Problem 13 - Algebra 2 and Trigonometry

Question

In \(11-17 :\) a. Draw a scatter plot. b. Does the data set show strong positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or strong negative linear correlation? c. If there is strong or moderate correlation, write the equation of the regression line that approximates the data. Jack Sheehan looked through some of his favorite recipes to compare the number of calories per serving to the number of grams of fat. The table below shows the results. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Calories } & {310} & {210} & {260} & {290} & {320} & {245} & {293} & {220} & {260} & {350} \\ \hline F a t & {5} & {11} & {12} & {14} & {16} & {7} & {10} & {8} & {8} & {15} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The scatter plot shows a moderate positive linear correlation. The regression line is approximately \( y = 0.05x + 7.2 \).

Step 1: Plotting the Scatter Plot

To create a scatter plot, each pair of values (Calories, Fat) will be a point on the graph. The `x`-axis represents the calories, and the `y`-axis represents the grams of fat. The pairs are: (310, 5), (210, 11), (260, 12), (290, 14), (320, 16), (245, 7), (293, 10), (220, 8), (260, 8), (350, 15). Plot these points to visualize the data distribution.

Step 2: Analyzing the Correlation

Examine the pattern formed by the plotted points. If the points appear to trend upwards forming a line from left to right, there is a positive correlation. If the points scatter without any clear trend, there is no correlation. For this data, the points generally move upwards, indicating a positive linear correlation. It can be described as moderate because the points do not align perfectly and there is some spread.

Step 3: Calculating the Regression Line

Using the formula for the line of best fit (least squares regression line), we calculate the slope and intercept. The formula for the slope (m) is given by \( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \) and for the y-intercept (b) by \( b = \frac{(\sum y) - m(\sum x)}{n} \). Input the values of calories as \(x\) and fat content as \(y\), calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\), and compute \(m\) and \(b\).

Step 4: Writing the Regression Equation

Substitute the calculated slope and y-intercept into the equation of a line \( y = mx + b \). Suppose calculations yield, for example, slope \(m = 0.05\) and intercept \(b = 7.2\), then the regression line is \( y = 0.05x + 7.2 \).

Key Concepts

Scatter Plot

A scatter plot is a type of graph that displays individual data points representing the relationship between two quantitative variables. In this context, we plot the number of calories against grams of fat.
Each recipe from the table becomes a point on the scatter plot, with calories on the x-axis and fat on the y-axis. By observing the arrangement of these points, we get a visual idea of how one variable affects the other.
Creating a scatter plot helps in visually assessing the type of correlation between the two sets of data, making it a powerful first step in data analysis.

Correlation Analysis

Correlation analysis involves examining the data plotted on the scatter plot to identify the pattern or direction of the relationship. This tells us how strongly two variables are related.
Here, we look for trends among the points. If they rise together, there's a positive correlation; if one descends as the other rises, it's negative.

• Positive Correlation: As one variable increases, the other also increases.
• Negative Correlation: As one variable increases, the other decreases.
• No Correlation: No discernible pattern.

In our exercise, the points tend to rise as calories increase, suggesting a moderate positive linear correlation as they don't form a perfect line.

Regression Equation

The regression equation represents the line of best fit among the data points on a scatter plot. It predicts one variable based on the other.
The general form of a regression equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This line minimizes the distance between itself and all points in the scatter plot.
A correct regression equation allows for estimating the fat content based on the number of calories in a recipe. It speaks to the strength and nature of the relationship analyzed in the correlation assessment.

Least Squares Method

The least squares method is a mathematical approach to finding the best fit line in a scatter plot by minimizing the sum of squared deviations from the line.
To calculate, you'll need:

• Sum of x values \( \,\sum x \) and y values \( \,\sum y \)
• Sum of products \( \,\sum xy \)
• Sum of squares \( \,\sum x^2 \)

The formula for the slope \( m \) is \( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \) and the intercept \( b \) is \( b = \frac{(\sum y) - m(\sum x)}{n} \).
Through this method, we ensure accuracy in predicting the relationship between variables, yielding a more reliable model for further analysis.