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Chapter 4: Problem 64

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?

Short Answer

Expert verified
The result would be a scatterplot with calories as a dependent variable, alcohol content as an independent variable and a line of best fit. The correlation coefficient would indicate whether a linear relationship exists, with its sign indicating a positive or negative relationship. The regression line equation will establish the relationship between alcohol % and calorie count, and its slope will indicate how much calorie count changes per unit increase in alcohol content. The coefficient of determination will indicate the proportion of variability in calorie count that can be explained by the variability in alcohol content. And finally, the inclusion of an outlier will show the effect on these calculated values.

Step by step solution

01

Creating Scatterplot

First, to create a scatterplot, plot the six given pairs of alcohol content and calorie data points. Alcohol content will be on the x-axis and calories on the y-axis. Draw a line of best fit (regression line) through the data points to visualize the relationship between alcohol content and calorie count.
02

Calculate Correlation

Next, find the correlation between alcohol percentage and calorie count by using the Pearson's correlation formula. This involves summing the products of corresponding standard scores of the two variables, and dividing by the number of data points.
03

Interpret Correlation

Interpret the sign of the correlation coefficient. A positive value indicates a positive linear relationship between calorie count and alcohol content, and a negative value indicates a negative linear relationship.
04

Find the Regression Line

The regression line can be calculated using the formula \(y = mx + c\), where \(m\) is the slope of the line (change in y over change in x) and \(c\) is the y-intercept. The slope (\(m\)) and intercept (\(c\)) can be calculated using the given data. The slope of the line is the correlation times the standard deviation of y values divided by the standard deviation of x values, and the intercept is the mean of y values minus the slope times the mean of x values.
05

Interpret the Slope

The slope represents the estimated increase or decrease in the dependent variable (calories) for a one unit increase in the independent variable (alcohol content). In context, this is how many additional calories are expected for each additional 1% of alcohol content.
06

Calculate Coefficient of Determination

The coefficient of determination (R-squared) can be calculated by squaring the Pearson's correlation value. It represents the proportion of the variance in the dependent variable that is predictable from the independent variable.
07

Add New Data Point and Repeat Steps 2,4

Add the new data point (20%, 0 calories) to the dataset and repeat the calculation of correlation and regression line. Observe any changes in these values.
08

Discuss the Effect of This New Data Point

One outlying point can have a large effect on the correlation coefficient and the slope of the regression line. Discuss how this point affected these values in your data set.

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Key Concepts

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

Scatterplot
When exploring the relationship between two variables, a scatterplot is an essential tool. It is a type of graph that shows individual data points positioned according to their values along the X-axis (independent variable) and the Y-axis (dependent variable). For instance, when plotting the percentage of alcohol (independent variable) against calories (dependent variable) in various wines, each dot on the scatterplot represents a different wine characterized by these two features.

Creating a scatterplot offers immediate visual insight into the data. You can observe patterns, trends, and even outliers that do not fit the general pattern. When analyzing our example of alcohol content versus calories, a scatterplot will reveal if there's a general trend indicating whether wines with higher alcohol content tend to have more calories. By including a regression line, or 'line of best fit', you can further visualize the average relationship between the two variables and make predictions accordingly.
Correlation Coefficient
The correlation coefficient, typically denoted as 'r', quantifies the strength and direction of the linear relationship between two variables. Its value ranges from -1 to 1. A value close to 1 indicates a strong positive linear relationship, meaning that as one variable increases, the other tends to increase as well. Conversely, a value close to -1 implies a strong negative linear relationship, signifying that as one variable increases, the other tends to decrease. When 'r' is around 0, there's little to no apparent linear relationship.

For our wine data, calculating 'r' will reveal how strongly alcohol content is related to calorie count. The sign of the correlation coefficient also carries meaning. A positive sign would indicate that wines with higher percentages of alcohol typically contain more calories, reflecting a direct relationship. On the other hand, a negative sign would illustrate an inverse relationship - not something one might expect in the case of alcohol and calorie content.
Regression Line Equation
The equation of the regression line is a mathematical representation of the average relationship between the independent and dependent variables. It is usually written as \( y = mx + c \), where \( m \) is the slope of the line and \( c \) is the y-intercept. The slope indicates how much the dependent variable is expected to change for each one-unit change in the independent variable.

In the context of our wine dataset, the regression line equation provides a formula to estimate the calories based on the alcohol content. The slope here would tell us the expected change in calorie count for each 1% increase in alcohol content. This information is especially useful for nutritionists or diet-conscious individuals who wish to understand the caloric impact of alcohol consumption. For a winemaker or consumer, this knowledge can aid in making informed decisions regarding wine selection based on dietary preferences or restrictions.
Coefficient of Determination
The coefficient of determination, denoted as \( R^2 \), is a key output of regression analysis. It represents how much of the variance in the dependent variable can be explained by the independent variable. Calculated by squaring the correlation coefficient, \( R^2 \) varies between 0 and 1. A higher value indicates a better fit of the model to the data, meaning that the independent variable does an excellent job in predicting the dependent variable.

For the alcohol content and calories example, the \( R^2 \) value will indicate how well the percentage of alcohol predicts the calorie amounts in the wines. A high \( R^2 \) means that a significant portion of the variation in calories among different wines can be attributed to their alcohol content, giving us confidence in our regression model. This statistic is crucial for making predictions but also necessary for understanding the limitations of the data's explanatory power.

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Most popular questions from this chapter

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. $$ \begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array} $$ $$ \begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} $$

The table shows the heights (in inches) and weights (in pounds) of 14 college men. The scatterplot shows that the association is linear enough to proceed. $$ \begin{array}{c|c|c|c|} \hline \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} & \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 68 & 205 & 70 & 200 \\ \hline 68 & 168 & 69 & 175 \\ \hline 74 & 230 & 72 & 210 \\ \hline 68 & 190 & 72 & 205 \\ \hline 67 & 185 & 72 & 185 \\ \hline 69 & 190 & 71 & 200 \\ \hline 68 & 165 & 73 & 195 \\ \hline \end{array} $$ a. Find the equation for the regression line with weight (in pounds) as the response and height (in inches) as the predictor. Report the slope and the intercept of the regression line, and explain what they show. If the intercept is not appropriate to report, explain why. b. Find the correlation between weight (in pounds) and height (in inches). c. Find the coefficient of determination and interpret it. d. If you changed each height to centimeters by multiplying heights in inches by \(2.54\), what would the new correlation be? Explain. e. Find the equation with weight (in pounds) as the response and height (in inches) as the predictor, and interpret the slope. f. Summarize what you found: Does changing units change the correlation? Does changing units change the regression equation?

Construct a set of numbers (with at least three points) with a strong negative correlation. Then add one point (an influential point) that changes the correlation to positive. Report the data and give the correlation of each set.

Construct a small set of numbers with at least three points with a perfect negative correlation of \(-1.00\).

The table shows the Earned Run Average (ERA) and WHIP rating (walks plus hits per inning) for the top 40 Major League Baseball pitchers in the 2017 season. Top pitchers will tend to have low ERA and WHIP ratings. (Source: ESPN.com) a. Make a scatterplot of the data and state the sign of the slope from the scatterplot. Use WHIP to predict ERA. b. Use linear regression to find the equation of the best-fit line. Show the line on the scatterplot using technology or by hand. c. Interpret the slope. d. Interpret the \(y\) -intercept or explain why it would be inappropriate to do so. $$\begin{array}{|ll|} \hline \text { WHIP } & \text { ERA } \\ \hline 0.87 & 2.25 \\ \hline 0.95 & 2.31 \\ \hline 0.9 & 2.51 \\ \hline 1.02 & 2.52 \\ \hline 1.15 & 2.89 \\ \hline 0.97 & 2.9 \\ \hline 1.18 & 2.96 \\ \hline 1.04 & 2.98 \\ \hline1.31 & 3.09 \\ \hline 1.07 & 3.2 \\ \hline 1.13 & 3.28 \\ \hline 1.1 & 3.29 \\ \hline 1.35 & 3.32 \\ \hline 1.17 & 3.36 \\ \hline 1.32 & 3.4 \\ \hline 1.23 & 3.43 \\ \hline 1.25 & 3.49 \\ \hline 1.22 & 3.53 \\ \hline 1.19 & 3.53 \\ \hline 1.21 & 3.54 \\ \hline \end{array}$$ $$\begin{array}{|ll|} \hline \text { WHIP } & \text { ERA } \\ \hline 1.21 & 3.55 \\ \hline 1.22 & 3.64 \\ \hline 1.22 & 3.66 \\ \hline 1.27 & 3.82 \\ \hline 1.15 & 3.83 \\ \hline 1.16 & 3.86 \\ \hline 1.27 & 3.89 \\ \hline 1.35 & 3.9 \\ \hline1.28 & 3.92 \\ \hline 1.42 & 4.03 \\ \hline 1.26 & 4.07 \\ \hline 1.36 & 4.13 \\ \hline 1.28 & 4.14 \\ \hline 1.22 & 4.15 \\ \hline 1.33 & 4.16 \\ \hline 1.37 & 4.19 \\ \hline 1.2 & 4.24 \\ \hline 1.25 & 4.26 \\ \hline 1.3 & 4.26 \\ \hline 1.37 & 4.26 \\ \hline \end{array}$$
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