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Chapter 3: Problem 38

Example 9 related \(y=\) team scoring (per game) and \(x=\) team batting average for American League teams. For National League teams in 2010 , \(\hat{y}=-6.25+41.5 x\). (Data available on the book's website in the NL team statistics file.) a. The team batting averages fell between 0.242 and 0.272. Explain how to interpret the slope in context. b. The standard deviations were 0.00782 for team batting average and 0.3604 for team scoring. The correlation between these variables was 0.900 . Show how the correlation and slope of 41.5 relate in terms of these standard deviations. c. Software reports \(r^{2}=0.81 .\) Explain how to interpret this measure.

Short Answer

Expert verified
a. The slope 41.5 implies higher batting averages increase scoring. b. Slope relates to correlation and standard deviations as given by \( b = r \times (\frac{s_y}{s_x}) \). c. \( r^2 = 0.81 \) indicates 81% of scoring variability is explained by batting average.

Step by step solution

01

Contextual Interpretation of Slope

In the regression equation \( \hat{y} = -6.25 + 41.5x \), the slope is 41.5. This indicates that for every increase of 1 unit in team batting average \( x \), the team scoring \( \hat{y} \) is expected to increase by 41.5 runs per game on average. In simpler terms, higher team batting averages are strongly associated with more runs scored each game.
02

Relating Slope to Correlation and Standard Deviations

The formula relating the slope \( b \) of the regression line to the correlation \( r \) is given by \( b = r \times (\frac{s_y}{s_x}) \), where \( s_y \) and \( s_x \) are the standard deviations of team scoring and team batting average, respectively. Thus, \( 41.5 = 0.900 \times (\frac{0.3604}{0.00782}) \). This verifies that the slope can indeed be derived from the correlation and the standard deviations of the two variables.
03

Interpretation of \(r^2\)

The \( r^2 \) value, also known as the coefficient of determination, is 0.81 or 81%. This means that 81% of the variability in team scoring is accounted for by the variability in the team batting averages. In essence, the team batting average is a strong predictor of team scoring, explaining the majority of its variation.

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Key Concepts

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

Slope Interpretation
In regression analysis, the concept of slope is crucial to understand. It tells us how much the dependent variable changes with a change in the independent variable. In the equation \( \hat{y} = -6.25 + 41.5x \), the slope is 41.5. This means that for every 1 unit increase in the team batting average (\( x \)), the team scoring (\( \hat{y} \)) is expected to increase by 41.5 runs. Simply put:

- A higher team batting average results in more runs per game.
- This suggests a strong positive relationship between batting average and scoring.

The slope is essential because it quantifies the strength and direction of this relationship. A positive slope like 41.5 means that as one variable increases, the other does too.
Correlation and Standard Deviation Relationship
The relationship between the slope of a regression line, correlation, and standard deviation is captured through a specific formula. This formula is:

\[ b = r \times \left( \frac{s_y}{s_x} \right) \]

Where:
- \( b \) is the slope.
- \( r \) is the correlation coefficient.
- \( s_y \) is the standard deviation of the team scoring.
- \( s_x \) is the standard deviation of the team batting average.

In this exercise, we have:
- Correlation, \( r = 0.900 \)
- \( s_y = 0.3604 \)
- \( s_x = 0.00782 \)

Putting these into the formula verifies that:
\[ 41.5 = 0.900 \times \left( \frac{0.3604}{0.00782} \right) \]

This relationship means that the slope is heavily influenced by how closely the variables are related (correlation) and how spread out they are (standard deviations).
Coefficient of Determination (R-squared)
The coefficient of determination, often referred to as \( R^2 \), is a valuable statistic in regression analysis. It measures how well the independent variable predicts the dependent variable. In our example, \( R^2 = 0.81 \). This translates to 81%, which has a particular interpretation:

- 81% of the variability in team scoring is explained by team batting averages.
- It shows the strength of the prediction: how well the batting average predicts scoring.
- A higher \( R^2 \) means a more reliable model.

Simply, if \( R^2 \) is 1 or 100%, it means the model perfectly predicts all data points without error. However, \( R^2 = 0.81 \) is already a strong indicator that batting average is an excellent predictor of team scoring.

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Most popular questions from this chapter

In 2015, eighth-grade math scores on the National Assessment of Educational Progress had a mean of 283.56 in Maryland compared to a mean of 284.37 in Connecticut (Source: http://nces.ed.gov/nationsreportcard/ naepdata/dataset.aspx). a. Identify the response variable and the explanatory variable. b. The means in Maryland were respectively \(274,284,285,\) 291 and 294 for people who reported the number of pages read in school and for homework, respectively as \(0-5,6-10,11-15,15-20\) and 20 or more. These means were 270,281,284,289 and 293 in Connecticut. Identify the third variable given here. Explain how it is possible for Maryland to have the higher mean for each class, yet for Connecticut to have the higher mean when the data are combined. (This is a case of Simpson's paradox for a quantitative response.)

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In an introductory statistics course, \(x=\) midterm exam score and \(y=\) final exam score. Both have mean \(=80\) and standard deviation \(=10\). The correlation between the exam scores is 0.70 . a. Find the regression equation. b. Find the predicted final exam score for a student with midterm exam score \(=80\) and another with midterm exam score \(=90\).

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In 2015, an article published in the journal Breast Cancer Research and Treatment examined the impact of diabetes on the stages of breast cancer. The study concluded that diabetes is associated with advanced stages of breast cancer in patients and this could be a reason behind higher mortality rates. The researchers suggested looking at the possibility of race/ethnicity being a possible confounder. a. Explain what the last sentence means and how race/ ethnicity could potentially explain the association between diabetes and breast cancer. b. If race/ethnicity was not measured in the study and the researchers failed to consider its effects, could it be a confounding variable or a lurking variable? Explain the difference between a lurking variable and a confounding variable.
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