Math 147 Lecture Notes: Lecture 12

Transcription

1 Math 147 Lecture Notes: Lecture 12 Walter Carlip February, 2018 All generalizations are false, including this one.. Samuel Clemens (aka Mark Twain) ( ) Figures don t lie, but liars do figure. Samuel Clemens (aka Mark Twain) ( ) All we know about the world teaches us that the effects of A and B are always different in some decimal place for any A and B. Thus asking are the effects different? is foolish. John Tukey ( ) The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data. John Tukey ( ) Copyright c 2018 by Walter Carlip 1 Lecture 12

2 Today s Topics Announcements. Regression. RMS Error for Regression Copyright c 2018 by Walter Carlip 2 Lecture 12

3 Announcements Don t forget that Exam I will be held in class on Monday. Everything we have covered or is covered in the text, through Friday s lecture is fair game for the exam. That will include Chapters 1 through Chapter 10 of the text, and whatever portion of Chapter 11 we complete this week. The exam will certainly be closed book. Do not even bring your book or notes to class! You will be permitted to bring one 3 5 index card of notes no larger to the exam. You will need a simple calculator capable of performing basic arithmetic operations, including square roots. You may not borrow a calculator, and proctors will not give out loaners. You will be expected to use the tables (which will be provided) for computations involving the normal curve, and you will be expected to compute other values (mean, median, percentiles) directly not using a programmed calculator function. Using a programmed calculator function may result in an answer different and be marked wrong. You may not use a laptop or pad computer, cell phone, pda, or other electronic device. If you bring any such device to class, it must be powered off and stowed away out of sight. Photo IDs may be checked during the exam, so have your ID available and in sight. Be sure to read the class policy on cheating that appears on the syllabus on the web site! Copyright c 2018 by Walter Carlip 3 Lecture 12

4 Announcements Exams will be placed on seats in advance. These will be arranged by section, with your name already on the paper. You should come prepared, already knowing your section number and name (duh). Try to come a little early so that when we let students in, you can find your designated seat quickly and efficiently. Copyright c 2018 by Walter Carlip 4 Lecture 12

5 Ecological Correlations Ecological correlations are correlations that are based on averages or rates for groups. Correlations computed in this way are often misleading and overstate the strength of an association. The reason is rather straight forward: computing averages or rates for groups eliminates the spread by reducing the group to a single value. Since correlation is used to describe one kind of spread, using averages or rates to compute correlation tends to undervalue or ignore some of the spread that is present. Copyright c 2018 by Walter Carlip 5 Lecture 12

6 Ecological Correlations The text offers an example comparing income and education. If you compute the correlation between income and education for men aged surveyed in the Current Population Survey for 2005, you find the correlation is r However, if you compute the average income and average education level for the fifty states and the District of Columbia, and compute the correlation coefficient for these 50 pairs, you obtain r The scatter graphs tell the story. The text offers a scatter graph giving hypothetical data for three states. By collapsing the scattered data for each state to a single point, the scatter graph suggests stronger association than there actually is. Copyright c 2018 by Walter Carlip 6 Lecture 12

7 Copyright c 2018 by Walter Carlip 7 Lecture 12

8 Association is not Causation A Famous Example If we were to create a scatter graph pairing children s shoe size to their reading ability, we would find a strong association, and a correlation coefficient near 1. Does reading ability make children s feet grow? Do big feet improving learning ability? Obviously not. There is a third factor involved: age. As children age, their feet tend to grow, and they tend to improve their reading skills. In this example, age is a confounding variable. When analyzing a scatter graph, be careful to remember that the correlation coefficient measures association, but that association is not the same as causation. Copyright c 2018 by Walter Carlip 8 Lecture 12

9 Estimating y from x: Regression Suppose we are studying two properties x and y of a population of subjects. Visually, we can see the relationship between x and y with a scatter diagram, and often the data forms a football-shaped cloud around the SD-line. If we designate the variable x to be the independent variable and y the dependent, we may wish to estimate the value of y for a subject with a known x. We will consider this question first with a concrete example: height and weight data for men in the Hanes5 study. We will take height to be the independent variable x and weight to be the dependent variable y. The goal will be to make a sensible estimate of the weight of a man whose height is known. Copyright c 2018 by Walter Carlip 9 Lecture 12

10 The Hanes5 Height/Weight Data The Hanes5 data for heights and weights of men have the following summary statistics: average height 70 inches, SD 3 inches average weight 180 pounds, SD 45 pounds, r 0.4 A scatter diagram of the height and weight data for men from the Hanes5, with scales set up so that one SD for height is the same length as one SD for weight, is shown below. The dashed line is the SD line and it rises at 45 degrees, passing through the point of averages; the gray lines are new axes that pass through the point of averages. Copyright c 2018 by Walter Carlip 10 Lecture 12

11 Estimating Weight Suppose we wish to estimate the weight of a man whose height is 1 SD above average. Such a man will have height average height + 1 SD of height = 70 inches +3 inches = 73 inches. This is shown by the arrow in the diagram below. The vertical strip includes those men whose height is about 1 SD above average. Copyright c 2018 by Walter Carlip 11 Lecture 12

12 Estimating Weight Copyright c 2018 by Walter Carlip 12 Lecture 12

13 Estimating Weight Notice that a person whose height and weight are both 1 SD above average, would correspond to a point on the SD line. However, most of the points in the vertical strip lie below the SD line. Therefore an estimated weight one SD above average would be an over estimate. (We will see why this is true shortly.) Copyright c 2018 by Walter Carlip 13 Lecture 12

14 Estimating Weight It is clear from the picture that the average of the weights of men who are 1 SD above average, that is, the average weight of men whose dots appear in the vertical strip, is only a fraction of 1 SD in weight above average. The reason for this is that the correlation is only r = (If the data had r = 1, all the points would lie on the SD line, and every man with height 1 SD above average would have weight 1 SD above average.) Copyright c 2018 by Walter Carlip 14 Lecture 12

15 The Correct Estimate It turns out that the correct fraction to use is r. The average weight of men whose height is 1 SD above average height is about r SDs above average weight. In our example, the men in the vertical strip (those whose height is about 1 SD above average), have average weight approximately average weight+.40 SD of weight = 180 pounds pounds = 180 pounds +18 pounds = 198 pounds. Copyright c 2018 by Walter Carlip 15 Lecture 12

16 The analysis generalizes. The Correct Estimate Consider men whose height is 2 SDs above average: average height + 2 SD of height = 70 inches inches = 76 inches. The average weight of such men will not be 2 SDs above average, it will instead be 2r SDs above average in weight, that is, 2.40 =.80 SDs above average. Thus, the average weight of 76 inch tall men will be average weight+2.40 SD of weight = 180 pounds pounds = 180 pounds +36 pounds = 216 pounds. Copyright c 2018 by Walter Carlip 16 Lecture 12

17 The Regression Line The same analysis works for men whose heights are below average. For each SD above or below average height, the average weight will increase or decrease by.40 SDs in weight. The average weight of men who are 1 SD below average in height is.40 SDs below average in weight, that is, 70 3 = 67 inch tall men average = 162 pounds. The average weights for men 1 and 2 SDs above average and 3 SDs below average are marked with x s in the diagram below. These x s lie on a line: The Regression Line. Copyright c 2018 by Walter Carlip 17 Lecture 12

18 The Regression Line Each increase of one SD in a variable x results, on the average, in an increase of approximately r SDs in y. The regression line passes through points with a given x value and corresponding average y value. Therefore it is a line that passes through the point of averages and has slope equal to slope of regression line = r SD y SD x Copyright c 2018 by Walter Carlip 18 Lecture 12

19 The Regression Line Notice that we were careful to say average weight increases by approximately r SDs for each increase of one SD in height. If we graph the actual average weights, we obtain the following scatter graph. The line in the chart is the regression line. The regression line is a smooth approximation of the graph of averages. Copyright c 2018 by Walter Carlip 19 Lecture 12

20 Computational Examples: Height and Weight The summary statistics for height and weight of men in the Hanes5 study are as follows: height average 70 inches height SD 3 inches weight average 180 pounds r 0.40 weight SD 45 pounds Suppose a man from the study is chosen at random and you are asked to guess his weight without any other information about him. The best guess is the overall average 180 pounds. Now suppose you are told the man s height is 73 inches. You have more information about him. The man is tall (3 inches above average) and you should expect him to be heavier than average. Your best guess should be the average weight of those men in the study that are 73 inches tall. This average is best estimated using the regression line. Since 73 inch tall men are one standard deviation above average, and the correlation is.4, the average weight of 73 inch tall men in the study is approximately = = 198 pounds. Copyright c 2018 by Walter Carlip 20 Lecture 12

21 Why? When we examine the Hanes5 height/weight data for men, why don t we estimate the weight of a man who is 1 SD above average height with a weight that is also 1 SD above average? The key lies in the geometry of the scatter diagrams. A typical scatter diagram forms a football shaped oval, slanted in the direction of the SD line. Suppose that r is positive, so the SD line slopes upward. Because of the slant and the oval shape, a vertical strip to the right of center cuts off more of the football below the SD line and a strip to the left of the center cuts off less of the football below the SD line. A line that cuts each strip in half will intersect the SD line at the point of averages, but will have a smaller slope. Copyright c 2018 by Walter Carlip 21 Lecture 12

22 Typical Data Shape in Standard Units Copyright c 2018 by Walter Carlip 22 Lecture 12

23 Computational Examples: Do tall men marry tall women? Researchers studied 1000 families to compare heights of husbands and wives. They computed the average and standard deviations of the heights of the husbands and wives, and then found the correlation coefficient. These values are often grouped together and described as summary statistics for the data: Husbands average 68 inches Husbands SD 2.7 inches Wives average 63 inches Wives SD 2.5 inches r 0.25 The objective is to predict the height of a wife, knowing the height of the husband. Certainly we should predict that a husband of average height (68 inches) will have (on the average) a wife of average height (63 inches). This is because the point of averages should land in the center of the football-shaped cloud of data. What should we predict for a husband whose height is 72 inches? Copyright c 2018 by Walter Carlip 23 Lecture 12

24 Computational Examples: Do tall men marry tall women? What should we predict for a husband whose height is 72 inches? We compute: = 4, so the husband is 4 inches above average height. Since 4/ , this husband is 1.48 standard deviations taller than average (that is, his height in standard units is 1.48). We predict his wife will be =.37 standard deviations above average height. Thus, we predict a height of = = inches. What should we predict for a husband whose height is 64 inches? Now the husband is shorter than average: = 4, so the husband is 4 inches below average height, and therefore 1.48 standard deviations below average height. (Now his height in standard units is 1.48.) We predict his wife will be =.37 standard deviations below average height, that is, = = inches. Copyright c 2018 by Walter Carlip 24 Lecture 12

25 SATs and GPAs A university analyzes the relationship between Math SAT scores and Freshman GPAs for students completing the first year. The summary statistics are: SAT average 550 points GPA average 2.6 points r 0.4 SAT SD 80 points GPA SD.6 points The scatter diagram is football-shaped. A student is chosen at random and has an SAT score of 700. What should we predict for his first year GPA? The student is = 150 points above average, and hence 150/80 = standard deviations above average, i.e., his SAT score is in standard units. Since r 0.4, we predict his GPA to be =.75 standard deviations above average. Thus, we predict a GPA of = = Copyright c 2018 by Walter Carlip 25 Lecture 12

26 Percentiles The regression method can also be used to estimate percentile ranks. Suppose that the percentile rank of one student on the SAT is 90% among the first year students. What percentile rank should we predict for his GPA? The student s percentile rank can be used to determine how many standard deviations above average his score is. Remember, 90% of students have a score below his. We find the student s score in standard units by using the normal table (in reverse): Once we know the student s SAT in standard units, we can predict his GPA. In standard units we predict a GPA of about = (Note that the text rounds this to.5.) To find the predicted GPA, we compute predicted GPA = To compute the predicted percentile, however, we only need the predicted GPA of about 0.5 in standard units. We predict that this student will land in the 69-th percentile. When r = 0.4, we predict a student in the 90-th percentile for SAT will land in the 69-th percentile for GPA. Copyright c 2018 by Walter Carlip 26 Lecture 12

27 Percentiles Let s try it again for a student in the 10-th percentile. From the tables, a student in the 10-th percentile has an SAT score in standard units of z 1.3: Once we know the student s SAT in standard units, we can predict his GPA: 2.6 ( ) This time, the GPA is =.52 (about 0.5) standard deviations below average. Thus, we predict a GPA of 0.5 standard units. Again we consult the normal tables: When r = 0.4, we predict a student in the 10-th percentile for SAT will land in the 30-th percentile for GPA. Copyright c 2018 by Walter Carlip 27 Lecture 12

28 Why? If x and y values have perfect correlation, r = 1, then an x-value n standard units above average will correspond to a y value n standard units above average. In this case, all of the data lies exactly on the SD-line. If you know x in standard units, you guess y to have the same value in standard units. However, if the data are totally unrelated, r = 0, information about x tells you nothing about the corresponding y value. The cloud of data in the scatter diagram has no relation to the SD-line. If you know x, your best guess for y is average, i.e., zero standard units, whether x is above or below average. [Textbook example: if you know someone is above average in pottery, this will not help predict how good they are in mathematics.] If r lies between 0 and 1, then the x value tells you something about the y value. The text offers the example of skill in mathematics and physics. If a person is one standard deviation above average in physics, you expect them to be above average in mathematics, but perhaps not a full standard deviation. If the correlation between physics and math scores is r = 0.8, then you predict the student to be only 0.8 standard deviations above average in math. Copyright c 2018 by Walter Carlip 28 Lecture 12

29 Homework Please read Chapters 11 and 12 carefully Homework Week Date Assignment Chapter Page Set Problems 5 February 12, 2016 Reading 10 Set # A 1, 2, 3, B 2, 3, C 1, 2, D 1, 2, E 1, 2, Review 1, 2, 4, 6, 7, 9, 10 5 February 14, 2016 Reading 11 Set # A 1, 2, 3, 4, 5, B 1, 2, C 1, 2, D 1, E 1, Review 1, 2, 3, 7 Copyright c 2018 by Walter Carlip 29 Lecture 12