Related Example on Page(s) R , 148 R , 148 R , 156, 157 R3.1, R3.2. Activity on 152, , 190.

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1 Name Chapter 3 Learning Objectives Identify explanatory and response variables in situations where one variable helps to explain or influences the other. Make a scatterplot to display the relationship between two quantitative variables. Describe the direction, form, and strength of a relationship displayed in a scatterplot and recognize outliers in a scatterplot. Section Related Example on Page(s) Relevant Chapter Review Exercise(s) R , 148 R , 148 R3.1 Interpret the correlation R3.3, R3.4 Understand the basic properties of correlation, including how the correlation is influenced by outliers , 156, 157 R3.1, R3. Use technology to calculate correlation. 3.1 Activity on 15, 171 R3.4 Explain why association does not imply Discussion on 3.1 causation. 156, 190 R3.6 Interpret the slope and y intercept of a leastsquares regression line R3., R3.4 Use the least-squares regression line to predict 167, R3., R3.4, y for a given x. Explain the dangers of 3. Discussion on R3.5 extrapolation. 168 Calculate and interpret residuals R3.3, R3.4 Discussion on Explain the concept of least squares R3.5 Determine the equation of a least-squares regression line using technology or computer output. 3. Technology Corner on 171, 181 R3.3, R3.4 Construct and interpret residual plots to assess whether a linear model is appropriate. Interpret the standard deviation of the residuals and r and use these values to assess how well the least-squares regression line models the relationship between two variables. Describe how the slope, y intercept, standard deviation of the residuals, and r are influenced by outliers. Find the slope and y intercept of the leastsquares regression line from the means and standard deviations of x and y and their correlation. 3. Discussion on 175, 180 R3.3, R R3.3, R Discussion on 188 R R3.5 Can I do this? 1

2 3.1 Describing Relationships Read What is the difference between an explanatory variable and a response variable? Read On scatterplots, the explanatory variable goes on the horizontal axis and we don t necessarily start each axis at (0,0). Start each axis near the data. What is the easiest way to lose points when making a scatterplot? (xkcd.com/833) Track and Field Day! The table below shows data for 13 students in a statistics class. Each member of the class ran a 40-yard sprint and then did a long jump (with a running start). Make a scatterplot of the relationship between sprint time (in seconds) and long jump distance (in inches). Sprint Time (s) Long Jump Distance (in)

3 Four characteristics you should consider when interpreting a scatterplot: Always Form linear or nonlinear Direction positive or negative Strength very weak, weak, moderate, strong, very strong, perfect DUFS Sometimes Unusual stuff outliers, clusters The following scatterplot shows the lengths (in 1000s of feet) and the speeds (in miles/hour) for the 78 roller coasters I rode from Oct. 007 to Aug Describe the relationship between length and speed. Caution! A strong association between two variables does not automatically indicate a cause-and-effect relationship! Correlation is NOT causation! Examples: $ of ice cream sales/week in South Carolina vs. # of shark attacks each week in South Carolina # of kids with cell phones vs. # kids who pass AP Stats exam monthly sunblock sales vs. # of shark attacks monthly Ice cream sales vs. drowning deaths hand span vs. vocabulary size Read 150: Using technology to create scatterplots HW #6: page 159 (1, 5, 7, 9, 11, 13, 34) 3

4 3.1 Correlation Just like two distributions can have the same shape and center with different spreads, two associations can have the same direction and form, but very different strengths. These are both linear and positive, but which one is stronger? Read What is the correlation r? It is a numerical measure of direction and strength of linear patterns. Linear? Use r. Not linear? Don t use r. Draw an oval around the points. A near circle indicates nearly a 0 correlation (weak). Long skinny ovals indicate correlation closer to +1 or -1 (strong). Correlation won't tell you the form of a relationship. Graph first, then you might be able to use correlation to describe strength and direction. Correlation, r, is NOT a resistant measure of strength; outliers will affect it. The formula for correlation is on the formula packet, so don't memorize it. 4

5 Read Read Eleven important facts about correlation, r 1. Switching x & y won't change r.. x and y must be quantitative (if either variable isn't quantitative, use "association"). 3. Units don't affect r & r has no units. 4. Correlation does not imply causation. 5. sign of r tells direction r 1 7. If the scatterplot is linear, then r near ±1 means strong, near 0 means weak. 8. r measures linear relationships only. 9. Correlation is not resistant-- outliers will affect r. 10. r won't tell you whether something is linear. 11. r doesn't tell the whole story; identify means and standard deviations for both x and y. HW #7: page 161 (15 18, 1, 7 3) 3. Least-Squares Regression Read The general form of a regression equation: y = a + bx The difference between y and ŷ : y = an observed value, a real data value y = a predicted value, a value from the equation To interpret slope, you must address the ideas of: rate, units, prediction, & average. To interpret the intercept, you must address the ideas of: units, prediction when x is 0 (in context), & average. To predict a y for some x, you must address the ideas of: units, prediction when x is a # (in context) & average. Used Ford Escapes The following scatterplot shows the number of miles driven (in thousands) and advertised price (in thousands) for 11 used Ford Escapes from the model years. The regression line shown on the scatterplot is ŷ = x. Miles Driven Price a) Interpret the slope and y intercept of a regression line. b) Predict the price of a used Escape with 50,000 miles. 5

6 c) Predict the price of a used Escape with 50,000 miles. How confident are you in this prediction? Extrapolation can be a bad thing: least- Using the Track and Field data from earlier, the equation of the squares regression line is ŷ = x where y = long jump distance and x = sprint time. a) Interpret the slope. b) Does it make sense to interpret the y-intercept? Explain. HW #8 page 193 (35, 37, 39, 41, 45) 3. Residuals Read What is a residual? Interpreting a residual: The residual is how far the actual y is above or below what we predict. Positive residuals mean points are above the line. Negative residuals mean points are below the line. 6

7 Calculate and interpret the residual for the Ford Escape with 57,000 miles and an asking price of $15,599. How can we determine the best regression line for a set of data? The least-squares regression line is not resistant to outliers. Example: 1 of Taco Bell s chicken menu items: (a) Calculate the equation of the least-squares regression line using technology. Make sure to define variables! Sketch the scatterplot with the graph of the least-squares regression line. (b) Interpret the slope and y-intercept in context. (c) Calculate and interpret the residual for the first item listed, the Chicken Burrito Supreme, with 1g of fat and 50g of carbs. Read To know a line is the right model to use, we can examine a residual plot. What is a residual plot? A scatterplot is a graph of the explanatory variable (x) and the response variable (y). A residual plot is a graph of the explanatory variable (x) and the residuals. 7

8 What is the purpose of a residual plot? A residual plot can reveal patterns that were not apparent (not obvious) in the original scatterplot. What would we see in a residual plot that tells us that our linear model is not a good one? Bad: A curved pattern, meaning the relationship between x&y is not linear. A line is a bad model to choose. Bad: Changing vertical spread, meaning predictions will be less accurate for some values of x. What would we see in a residual plot that tells us that our linear model is appropriate? Good: A uniform, even, random scatter, meaning any errors in our predictions will be random and about equal for any value of x. Construct and interpret a residual plot for the Ford Escape data. HW #9: page 193 (43, 47, 49, 51) 3. Standard deviation of the residuals and Read page 177 What is the standard deviation of the residuals? How do you calculate and interpret it? r Calculate and interpret the standard deviation of the residuals for the Ford Escape data. Suppose that you see a used Ford Escape for sale. Predict the asking price for this Escape, using the average asking price. Would our predictions be better if we used miles driven to help estimate the sale price? Read

9 What is the coefficient of determination r? How do you calculate it? r is a numerical measure of how much better our predictions are by using the regression equation instead of just using the average of the y values. r is called the coefficient of determination. How do you interpret r? % of the variation in (put the response variable there) is accounted for by the LSRL relating (put the response variable there) to (put the explanatory variable there). How is r related to r? r () r If r 0.49, then r 0.7 To know whether r is +0.7 or -0.7, we will need more information. How is r related to s? both tell about how well the line fits our data neither tells about form As r 1, then s 0 (but r 1 s). s 0 0 r 1 because 1 r 1 r has no unites because r has no units s has the same units as the response variable HW #30: page 193 (55, 57) Interpreting Computer Output Read

10 Several factors contribute to the speed of a roller coaster: largest drop, maximum angle of drop, initial speed at the top of the largest hill, etc. Some parks report both height and drop, others only report the height. To investigate the relationship between height and speed for steel sit-down coasters, output from a regression analysis of 39 such American rides is shown below. (a) What is the equation of the least-squares regression line? Define any variables you use. Predictor Coef SE Coef T P Constant Height S = 6.54 R-Sq = 86.0% R-Sq(adj) = 87.% (b) Interpret the slope of the least-squares regression line. (c) What is the correlation? (d) Is a linear model appropriate for this data? Explain. (e) Would you be willing to use the linear model to predict the speed of a roller coaster that is 500 feet tall, if it were built? Explain. (f) Calculate and interpret the residual for Mr. Freeze, which is 18 ft. tall and has a speed of 70 mph. 10

11 (g) Interpret the values of r and s. (h) If the speed were measured in km/h instead of mph, how would the values of r and s be affected? Explain. HW #31 page 193 (59, 61, 1 9 below) In addition to other factors affecting climate, temperatures generally tend to decrease as one moves north from the equator to the North Pole. Here are average June high temperatures and latitude for 1 cities along the US east coast. 1. Interpret the scatterplot for this data. City Latitude Avg. June High Portland Boston Providence Philadelphia Baltimore Washington, DC Virginia Beach Wilmington, NC Savannah Jacksonville Daytona Miami Calculate the equation of the least squares regression line. 3. Interpret the slope and y-intercept in the context of the problem. 11

12 4. Calculate and interpret the value of the correlation coefficient. 5. If the temperature were measured in Celsius instead of Fahrenheit, how would the correlation change? Explain. 6. Calculate and interpret the residual for Washington, DC. 7. Interpret the residual plot. 8. Identify and interpret the value of s in the context of the problem. 9. Identify and interpret the value of r in the context of the problem. 1

13 Regression Wisdom, etc. Read Does it matter which variable is x and which is y? Which of the following has the highest correlation? Examine the hypothetical scatterplot heights & weights of H. S. students below (Form, Direction, Strength?) Add each of these men and consider: Outlier in height or weight? Still fit the overall pattern? Influential? Basketball player Shaquille O'Neal (7'1", 35 lbs.) Sumo wrestler Kasugao Katsumasa (6'0", 340 lbs.) Basketball player Manute Bol, (7'6" 00 lbs) How do outliers affect the correlation, least-squares regression line, and standard deviation of the residuals? How do outliers affect the correlation, least-squares regression line, & standard deviation of the residuals? Are all outliers influential? Outlier in pattern Outlier out of pattern r gets stronger (closer to 1 or -1) r gets stronger (closer to 0) LSRL won't move much LSRL will change a lot s won't change much s will increase 13

14 Read To calculate the equation of the least-squares regression line using summary statistics, use these facts: 1. Slope = b = r s y. The LSRL passes through the point (x, y ). s x Notice that for each increase of 1 standard deviation in x, the predicted value of y increases by r times the standard deviation of y. example: Given: Mean for x=3, mean for y=7, std. dev. for x=1.5, std. dev. for y=.5, and r =.89, what is the equation of the LSRL? HW #3 page 193 (65, 71 78) Review Chapter 3 HW #33 page 0 Chapter 3 Review Exercises Review / FRAPPY! FRAPPY: 005 #3 HW #34 page 03 Chapter 3 AP Statistics Practice Test Chapter 3 Test A note on predictions: consider these two interpretations and notice how the second one is better than the first one. The model tells us that a 6 th grader with a forearm circumference of 1 inches should have a 1 rep max of 50.8 pounds. The average 1 rep max for a 6 th grader (from the sampled population) with a forearm circumference of 1 inches is expected to be approximately 50.8 pounds. 14

15 Form? Direction? Strength? Shows relationship? Use for predicting or estimating? Plot, statistic, or equation Characteristics Other facts Scatterplot Correlation r Coefficient of determination r Least Squares Regression Line y = ax + b or y = b 0 + b 1 x Residual plot 15