91Ó°ÊÓ

lce Cream, Anyone? As much as Americans try to avoid high fat, high calorie foods, the demand for a cold, creamy ice cream cone on a hot day is hard to resist. The popular ice cream franchise Cold stone Creamery posted the nutritional information for its ice cream offerings in three serving sizes- "Like it", "Love it", and "Gotta Have it"-on their website. \({ }^{12}\) A portion of that information for the "Like it" serving size is shown in the table. $$ \begin{array}{lcc} \text { Flavor } & \text { Calories } & \text { Total Fat (grams) } \\ \hline \text { Cake Batter } & 340 & 19 \\ \text { Cinnamon Bun } & 370 & 21 \\ \text { French Toast } & 330 & 19 \\ \text { Mocha } & 320 & 20 \\ \text { OREO }^{B} \text { Crème } & 440 & 31 \\ \text { Peanut Butter } & 370 & 24 \\ \text { Strawberry Cheesecake } & 320 & 21 \end{array} $$ a. Should you use the methods of linear regression analysis or correlation analysis to analyze the data? Explain. b. Analyze the data to determine the nature of the relationship between total fat and calories in Colds tone Creamery ice cream.

Step by step solution

01

Identify the goal

The goal is to analyze the relationship between two sets of data: the total fat (in grams) and the calories for each ice cream flavor in the "Like it" serving size.

02

Correlation analysis or linear regression analysis?

Since we want to investigate the relationship between two variables, we need to consider whether to use correlation analysis or linear regression analysis. - Correlation analysis is used to determine the strength and direction of the relationship between two variables. - Linear regression analysis is used to create a model that predicts the value of one variable based on the value of another variable. In this case, we want to analyze the relationship between calories and total fat, so correlation analysis will be the appropriate method to use. Answer to (a): It's more suitable to use correlation analysis for analyzing the data as we want to investigate the strength and direction of the relationship between the total fat and the calories. #b. Analyzing the data using correlation analysis#

03

Calculate the mean

Before calculating the correlation, we need the mean of both total fat and calories. Total fat mean = (19 + 21 + 19 + 20 + 31 + 24 + 21) / 7 = 20.57 grams Calories mean = (340 + 370 + 330 + 320 + 440 + 370 + 320) / 7 = 355.71

04

Calculate standard deviation

Calculate the standard deviation for both total fat and calories: Total fat standard deviation = sqrt( [(19-20.57)^2 + (21-20.57)^2 + (19-20.57)^2 + (20-20.57)^2 + (31-20.57)^2 + (24-20.57)^2 + (21-20.57)^2] / 7 ) ≈ 3.92 grams Calories standard deviation = sqrt( [(340-355.71)^2 + (370-355.71)^2 + (330-355.71)^2 + (320-355.71)^2 + (440-355.71)^2 + (370-355.71)^2 + (320-355.71)^2] / 7 ) ≈ 37.50

05

Calculate the correlation coefficient

Now we can calculate the correlation coefficient (r): r = [ (19-20.57)(340-355.71) + (21-20.57)(370-355.71) + (19-20.57)(330-355.71) + (20-20.57)(320-355.71) + (31-20.57)(440-355.71) + (24-20.57)(370-355.71) + (21-20.57)(320-355.71) ] / [ 7 * 3.92 * 37.50 ] r ≈ 0.94 Answer to (b): The correlation coefficient (r) is approximately 0.94, which indicates a strong positive relationship between total fat and calories in Cold Stone Creamery ice cream.

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Key Concepts

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

Linear Regression

Linear regression is a fundamental concept in statistics used to model the relationship between two quantitative variables. It's about finding the straight line, known as the regression line, that best fits the data.
The general form of a linear regression equation is:
\[ y = mx + b \]where:

• \( y \) is the dependent variable (the outcome variable we try to predict),
• \( x \) is the independent variable (the predictor variable used for prediction),
• \( m \) is the slope of the line which represents the rate of change,
• \( b \) is the y-intercept, representing the value of \( y \) when \( x \) is 0.

In simple terms, linear regression helps us understand how changes in the predictor variable \( x \) will affect the outcome variable \( y \). In the context of our ice cream example, if we'd used linear regression, we'd be trying to predict calories based on total fat content. However, in this exercise, correlation analysis was deemed more suitable as we were more interested in the strength and direction of the relationship rather than prediction.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It gives us an idea of how spread out the values are around the mean.
The formula for standard deviation is:
\[ s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n}} \]where:

• \( x_i \) represents each data point,
• \( \bar{x} \) is the mean of the data,
• \( n \) is the number of observations.

A higher standard deviation indicates that the data points are spread out over a wider range of values, while a lower standard deviation indicates that they are closer to the mean.
In our ice cream data analysis, the standard deviation for `total fat` and `calories` was calculated to understand how much these values varied from their respective means. Knowing the standard deviation helped provide context for the correlation coefficient, allowing for a better understanding of data consistency and reliability of the correlation.

Correlation Coefficient

The correlation coefficient, often represented as \( r \), measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1.
A value of:

• +1 indicates a perfect positive linear relationship,
• -1 indicates a perfect negative linear relationship,
• 0 indicates no linear relationship.

The formula for the correlation coefficient is:
\[ r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \, \sum{(y_i - \bar{y})^2}}}\]where:

• \( x_i, y_i \) are the data points,
• \( \bar{x}, \bar{y} \) are the means of the data sets.

In the ice cream example, the calculated correlation coefficient was approximately 0.94, indicating a strong positive relationship between total fat and calories. This means that as the total fat content increases, the calorie count tends to increase as well. Such a strong correlation suggests a predictable connection, helping consumers and producers understand the nutritional implications of fat content in ice creams.