Introduction When linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context. Recall that data in a scatter plot can be approximated using a linear fit, or linear function that models real-world relationships. A linear fit is the approximation of data using a linear function. 4.3.1: Interpreting Slope and y-intercept

Introduction, continued If the x-axis of a graph represents hours and the y-axis represents miles traveled, the slope of a linear function graphed on these axes would be , or the miles traveled each hour. 4.3.1: Interpreting Slope and y-intercept

Key Concepts The slope of a line with the equation y = mx + b is m. The slope between two points (x1, y1) and (x2, y2) is . In context, the slope describes how much the dependent variable changes each time the independent variable changes by 1 unit. 4.3.1: Interpreting Slope and y-intercept

Key Concepts, continued The y-intercept of a line with the equation y = mx + b is b. The y-intercept is the value of y at which a graph crosses the y-axis. In context, the y-intercept is the initial value of the quantity represented by the y-axis, or the quantity of y when the quantity represented by the x-axis equals 0. 4.3.1: Interpreting Slope and y-intercept

Common Errors/Misconceptions incorrectly calculating the slope confusing the y- and x-intercepts, both in context and when calculating using a graph or equation 4.3.1: Interpreting Slope and y-intercept

Guided Practice Example 1 The graph on the next slide contains a linear model that approximates the relationship between the size of a home and how much it costs. The x-axis represents size in square feet, and the y-axis represents cost in dollars. Describe what the slope and the y-intercept of the linear model mean in terms of housing prices. 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 1, continued Cost in dollars ($) Size in square feet 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 1, continued Find the equation of the linear fit. Remember that in order to do that, you must choose 2 points on the line and find the slope between those 2 points. Then, use the slope and one of those points to determine the y-intercept. My two points are: ______________ The slope is: ______________ 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 1, continued Next, find the y-intercept. Use the equation for slope-intercept form, y = mx + b, where b is the y-intercept. Replace x and y with values from a single point on the line. Let’s use ____________. Replace m with the slope. Solve for b below: y = m = x = 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 1, continued What does the slope mean in context of the problem? Divide the units on the y-axis by the units on the x-axis: . The units of the slope are dollars per square foot. 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 1, continued Describe what the slope means in context. A positive slope means the quantity represented by the y-axis increases when the quantity represented by the x-axis also increases. The cost of the home increases by $200 for each square foot. 4.3.1: Interpreting Slope and y-intercept

✔ Guided Practice: Example 1, continued 4. Describe what the y-intercept means in context. The y-intercept in our equation is ______. For a home with no area, or for no home, the cost is $0. ✔ 4.3.1: Interpreting Slope and y-intercept

Guided Practice Example 2 A teller at a bank records the amount of time a customer waits in line and the number of people in line ahead of that customer when he or she entered the line. Describe the relationship between waiting time and the people ahead of a customer when the customer enters a line. Use the table on the following slide. 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 2, continued People ahead of customer Minutes waiting 1 10 2 21 3 32 5 35 8 42 9 45 61 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 2, continued Minutes spent waiting Number of people ahead 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 2, continued 1. Find the equation of a linear model to represent the data. We need to draw a line that accurately models the data. Which 2 points should we use in this situation? (1, 10) and (10, 61) What is the slope between these 2 points? 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 2, continued Find the y-intercept. Replace x and y with values from a single point on the line. Let’s use _______. Replace m with the slope, 5.67. Solve for b below. 4.3.1: Interpreting Slope and y-intercept

Guided Practice: Example 2, continued 2. Describe what the slope means in context. The slope describes how the waiting time increases for each person in line ahead of the customer. A customer waits approximately 5.67 minutes for each person who is in line ahead of the customer. 4.3.1: Interpreting Slope and y-intercept

✔ Guided Practice: Example 2, continued 3. Describe what the y-intercept means in context. The y-intercept is the value of the equation when x = 0, or when the number of people ahead of the customer is 0. The y-intercept is 4.33. In this context, the y-intercept isn’t relevant, because if no one was in line ahead of a customer, the wait time would be 0 minutes. Creating a linear model that matched the data resulted in a y-intercept that wasn’t 0, but this value isn’t related to the context of the situation. ✔ 4.3.1: Interpreting Slope and y-intercept

Correlation Coefficient The strength of the relationship between data that has a linear trend can be analyzed using the correlation coefficient. A correlation is a relationship between two events, such as x and y, where a change in one event implies a change in another event. The correlation coefficient, r, is a quantity that allows us to determine how strong this relationship is between two events. It is a value that ranges from –1 to 1; a correlation coefficient of –1 indicates a strong negative correlation, a correlation coefficient of 1 indicates a strong positive correlation, and a correlation coefficient of 0 indicates a very weak or no linear correlation. To determine if the correlation coefficient is positive or negative, look at the slope of the data. A correlation coefficient of 1 or -1 means that the data very strongly follows a trend, .5 means the data slightly follows a trend, and 0 means the data does not follow any trend. 4.3.1: Interpreting Slope and y-intercept

Describe the data as having a positive correlation, a negative correlation, or approximately no correlation. Tell whether the correlation coefficient for the data is closest to -1, -0.5, 0, 0.5, or 1. 1. 2. Positive Correlation correlation coefficient: 1 No Correlation correlation coefficient: 0

3. 4. Positive Correlation correlation coefficient: 0.5 Negative Correlation correlation coefficient: -1

When two variables in a relationship are correlated, a change in one variable suggests a change in the other. However, correlation does not always imply causation. In other words, just because a change in variable x suggests a change in variable y, the change in x does not necessary cause the change in y. Both the change in x and the change in y could be caused by a third factor.

10. (a) A study has shown the as the number of hours a person spends dancing increases, the probability that the person will experience hearing loss also increases. Because there is an obvious correlation between hours spent dancing and probability of hearing loss, the study concluded that dancing causes hearing loss. Is it likely that the study’s conclusion is correct? (b). What else could have caused both an increase in the number of hours spent dancing and an increase in the probability of hearing loss? loud music (c). Is the relationship between variables likely one of causation or correlation? correlation Therefore, conclusion is incorrect.

11. a. Give an example of a variable that could be in correlation with the amount of hot cocoa a person consumes. The more hot cocoa a person consumes, the more likely that person is to experience frostbite. b. Explain whether or not this is an example of causation. Drinking hot cocoa probably doesn’t cause frostbite. However, cold whether can cause both an increase in hot cocoa consumption and frostbite. Therefore, the correlation between hot cocoa consumed and frostbite is NOT an example of causation.

Decide if each of the following relationships is likely one of causation, or is it only a correlation. 12. As a person’s corrected vision improves, his reading ability improves. could be causation 13. The more umbrellas are in use, the greener the grass becomes. correlation 14. The more skiers on the slopes, the more school days are cancelled. 15. The more a person types, the more likely the person is to experience carpal tunnel syndrome.