91Ó°ÊÓ

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?

Step by step solution

01

Plot the Data

To start, we'll plot the given pairs of (% alcohol, calorie) data on a scatter plot with % alcohol as the independent variable and calories as the dependent variable.

02

Calculate the Regression Line

Next, we'll calculate the regression line. This line represents the best fit to our given data and is calculated using the least squares method.

03

Analyze Linear Relationship

Look at the scatterplot created in Step 1 and analyze whether there is a strong linear relationship between the variables. By judging the proximity of the data points to the regression line calculated in Step 2, you can determine the degree of linear relationship.

04

Calculate the Correlation

Next, calculate the correlation using the Pearson correlation coefficient formula. This will give us a numerical value that represents the strength and direction of the linear relationship.

05

Interpret the Correlation

Based on the value of the correlation calculated in Step 4, you can interpret what the sign of the correlation says about the trajectory of the relationship between calories and % alcohol.

06

Interpret the Slope of Regression Line

The slope of the regression line represents the average change in calories for each 1% increase in alcohol content. Record the equation of the regression line and round to two decimal places.

07

Calculate the Coefficient of Determination

The coefficient of determination is the square of the correlation and represents the proportion of variance in the dependent variable (calories) that is predictable from the independent variable (% alcohol).

08

Add a New Data Point

Next, we'll add a new data point to the set: a wine that is 20% alcohol and contains 0 calories.

09

Recalculate \(r\) and Regression Equation

Calculate the new correlation coefficient and regression equation with the added data point, and compare these new results to the original values to identify the effect of this new data point.

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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 visual representation of data where each point on the plot represents a pair of values. In our exercise, each point corresponds to a specific wine, with the x-axis representing the % alcohol content and the y-axis showing the calories. By plotting each wine's data, you can easily see any potential trends or patterns.
This scatterplot helps us understand whether there's a linear relationship between the variables. If the points seem to form a pattern that moves in a specific direction, it indicates that there could be a correlation. Additionally, overlaying a regression line helps to visualize how close the data fit along a line of best fit, further illustrating the strength of any potential relationship.

Correlation Coefficient

The correlation coefficient, often denoted as r, is a numerical measure that showcases the strength and direction of a linear relationship between two variables. In the context of our wine data, the correlation coefficient allows us to quantify the relationship between % alcohol content and calories.
An r value close to 1 or -1 indicates a strong correlation, with positive values meaning that as one variable increases, the other also increases, while negative values suggest an inverse relationship. A value close to 0 means little to no linear relationship. By calculating this, we interpret how well our data tends to form a straight line when plotted.

Regression Equation

A regression equation represents the relationship between an independent variable and a dependent variable. It's usually written in the form of: \[ Calories = a + b \cdot \% Alcohol \] where \( a \) is the y-intercept and \( b \) is the slope.
The slope \( b \) indicates how much the calories are expected to change with a percentage change in alcohol content. For example, if the slope is positive, it means that for each 1% increase in alcohol, the calories increase by the value of the slope. By finding this equation, you can predict calories for any given % alcohol, making it a powerful tool for analysis and prediction.

Coefficient of Determination

The coefficient of determination, denoted as \( R^2 \), is a key metric in regression analysis. It tells us how well the independent variable (in this case, % alcohol) explains the variability of the dependent variable (calories).
In simpler terms, \( R^2 \) is the square of the correlation coefficient \( r \). It provides insight into how confidently you can use the regression equation to predict the dependent variable. A higher \( R^2 \) value means a greater portion of the variance is accounted for by the model, indicating a strong relationship. Understanding \( R^2 \) helps in evaluating the reliability of your regression analysis, especially when mixed with back-checks against data.