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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

Suppose that the growth rate of children looks like a straight line if the height of a child is observed at the ages of 24 months, 28 months, 32 months, and 36 months. If you use the regression obtained from these ages and predict the height of the child at 21 years, you might find that the predicted height is 20 feet. What is wrong with the prediction and the process used?

The following table shows the number of text messages sent and received by some people in one day. (Source: StatCrunch: Responses to survey How often do you text? Owner: Webster West. A subset was used.) a. Make a scatterplot of the data, and state the sign of the slope from the scatterplot. Use the number sent as the independent variable. b. Use linear regression to find the equation of the best-fit line. Graph the line with technology or by hand. c. Interpret the slope. d. Interpret the intercept. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array}\\\ &\begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} \end{aligned} $$

The following table gives the distance from Boston to each city and the cost of a train ticket from Boston to that city for a certain date. $$ \begin{array}{lcc} \hline \text { City } & \text { Distance (in miles) } & \text { Ticket Price (in \$) } \\ \hline \text { Washington, } & 439 & 181 \\ \text { D.C. } & & \\ \hline \text { Hartford } & 102 & 73 \\ \hline \text { New York } & 215 & 79 \\ \hline \text { Philadelphia } & 310 & 293 \\ \hline \text { Baltimore } & 406 & 175 \\ \hline \text { Charlotte } & 847 & 288 \\ \hline \text { Miami } & 1499 & 340 \\ \hline \text { Roanoke } & 680 & 219 \\ \hline \text { Atlanta } & 1086 & 310 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { City } & \text { Distance (in miles) } & \text { Ticket Price (in \$) } \\ \hline \text { Tampa } & 1349 & 370 \\ \text { Montgomery } & 1247 & 373 \\ \text { Columbus } & 776 & 164 \\ \hline \text { Indianapolis } & 950 & 245 \\ \hline \text { Detroit } & 707 & 189 \\ \hline \text { Nashville } & 1105 & 245 \\ \hline \end{array} $$ a. Use technology to produce a scatterplot. Based on your scatterplot do you think there is a strong linear relationship between these two variables? Explain. b. Compute \(r\) and write the equation of the regression line. Use the words "Ticket Price" and "Distance" in your equation. Round off to two decimal places. c. Provide an interpretation of the slope of the regression line. d. Provide an interpretation of the \(y\) -intercept of the regression line or explain why it would not be appropriate to do so. e. Use the regression equation to predict the cost of a train ticket from Boston to Pittsburgh, a distance of 572 miles.

The computer output shown below is for predicting foot length from hand length (in centimeters) for a group of women. Assume the trend is linear. Summary statistics for the data are shown in the table below. $$ \begin{array}{|l|l|c|} \hline & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Hand, } x & 17.682 & 1.168 \\ \hline \text { Foot, } y & 23.318 & 1.230 \\ \hline \end{array} $$

Answer the questions using complete sentences. a. What is an influential point? How should influential points be treated when doing a regression analysis? b. What is the coefficient of determination and what does it measure? c. What is extrapolation? Should extrapolation ever be used?
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