Do Now What is the predicted English score when the math score is 10?

Lesson 2.5 Regression lines

Objectives Make predictions using regression lines, keeping in mind the dangers of extrapolation. Calculate and interpret a residual. Interpret the slope and y intercept of a regression line.

Make a prediction We have a hole in our data because someone in the group forgot to record the value for 5 rubber bands….so we are just going to make our best prediction.  What do you think the lowest point could be for 5 rubber bands? Discuss with your groups and come up with a prediction

Let’s Calculate it! We’re going to graph a line to fit this data and use that to make a prediction next

Put your data into L1 and L2 (Stat-Edit) Go to Stat-Calc- #8 LinReg (a+bx), enter Do NOT hit enter again yet! After LinReg(a+bx) comes on your screen we must enter L1,L2,Y1 after it and then hit enter To get to L1- 2nd 1 To get to L2 – 2nd 2 To get to Y1 – VARS- Y-VARS-Function –Y1 Zoom- #9

Predict using our graph On your graph… Hit 2nd Trace Hit value Enter in the x value you would like to find and hit enter The y shows in the bottom right corner

Complete #3-6

When the relationship between two quantitative variables is linear, we can use a regression line to model the relationship and make predictions

Definition A regression line is a line that describes how a response variable y changes as an explanatory variable x changes. Regression lines are expressed in the form 𝑦 = a + bx, where 𝑦 (pronounced “y hat”) is the predicted value of y for a given value of x.

Definition The y intercept, a is the predicted value of y when x = 0. The slope b of a regression line describes the predicted change in the y variable for each 1-unit increase in the x variable. It is very important to include the word “predicted” (or its equivalent) in the interpretation of the slope and y intercept.

Definition Extrapolation is the use of a regression line for prediction far outside the interval of x values used to obtain the line. Such predictions are often not accurate.

The prediction we make using the regression line is called an extrapolation

Even when we are not extrapolating, our predictions are seldom perfect Even when we are not extrapolating, our predictions are seldom perfect. For a specific point, the difference between the actual value of y and the predicted value of y is called a residual.

Definition A residual is the difference between an actual value of y and the value of y predicted by the regression line. That is,

Alligator Investigation

Check Understanding- Exit Ticket

Roller coasters with larger maximum heights usually go faster than shorter ones. Here is a scatterplot of x = height (in feet) versus y = maximum speed (in miles per hour) for nine roller coasters that opened in 2012. The equation of the regression line for this relationship is 𝑦 =28.17+0.2143𝑥. Calculate, and interpret the residual for the Iron Shark, which has maximum height of 100 feet and a top speed of 52 miles per hour. Interpret the slope of the regression line. Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

Do Now Data on x = size of a house (in square feet) and y= amount of natural use (therms) during a specified period were used to fit the least squares regression line. The slope was 0.017 and the y-intercept was -5. Houses in this data set ranged in size from 1000 to 3000 square feet. What is the equation of the least squares regression line? What would you predict for gas usage for a 2100 sq. ft. house? What is the approximate change in gas usage associated with a 1 sq. ft. increase is size? Would you use the least squares regression line to predict gas usage for a 500 sq. ft. house? Why or why not?

Do Now Complete #1-3 It’s exactly what we’ve been doing in class. Use that note sheet to follow directions

Lesson 2.6 Least Squares Regression Lines Objectives: Calculate the equation of the least squares regression line using technology Calculate the equation of the least squares regression line using summary statistics Describe how outliers affect the least-squares regression line

Complete #4-6

Discuss How did outliers affect our least squares regression line?

Definition The least squares regression line is the line that makes the sum of the squared residuals as small as possible

What is a residual? The magnitude of a typical residual can give us a sense of generally how close our estimates are.

What could the standard deviation of a residual tell us about our regression line? The smaller the residual standard deviation, the closer is the fit to the data. The smaller the residual standard deviation is compared to the sample standard deviation, the more predictive the model is. The standard deviation of your residuals, S, can also be thought of as the "typical" residual, so most points should be within S points from the line of best fit.

What do residuals look like?

Desmos Activity

Important Ideas from the Text “Least squares” means smallest squares of residuals Outliers pull lines towards themselves Outliers often change slope and y-intercept

Complete the Application problem on the back

Check your understanding

Researchers have investigated if pomegranate’s antioxidant properties are useful in the treatment of cancer. One study investigated whether pomegranate fruit extract (PPE) was affective in slowing the growth of prostate cancer tumors. In this study, 24 mice were injected with cancer cells. The mice were then randomly assigned to one of three treatment groups. One group of eight mice received normal drinking water, the second group received drinking water supplemented with 0.1% PPE, and the third received drinking water supplemented with 0.2% PPE. The average tumor volume for the mice in each group was recorded at several points in time.

x = number of days after injection of cancer cells 11 15 19 23 27 y 150 270 450 580 740 x = number of days after injection of cancer cells y = average tumor volume (in 𝑚𝑚 3 ) The summary quantities necessary to calculate the equation of the least squares regression line are: 𝑥 =95 𝑦 =2190 (𝑥− 𝑥 )(𝑦− 𝑦) =5960 (𝑥− 𝑥 ) 2 =160

If the goal is to learn about how birth weight is related to mother’s age, which of these two variables response variable and the explanatory variable? Construct a scatterplot of these data. Would it be reasonable to use a line to summarize the relationship between birth weight and mother’s age? Find the equation of the least squares regression line.