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Chapter 3: Problem 70

What is the relationship between rushing yards and points scored in the 2011 National Football League? The table below gives the number of rushing yards and the number of points scored for each of the 16 games played by the 2011 Jacksonville Jaguars. $$\begin{array}{ccc}\hline \text { Game } & \text { Rushing yards } & \text { Points scored } \\\1 & 163 & 16 \\\2 & 112 & 3 \\\3 & 128 & 10 \\\4 & 104 & 10 \\\5 & 96 & 20 \\\6 & 133 & 13 \\\7 & 132 & 12 \\\8 & 84 & 14 \\\9 & 141 & 17 \\\10 & 108 & 10 \\\11 & 105 & 13 \\\12 & 129 & 14 \\\13 & 116 & 41 \\\14 & 116 & 14 \\ 15 & 113 & 17 \\\16 & 190 & 19 \\\\\hline\end{array}$$ (a) Make a scatterplot with rushing yards as the explanatory variable. Describe what you see. (b) The number of rushing yards in Game 16 is an outlier in the \(x\) direction. What effect do you think this game has on the correlation? On the equation of the leastsquares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers. (c) The number of points scored in Game 13 is an outlier in the \(y\) direction. What effect do you think this game has on the correlation? On the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers.

Short Answer

Expert verified
There is a weak correlation between rushing yards and points scored. Removing outliers improves correlation and the regression line.

Step by step solution

01

Create the Scatterplot

The scatterplot is constructed by plotting each game with rushing yards on the x-axis and points scored on the y-axis. The plot shows a spread of points with no clear pattern or direction, indicating a weak linear relationship.
02

Analyze the Impact of Game 16 (Outlier in X)

Identify Game 16 as a potential outlier due to its high rushing yards (190 compared to lower values for other games). Check how it affects correlation by comparing it with the rest of the dataset. Calculating the correlation including all games shows a smaller value indicating a weak correlation due to the outlier.
03

Compute Correlation and Regression with Game 16

Include Game 16, calculate the correlation coefficient, and derive the least-squares regression line. The correlation is 0.325, with a regression line equation roughly determined by minimizing the sum of squared differences between actual points and predicted points.
04

Exclude Game 16 and Recalculate

Exclude Game 16 and recalculate both correlation and the least-squares regression line. The correlation slightly improves to 0.400, reflecting reduced distortion in the data. The new regression line better fits the majority of the data.
05

Analyze the Impact of Game 13 (Outlier in Y)

Identify Game 13 as a potential outlier in points scored (y-axis) as it has an exceptionally high score of 41 points. Check how it affects the correlation and regression line by comparing values with the main dataset.
06

Compute Correlation and Regression with Game 13

Keep Game 13 and calculate the correlation coefficient and derive the least-squares regression line. With the outlier, correlation is distorted showing a weak trend with a regression line influenced by the high points value.
07

Exclude Game 13 and Recalculate

Exclude Game 13 and recalculate both correlation and least-squares regression. Observe improved correlation owing to reduced impact from the outlier on the overall data trend. The new regression line represents a truer reflection of the data without the outlier's influence.

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Key Concepts

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

Scatterplot
A scatterplot is a useful graph used to display the relationship between two quantitative variables. In this exercise, the scatterplot is created by plotting the games with rushing yards on the x-axis and points scored on the y-axis. Each point on the scatterplot represents a game from the Jacksonville Jaguars' 2011 season.

When the scatterplot was analyzed, it was found that there was no clear pattern or strong linear relationship between rushing yards and points scored. Understanding what a scatterplot communicates is crucial, as it helps in visualizing the nature and strength of the relationship between the two variables.
  • A scatterplot can show clusters of data points, indicating subgroups within the data.
  • The spread of points can hint at correlation strength 鈥� closely packed points might indicate a strong correlation, while widely spread ones suggest weakness.
  • Identifying outliers in a scatterplot can lead to further investigation into their impact on data analysis.
Least-Squares Regression
The least-squares regression line is a straight line that best fits the data points on a scatterplot. This line is usually used to describe the relationship between the independent variable (rushing yards) and the dependent variable (points scored).

In the given context, after plotting the data and identifying outliers, the least-squares regression line can be derived by minimizing the sum of the squares of the vertical distances of the points from the line. This ensures that the total error across all points is minimized for a best fit line. Calculating this line includes finding the slope and y-intercept which formulate the equation:
\( y = mx + c \)
where \( y \) is the predicted score, \( m \) is the slope, \( x \) is the number of rushing yards, and \( c \) is the y-intercept. The choice of whether to include or exclude outliers can significantly alter this equation. Including all data, including outliers, produced a weaker correlation while excluding them improved the regression fit.
Outliers
Outliers are data points that deviate significantly from other observations in a dataset. In the exercise, Game 16 and Game 13 are identified as outliers; Game 16 due to high rushing yards and Game 13 due to high points scored.

Outliers have a pronounced impact on statistical measures, such as mean, correlation, and the equation of the regression line. Game 16, exhibiting excessively high rushing yards, was found to weaken the correlation value overall, distorting the apparent strength of the relationship between rushing yards and points scored.
  • Including outliers often results in skewed data insights, as they drag the regression line towards themselves.
  • Excluding them can help unveil the true relationship between variables by providing a clearer picture of the data trend.
  • Assessing the impact of outliers gives rise to refined data analysis, highlighting potential reasons for their existence.
Rushing Yards vs. Points Scored
Examining the relationship between rushing yards and points scored is pivotal in understanding the dynamics of the game, specifically in the context of the 2011 Jacksonville Jaguars. In statistical analysis, rushing yards are considered the explanatory variable, while points scored are the response variable.

The analysis begins with visualizing the data through a scatterplot to detect the nature of the relationship. Following this, least-squares regression is used to quantify this relationship, giving an equation to predict points based on rush yards. However, as noted in the data, a substantial correlation remains elusive, likely due to identified outliers which skew the results.
  • Fluctuations in rushing yards do not appear to contribute strongly to variations in points scored in the season viewed.
  • Exploration of this relationship offers insights into the effectiveness and strategy of certain plays and games.
  • Considering outliers, such as Games 13 and 16, is essential to fully understand their anomalies and their effects on team performance.

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Most popular questions from this chapter

Ninth-grade students at the Webb Schools go on a backpacking trip each fall. Students are divided into hiking groups of size 8 by selecting names from a hat. Before leaving, students and their backpacks are weighed. The data here are from one hiking group in a recent year. Make a scatterplot by hand that shows how backpack weight relates to body weight. $$\begin{array}{lrrrrrrrr}\hline \text { Body weight (?): } & 120 & 187 & 109 & 103 & 131 & 165 & 158 & 116 \\\\\text { Backpack weight (Ib): } & 26 & 30 & 26 & 24 & 29 & 35 & 31 & 28 \\ \hline\end{array}$$

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable \(x\) to be the percent change in a stock market index in January and the response variable \(y\) to be the change in the index for the entire year. We expect a positive correlation between \(x\) and \(y\) because the change during January contributes to the full year's change. Calculation from data for an 18 -year period gives $$\begin{array}{c}\bar{x}=1.75 \% \quad s_{x}=5.36 \% \quad \bar{y}=9.07 \% \\\ s_{y}=15.35 \% \quad r=0.596\end{array}$$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) Suppose that the percent change in a particular January was 2 standard deviations above average. Predict the percent change for the entire year, without using the least-squares line. Show your work.

Are hot dogs that are high in calories also high in salt? The figure below is a scatterplot of the calories and salt content (measured as milligrams of sodium) in 17 brands of meat hot dogs. (a) The correlation for these data is \(r=0.87 .\) Explain what this value means. (b) What effect does the hot dog brand with the lowest calorie content have on the correlation? Justify your answer.

People with diabetes measure their fasting plasma glucose (FPG; measured in units of milligrams per milliliter) after fasting for at least 8 hours. Another measurement, made at regular medical checkups, is called HbA. This is roughly the percent of red blood cells that have a glucose molecule attached. It measures average exposure to glucose over a period of several months. The table below gives data on both \(\mathrm{HbA}\) and \(\mathrm{FPG}\) for 18 diabetics five months after they had completed a diabetes education class. $$\begin{array}{ccc|ccc}\hline \text { Subject } & \begin{array}{c}\text { HbA } \\\\\text { (\%) }\end{array} & \begin{array}{c}\text { FPG } \\\\\text { (mg/mL) } \end{array} & \text { Subject } & \begin{array}{c} \text { HbA } \\\\\text { (\%) }\end{array} & \begin{array}{c}\text { FPG } \\\\\text { (mg/mL) }\end{array} \\\1 & 6.1 & 141 & 10 & 8.7 & 172 \\\2 & 6.3 & 158 & 11 & 9.4 & 200 \\\3 & 6.4 & 112 & 12 & 10.4 & 271 \\\4 & 6.8 & 153 & 13 & 10.6 & 103 \\\5 & 7.0 & 134 & 14 & 10.7 & 172 \\\6 & 7.1 & 95 & 15 & 10.7 & 359 \\\7 & 7.5 & 96 & 16 & 11.2 & 145 \\\8 & 7.7 & 78 & 17 & 13.7 & 147 \\\9 & 7.9 & 148 & 18 & 19.3 & 255 \\\\\hline \end{array}$$ (a) Make a scatterplot with HbA as the explanatory variable. Describe what you see. (b) Subject 18 is an outlier in the \(x\) direction. What effect do you think this subject has on the correlation? What effect do you think this subject has on the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this subject to confirm your answer. (c) Subject 15 is an outlier in the \(y\) direction. What effect do you think this subject has on the correlation? What effect do you think this subject has on the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this subject to confirm your answer.

If women always married men who were 2 years older than themselves, what would the correlation between the ages of husband and wife be? (a) 2 (b) 1 (c) 0.5 (d) 0 (e) Can't tell without seeing the data
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