Overview. 4.1 Tables and Graphs for the Relationship Between Two Variables. 4.2 Introduction to Correlation. 4.3 Introduction to Regression 3.

Transcription

1 3.1-1

2 Overview 4.1 Tables and Graphs for the Relationship Between Two Variables 4.2 Introduction to Correlation 4.3 Introduction to Regression 3.1-2

3 4.1 Tables and Graphs for the Relationship Between Two Variables Objectives: By the end of this section, I will be able to 1) Construct and interpret crosstabulations for two categorical variables. 2) Construct and interpret clustered bar graphs for two categorical variables. 3) Construct and interpret scatterplots for two quantitative variables

4 Crosstabulations Tabular method for simultaneously summarizing the data for two categorical (qualitative) variables Constructing a Crosstabulation Step 1 Put the categories of one variable at the top of each column, and the categories of the other variable at the beginning of each row

5 Crosstabulations Steps for Constructing a Crosstabulation Step 2 For each row and column combination, enter the number of observations that fall in the two categories. Step 3 The bottom of the table gives the column totals, and the right-hand column gives the row totals

6 Table 4.1 Prestigious career survey data set 3.1-6

7 Example Crosstabulation of the prestigious career survey Construct a crosstabulation of career and gender

8 Example 4.1 continued Solution Step 1 Crosstabulation given in Table 4.2. Categories for the variable gender are at the top Categories for the variable career are on the left. Each student in the sample is associated with a certain cell For example, a male student who reported Military Officer appears as one of the four students in the Male column and the Military Officer row

9 Example 4.1 continued Step 2 For each row and column combination, enter the number of observations that fall in the two categories. This is shown in Table 4.2. Step 3 Total column contains the sum of the counts of the cells in each row (category) of the career variable This sum represents the frequency distribution for this variable

10 Example 4.1 continued Step 3 - continued Total contains sums the counts of the cells in each column (category) of the gender variable This sum represents the frequency distribution for this variable. Thus, we see that crosstabulations contain the frequency distributions of each of the two variables

11 Example 4.1 continued Table 4.2 Crosstabulation of prestigious career survey data set

12 Example 4.1 continued Use the crosstabulation to look for patterns For example, does there appear to be a difference between males and females responses? Most of the students who responded Doctor were females, and most of the students who responded Military Officer were males

13 Clustered Bar Graphs Useful for comparing two categorical variables Used in conjunction with crosstabulations Each set of bars in a clustered bar graph represents a single category of one variable One can construct clustered bar graphs using either frequencies or relative frequencies

14 Scatterplots Used to summarize the relationship between two quantitative variables that have been measured on the same element Graph of points (x, y) each of which represents one observation from the data set One of the variables is measured along the horizontal axis and is called the x variable The other variable is measured along the vertical axis and is called the y variable

15 Predictor Variable and Response Variable The value of the x variable can be used to predict or estimate the value of the y variable The x variable is referred to as the predictor variable The y variable is called the response variable

16 Scatterplot Terminology Note the terminology in the caption to Figure 4.2. When describing a scatterplot, always indicate the y variable first and use the term versus (vs.) or against the x variable. This terminology reinforces the notion that the y variable depends on the x variable

17 FIGURE 4.2 Scatterplot of sales price versus square footage

18 Positive relationship As the x variable increases in value, the y variable also tends to increase. FIGURE 4.3 (a) Scatterplot of a positive relationship

19 Negative relationship As the x variable increases in value, the y variable tends to decrease FIGURE 4.3 (b) scatterplot of a negative relationship

20 No apparent relationship As the x variable increases in value, the y variable tends to remain unchanged FIGURE 4.3 (c) scatterplot of no apparent relationship

21 Example Relationship between lot size and price in Glen Ellyn, Illinois Using Figure 4.2, investigate the relationship between lot square footage and lot price

22 Example 4.4 continued Solution The scatterplot in Figure 4.2 most resembles Figure 4.3a, where a positive relationship exists between the variables. Thus, smaller lot sizes tend to be associated with lower prices, and larger lot sizes tend to be associated with higher prices. Put another way, as the lot size increases, the lot price tends to increase as well

23 Summary Crosstabulation summarizes the relationship between two categorical variables. A crosstabulation is a table that gives the counts for each row-column combination, with totals for the rows and columns. Clustered bar graphs are useful for comparing two categorical variables and are often used in conjunction with crosstabulations. For two numerical variables, scatterplots summarize the relationship by plotting all the (x, y) points

24 4.2 Introduction to Correlation Objective: By the end of this section, I will be able to 1) Calculate and interpret the value of the correlation coefficient

25 Correlation Coefficient r Measures the strength and direction of the linear relationship between two variables. r ( x x ( n 1) s x s y )( y y) s x is the sample standard deviation of the x data values. s y is the sample standard deviation of the y data values

26 Example Page

27 Example

28 Example Positively correlated, negatively correlated, not correlated?

29 Example ANSWER: positively correlated

30 Example

31 Example do 20(b) first: 20(a) x 2553 y 2583 x x 2553 y y 516. n 5 n

32 Example 20(c) s x x n x s y y n y

33 Example 20(d) r ( x ( n x)( y 1) s x s y y) ( )(10.968)(5.320)

34 Equivalent Computational Formula for Calculating the Correlation Coefficient r r xy x y / n x x / n y y / n

35 Calculate r Directly Using Calculator Option 1 Enter the data in two lists. Press STAT and select TESTS LinRegTTest is option F (scroll arrow up 3 places) Enter the names of the lists from step 1. Arrow down to Calculate and then press Enter The r value is the last value displayed; round this value to three decimal places

36 Calculate r Directly Using Calculator Option 2 (page 193)

37 Example Page (e) Use the calculator to verify the answer in 20(d)

38 Interpreting the Correlation Coefficient r 1) Values of r close to 1 indicate a positive relationship between the two variables. The variables are said to be positively correlated. As x increases, y tends to increase as well

39 Interpreting the Correlation Coefficient r 2) Values of r close to -1 indicate a negative relationship between the two variables. The variables are said to be negatively correlated. As x increases, y tends to decrease

40 Interpreting the Correlation Coefficient r 3) Other values of r indicate the lack of either a positive or negative linear relationship between the two variables. The variables are said to be uncorrelated As x increases, y tends to neither increase nor decrease linearly

41 Guidelines for Interpreting the Correlation Coefficient r If the correlation coefficient between two variables is greater than 0.7, the variables are positively correlated. between 0.33 and 0.7, the variables are mildly positively correlated. between 0.33 and 0.33, the variables are not correlated. between 0.7 and 0.33, the variables are mildly negatively correlated. less than 0.7, the variables are negatively correlated

42 Example Page

43 Solution continued we found the correlation coefficient for the relationship between SAT I verbal and math scores to be r = r = is close to 1. We would therefore say that SAT I verbal and math scores are strongly positively correlated. As verbal score increases, math score also tends to increase

44 Example ANSWER:

45 Example Page

46 Example Page

47 ANSWER: positive Example

48 Example Page

49 Example ANSWER: somewhere in the middle

50 Example Page

51 Example ANSWER:

52 Common Error Interpreting Correlation correlation does not imply causality EXAMPLE: Umbrella sales are negatively correlated with attendance at baseball games in outdoor stadiums (that is, as the amount of umbrella sales increases, the attendance at baseball games in outdoor stadiums tend to decrease). It is not correct to conclude that increased umbrella sales causes a decrease in attendance. Both of these are probably caused by a hidden variable: rainfall

53 Summary Section 4.2 introduces the correlation coefficient r, a measure of the strength of linear association between two numeric variables. Values of r close to 1 indicate that the variables are positively correlated. Values of r close to 1 indicate that the variables are negatively correlated. Values of r close to 0 indicate that the variables are not correlated

54 4.3 Introduction to Regression Objectives: By the end of this section, I will be able to 1) Calculate the value and understand the meaning of the slope and the y intercept of the regression line. 2) Predict values of y for given values of x

55 Interpreting the Slope of a Line For a line with equation: y b mx we interpret a nonzero slope m as y increases (if m is positive) or decreases (if m is negative) by m units for every one unit increase in x

56 Interpreting the y-intercept of a Line For a line with equation: y b mx we interpret a y-intercept b as The y value is b when the x value is

57 Equation of the Regression Line Approximates the relationship between two random variables x and y The equation is yˆ b b x 0 1 where the regression coefficients are the slope, b 1, and the y-intercept, b 0. The hat over the y (pronounced y-hat ) indicates that this is an estimate of y and not necessarily an actual value of y

58 Relationship Between Slope and Correlation Coefficient The slope b 1 of the regression line and the correlation coefficient r always have the same sign. b 1 is positive if and only if r is positive. b 1 is negative if and only if r is negative

59 Regression coefficients b 0 and b 1 b 1 x x x y x 2 y b y b 0 1 x All of the quantities needed to calculate b 0 and b 1 have already been computed in the formula for r. Numerators for b 1 and r are exactly the same

60 Example Page 194, problems use this table

61 Example x 2553 y 2583 x 2553 y 2583 x y n 5 n

62 Example Page 194, problem 10(a) b 1 x x x y x 2 y

63 Example Page 194, problem 10(b) b 0 y b1 x

64 Example Page 194, problem 10(c) yˆ b b x 0 1 yˆ x

65 Calculator 1. Enter the data in two lists. 2. Make a scatter plot of the data (use 2 nd Y= to get STAT PLOT, choose Plot1 On, first scatterplot icon, then zoom 9) (your plot will look different)

66 Calculator 3. Plot the regression line. Choose: STAT CALC #4 LinReg(ax+b) 4. Include the parameters L 1, L 2, Y 1. NOTE: Y 1 comes from VARS YVARS, #Function, Y

67 Calculator 5. Choose Y= and the equation for the regression line will be stored in Y 1 Then choose GRAPH and the regression line will be plotted

68 Calculator 6. Choose TRACE and you can see X and Y values on scatterplot or regression line

69 Example Page 194, problem 11(a) A slope of means that the estimated SAT I Math score increases by points for every increase of 1 point in the SAT I verbal score

70 Example Page 194, problem 11(b) The y-intercept of means that the estimated SAT I Math score is when the SAT I Verbal score is

71 Example Page 194, problem 12(a) Find yˆ when x 500 ŷ

72 Example Page 194, problem 12(b) Find yˆ when x 510 ŷ

73 Example Page 194, problem 12(c) Find yˆ when x 490 The x values in the data set range from 497 to 522. Since 490 is not in the range of the x values in the data set, it is not appropriate to use the regression equation in this case

74 Summary Section 4.3 introduces regression, where the linear relationship between two numerical variables is approximated using a straight line, called the regression line. The equation of the regression line is written as y b0 b1x where the regression coefficients are the y intercept, b 0, and the slope, b

75 Summary The regression equation can be used to make predictions about values of y for particular values of x