Prob/Stats Questions? /32

Transcription

1 Prob/Stats 10.4 Questions? 1 /32

2 Prob/Stats 10.4 Homework Apply p551 Ex 10-4 p 551 7, 8, 9, 10, 12, 13, 28 2 /32

3 Prob/Stats 10.4 Objective Compute the equation of the least squares 3 /32

4 Regression A scatter plot may suggest a pattern to the relationship between two variables. That relationship allows for prediction between the variables. If the pattern is strong, knowing the value of one variable will suggest a value for the other variable. The form of the relationship might be linear, but not all relationships are linear. Correlation and linear regression are strictly measures of a linear relationship. It is possible to have a very strong relationships that is not linear. The relationship might be a curve (quadratic, power, or exponential). Manipulating the value of one variable by taking a logarithm or root may result in a linear relationship. We will restrict ourselves to naturally linear relationships. 4 /32

5 Regression If the linear relationship between variables is strong, we mean that the variables tend to cluster to a line. That suggests the ability to predict one variable from the value of the other variable. What we then do is find the equation of the line that best fits the data. That begs the question what is the best fit? The line of best fit in which we are interested, is called the least squares regression line (LSRL). 5 /32

6 How Well Do the Data Fit? There can be several possible lines that appear to fit the data. How do we determine which is best? The line with best fit minimizes the amount of error (residual) between each point and the line. As we did with variance, this requires minimizing the square of the vertical distance from each point to the line. 6 /32

7 Least Squares Regression Line (LSRL) 100 Predicted Value Y Actual Data value Y } Error (Y - Y ) Since the line of best fit will have an equal amount of error above and below the line, the line of best fit minimizes the squared distances from the line to the actual data values. This is called the least squares line of best fit Predictor value x We wish to minimize the value: n i =1 ( Y Y ') 2 i i 7 /32

8 Least Squares Least Squares Applet 8 /32

9 Least Squares Regression Line (LSRL) From basic algebra we know one form of a linear equation is y = mx + b. For statistics we change that formula to a model: y (or y)= ^ a + bx y ( or ^y) is the predicted value of the variable for a given value of the predictor variable x. The value a is the value of y when x is 0 (y intercept). The value b is the slope of the regression line. b = Δy ' Δx 9 /32

10 Least Squares Regression Line (LSRL) The determination of values for a and b is left to a computer in the real world. We will rely heavily on the TI-84. Remember this r = n n i =1 (X i X )(Y i Y i ) (X X ) 2 (Y Y ) 2 i i i =1 n i =1 Using similar logic and the same values of X and Y we have 10/32

11 Least Squares Regression Line (LSRL) These a = n i =1 Y i n i =1 X 2 i n i =1 n X 2 i n i =1 X i n i =1 X i n i =1 2 X Y i i Note the values we need. Once again we might construct a table to find the values and sums for X, Y, X 2, and XY. b = n n i =1 X Y i i n i =1 n X 2 i n i =1 X i n i =1 X i n i =1 2 Y i Are you beginning to appreciate your new best friend the calculator? 11/32

12 Example Redux Grade First Test Final Grade Note that the points come close to forming a line. 12/32

13 Example Redux Grade First Test Final Grade Previously we found r =.933, indicating a strong, positive, linear relationship. So now we ask; What is that relationship? Enter the data into two lists in the TI Now run the regression: STAT CALC 4:LinReg(ax+b) 8:LinReg(a+bx) XList: YList: FreqList: Store RegEQ: Calculate 13/32

14 Example Redux Grade First Test Final Grade Results LinReg y = ax+b a = b = r 2 = r = The relationship between grade on first test and final grade can be modeled by: y = ax + b y =.8212x y = predicted final grade, x = grade on first test. 14/32

15 Example Redux Grade First Test Final Grade Prediction y =.8212x /32

16 Conclusion There is a strong, linear, positive relationship (r =.9331) between grade on first exam and final grade in class. 87% (r 2 ) of the variability in final grade is explained by changes in the first exam grade. The relationship between grade on first exam and final grade can be modeled by: predicted final grade =.8212(grade on first exam) For every increase of one point on first test, final grade tends to increase by.8212 points. 16/32

17 So? The preceding work and the resultant regression equation are only significant if the correlation coefficient (r) is significant. If r is not significant there is no point in using x to predict y. In this case the best predictor of y is. Y 17/32

18 A Kicker There are two assumptions that make using regression chancy. One: At each value of X the possible values of Y are normally distributed. In other words X and Y are bivariate normal. Two: At each value of X the possible values of Y have the same variance. In other words, each distribution of y (at every value of x) is normally distributed with the same standard deviation. 18/32

19 Finally It is very, very risky to extrapolate regression predictions beyond the data collected. In fact extrapolating is just plain a bad idea. A regression model may be very valuable for the range of data collected but the model is not necessarily valid for data beyond that collected. There is no guarantee that the data will continue to exhibit the same pattern beyond what has been observed. We can find many examples of correlation and regression in the students currently attending CHHS, but we must be very careful to extend those predictions to students that will attend CHHS in the future. 19/32

20 Cautionary Tale Suppose we collected the weight of a male white lab rat for the first 25 weeks after its birth. A scatterplot of the weight (grams) and time since birth (weeks) shows a fairly strong, positive, linear relationship. The linear regression equation models the data fairly well. weight ' = (time in weeks) 1. What is the slope of the regression line? Explain what it means in context. The slope of 40 means that as the rat ages one week the model predicts an increase in weight of 40 grams. 2. What is the y intercept? Explain what it means in context. The y-intercept is 100 grams, representing the predicted birth weight of the rat pup. 20/32

21 Cautionary Tale weight ' = (time in weeks) 3. Predict the rat s weight after 16 weeks. weight ' = (16) = 740 grams 4. Should you use this line to predict the rat s weight at age 2 years? Use the equation to make the prediction and think about the reasonableness of the result. There are 52 weeks in a year and 454 grams in a pound. weight ' = (104) = 4260 grams = 9.4 pounds Obviously the result is not reasonable and highlights the danger in extrapolating a regression line beyond the data collected. 21/32

22 Well, maybe not Obviously the result is not reasonable and highlights the danger in extrapolating a regression line beyond the data collected. Well, maybe not so ridiculous 22/32

23 Real Example Here is some real data, collected right here in this room in a previous year. Find the model. Grade First Te s t Final Grade Grade First Te s t Final Grade Final Grade Grade First Test The scatter plot shows a moderate, positive, linear relationship, with some potentially unusual results at (34, 26), (31, 66), (40, 37) and (48, 81). 23/32

24 Real Example Predicted final grade =.7128(score on 1st test) Grade First Te s t Final Grade Grade First Te s t Final Grade Final Grade Grade First Test r =.7846, r 2 = % of the variability in Final Grade is accounted for by the first test grade. 24/32

25 Real Example Predicted final grade =.7128(score on 1st test) r =.7846, R 2 =.6156 Is this relationship statistically significant? H0: There is no relationship between score on 1st test and final grade. Ha: There is a relationship between score on 1st test and final grade. t = r n 2 1 r 2 = = p(t ) = tcdf(6.1996, 99, 24) = x 2 = tail test We reject the null. There is a relationship between score on 1st test and final grade. 25/32

26 Real Example Predicted final grade =.7128(score on 1st test) The slope of the line is That means the predicted value for final grade goes up by.7128 points for every increase of 1 point on 1st test grade. The y-intercept is , that is the predicted final grade when the grade on the first test is 0. Predict the final grade of a student scoring 75 on the first test. Predict the final grade of a student scoring 60 on the first test. y =.7128(75) y =.7128(60) /32

27 One More Example Number of Alcoholic Drinks Dexterity Score Guess a line of best fit Dexterity Number of Alcoholic Drinks The scatter plot shows a strong, negative, linear relationship, with no significant outliers. 27/32

28 One More Example Now find r and the least squares regression line (lsrl). Number of Alcoholic Drinks Dexterity Score STAT CALC 4:LinReg(ax+b) 8:LinReg(a+bx) LinReg y = ax+b a = b = r 2 = r = XList: YList: FreqList: Store RegEQ: Calculate 1 16 Predicted dexterity = (number of drinks) r = , r 2 = % of the variability in Dexterity is accounted for by the number of alcoholic drinks consumed. 28/32

29 One More Example Predicted dexterity = (number of drinks) Is the correlation significant? t = r n 2 = r 2 = p (t 8.289) = tcdf ( 99, 8.289,6) =.0000 Yep, significant. The model is legit. 29/32

30 TI-84 Predicted dexterity = (number of drinks) It is possible to run the regression analysis, find r, r 2, the regression model, and do the hypothesis test in one move on the TI-84. STAT CALC F:LinRegTTest XList: L1 YList: L2 FreqList: β & ρ: 0 <0 >0 RegEQ: Y1 Calculate VARS Y-VARS 1 1:FuncHon ENTER 1:Y 1 ENTER 2-SampZTest y=a+bx β 0 and ρ 0 t= p= df=6 a=19.25 b= s= r 2 = r= /32

31 Predicted dexterity = (number of drinks) The slope of the line is That means the predicted value for dexterity goes down by points for every additional drink consumed. The y-intercept is 19.25, that is the predicted dexterity score when the number of drinks consumed is 0. 31/32

32 One More Example Predicted dexterity = (number of drinks) Predict the dexterity score after 6 drinks. y = x y = (6) y = VARS Y-VARS 1:Function 1:Y1 ENTER ( 6 ) 16 Predict the dexterity score after 2 drinks. Dexterity 12 8 y = (2) y = VARS Y-VARS 1:Function 1:Y1 ENTER ( 2 ) Number of Alcoholic Drinks 32/32

33 Conclusion There is a strong, linear, negative relationship (r = ) between number of alcoholic drinks and dexterity. 92% (R 2 ) of the variability in dexterity is explained by changes in number of alcoholic drinks. The relationship between alcoholic drinks and dexterity can be modeled by: Predicted dexterity = (number of drinks) y = x y = predicted dexterity x = number of drinks For every increase of one drink, dexterity tends to decrease by 2.9 points. 33/32