch
StatCrunch is a comprehensive statistical software that aids in performing various data analyses, making statistical processes more streamlined and accessible. It is especially beneficial for students and educators, providing robust computational power for complex mathematical and statistical models.

In our exercise, StatCrunch was utilized to conduct linear regression analyses. It provided precise calculations for both the correlation coefficient \( R \) and the coefficient of determination \( R^2 \), which are crucial for interpreting and comparing relationships between variables.

Using tools like StatCrunch can significantly enhance understanding and interaction with data:
  • It simplifies the performance of statistical tests.
  • It offers easy-to-read outputs, making complex results more comprehensible.
  • It enhances learning by offering visual and numerical insights into data relationships.
For any student looking to dive deeper into statistics, becoming familiar with software like StatCrunch can improve analytical skills and aid in a more nuanced understanding of data-driven decision-making.

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Most popular questions from this chapter

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: 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.

Does a correlation of \(-0.70\) or \(+0.50\) give a larger coefficient of determination? We say that the linear relationship that has the larger coefficient of determination is more strongly correlated. Which of the values shows a stronger correlation?

The following table shows the fat content (in grams) and calories for a sample of granola bars. (Source: calorielab. com ) $$ \begin{array}{|c|l|} \hline \text { Fat (in grams) } & \text { Calories } \\ \hline 7.6 & 370 \\ \hline 3.3 & 106.1 \\ \hline 18.7 & 312.4 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Fat (in grams) } & \text { Calories } \\ \hline 3.8 & 113.1 \\ \hline 5 & 117.8 \\ \hline 5.5 & 131.9 \\ \hline 7.2 & 140.6 \\ \hline 6.1 & 118.8 \\ \hline 4.6 & 124.4 \\ \hline 3.9 & 105.1 \\ \hline 6.1 & 136 \\ \hline 4.8 & 124 \\ \hline 4.4 & 119.3 \\ \hline 7.7 & 192.6 \\ \hline \end{array} $$ a. Use technology to make a scatterplot of the data. Use fat as the independent variable \((x)\) and calories as the dependent variable \((y)\). Does there seem to be a linear trend to the data? b. Compute the correlation coefficient and the regression equation, using fat as the independent variable and calories as the dependent variable. c. What is the slope of the regression equation? Interpret the slope in the context of this problem. d. What is the \(y\) -intercept of the regression equation? Interpret the \(y\) -intercept in the context of this problem or explain why it would be inappropriate to do so. e. Find and interpret the coefficient of determination. f. Use the regression equation to predict the calories in a granola bar containing 7 grams of fat. g. Would it be appropriate to use the regression equation to predict the calories in a granola bar containing 25 grams of fat? If so, predict the number of calories in such a bar. If not, explain why it would be inappropriate to do so. h. Looking at the scatterplot there is a granola bar in the sample that has an extremely high number of calories given the moderate amount of fat it contains. Remove its data from the sample and recalculate the correlation coefficient and regression equation. How did removing this unusual point change the value of \(r\) and the regression equation?

A doctor is studying cholesterol readings in his patients. After reviewing the cholesterol readings, he calls the patients with the highest cholesterol readings (the top \(5 \%\) of readings in his office) and asks them to come back to discuss cholesterol-lowering methods. When he tests these patients a second time, the average cholesterol readings tend to have gone down somewhat. Explain what statistical phenomenon might have been partly responsible for this lowering of the readings.

The following table gives the distance from Boston to each city (in thousands of miles) and gives the time for one randomly chosen, commercial airplane to make that flight. Do a complete regression analysis that includes a scatterplot with the line, interprets the slope and intercept, and predicts how much time a nonstop flight from Boston to Seattle would take. The distance from Boston to Seattle is 3000 miles. See page 209 for guidance. $$ \begin{array}{|lcc|} \hline \text { City } & \begin{array}{c} \text { Distance } \\ \text { (1000s of miles) } \end{array} & \text { Time (hours) } \\ \hline \text { St. Louis } & 1.141 & 2.83 \\ \hline \text { Los Angeles } & 2.979 & 6.00 \\ \hline \text { Paris } & 3.346 & 7.25 \\ \hline \text { Denver } & 1.748 & 4.25 \\ \hline \text { Salt Lake City } & 2.343 & 5.00 \\ \hline \text { Houston } & 1.804 & 4.25 \\ \hline \text { New York } & 0.218 & 1.25 \\ \hline \end{array} $$
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