Because you want to prepare and serve the healthiest food possible, you monitor the fat and calorie content of items on your menu. Some of the menu items are included in the graph below.

Question

a) Your business partner describes this as a high positive correlation. Is your partner correct? Why or why not? (2 points)

b) Using the drawing tools, draw a trend line (line of best fit) on the graph above. (2 points)

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c) Judge the closeness of your trend line to the data points. Do you notice a relationship between the data points? (2 points)

d) Is the trend line linear? If so, write a linear equation that represents the trend line. Show your work. (3 points)

Answer 1

Without having the actual graph referenced in your question, I can provide you with general guidance on how to approach each of these questions based on common scenarios involving correlation, trend lines, and linear relationships.

a) High Positive Correlation

Your Partner's Description: To determine if your partner is correct, check the direction and strength of the data points.

Correct or Not:

• If the data points show a clear upward trend, with most points closely clustered around an imaginary line that rises from left to right, then your partner's statement about a high positive correlation could be correct.
• Conversely, if the data points are widely scattered or follow a different pattern (e.g., random distribution, little to no trend), then this would not be a high positive correlation.

Why or Why Not: You can justify your answer based on how closely the data points follow a straight line and how well they fit an upward trend.

b) Trend Line (Line of Best Fit)

Drawing the Trend Line:

• To create a trend line on a graph, you generally want to draw a straight line that best represents the data points.
• The line should be positioned so that roughly an equal number of points lie above and below it.
• The slope of the line indicates the direction of the trend (positive slope for increasing values, negative for decreasing values).

Since I cannot draw here, you would typically use a ruler or a straight edge to achieve this on your graph.

c) Closeness of Trend Line to Data Points

Judging the Closeness:

• After drawing your trend line, look at how many points fall near the line and how closely they cluster around it.
• If most data points are close to the line, this suggests a strong relationship.
• If points are scattered away from the line, this may indicate a weaker relationship or other influencing factors.

Noticing the Relationship:

• If the trend line closely follows the pattern of the points, you might notice that as one variable increases (e.g., fat content), the other variable (e.g., calorie content) also tends to increase, suggesting a correlation.

d) Linear Trend Line

Determining if the Trend Line is Linear:

• If you successfully drew a straight line that fits among the points, then yes, it is linear.

Finding the Linear Equation: To write a linear equation, you will need to determine two things:

1. Slope (m): Find the slope using the formula \((y_2 - y_1) / (x_2 - x_1)\) using two points on the trend line.
2. Y-intercept (b): This is where the line crosses the y-axis.

The equation will follow the format: \[ y = mx + b \]

Example Steps:

1. Identify two points on your trend line, say (x1, y1) and (x2, y2).
2. Calculate the slope \(m\).
3. Use one of your points to solve for the y-intercept \(b\) using the equation \(y = mx + b\).

Combine these to write your final equation.

If you provide specific data or more context, I would be able to assist you with more detailed calculations or explanations!