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3 Chapter 4 Describing the Relation between Two Variables © 2010 Pearson Prentice Hall. All rights reserved Section 4.1 Scatter Diagrams and Correlation © 2010 Pearson Prentice Hall. All rights reserved 4-2 © 2010 Pearson Prentice Hall. All rights reserved 4-3 3 © 2010 Pearson Prentice Hall. All rights reserved 4-4 © 2010 Pearson Prentice Hall. All rights reserved 4-5 5 EXAMPLE Drawing and Interpreting a Scatter Diagram The data shown to the right are based on a study for drilling rock. The researchers wanted to determine whether the time it takes to dry drill a distance of 5 feet in rock increases with the depth at which the drilling begins. So, depth at which drilling begins is the explanatory variable, x, and time (in minutes) to drill five feet is the response variable, y. Draw a scatter diagram of the data. Source: Penner, R., and Watts, D.G. “Mining Information.” The American Statistician, Vol. 45, No. 1, Feb. 1991, p. 6. © 2010 Pearson Prentice Hall. All rights reserved 4-6 © 2010 Pearson Prentice Hall. All rights reserved 4-7 Various Types of Relations in a Scatter Diagram © 2010 Pearson Prentice Hall. All rights reserved 4-8 © 2010 Pearson Prentice Hall. All rights reserved 4-9 © 2010 Pearson Prentice Hall. All rights reserved 4-10 © 2010 Pearson Prentice Hall. All rights reserved 4-11 © 2010 Pearson Prentice Hall. All rights reserved 4-12 © 2010 Pearson Prentice Hall. All rights reserved 4-13 © 2010 Pearson Prentice Hall. All rights reserved 4-14 © 2010 Pearson Prentice Hall. All rights reserved 4-15 EXAMPLE Determining the Linear Correlation Coefficient Determine the linear correlation coefficient of the drilling data. © 2010 Pearson Prentice Hall. All rights reserved 4-16 © 2010 Pearson Prentice Hall. All rights reserved 4-17 xi x yi y s s x y r n 1 8.501037 12 1 0.773 © 2010 Pearson Prentice Hall. All rights reserved 4-18 18 © 2010 Pearson Prentice Hall. All rights reserved 4-19 19 © 2010 Pearson Prentice Hall. All rights reserved 4-20 20 EXAMPLE Does a Linear Relation Exist? Determine whether a linear relation exists between time to drill five feet and depth at which drilling begins. Comment on the type of relation that appears to exist between time to drill five feet and depth at which drilling begins. The correlation between drilling depth and time to drill is 0.773. The critical value for n = 12 observations is 0.576. Since 0.773 > 0.576, there is a positive linear relation between time to drill five feet and depth at which drilling begins. © 2010 Pearson Prentice Hall. All rights reserved 4-21 21 © 2010 Pearson Prentice Hall. All rights reserved 4-22 22 According to data obtained from the Statistical Abstract of the United States, the correlation between the percentage of the female population with a bachelor’s degree and the percentage of births to unmarried mothers since 1990 is 0.940. Does this mean that a higher percentage of females with bachelor’s degrees causes a higher percentage of births to unmarried mothers? Certainly not! The correlation exists only because both percentages have been increasing since 1990. It is this relation that causes the high correlation. In general, time series data (data collected over time) will have high correlations because each variable is moving in a specific direction over time (both going up or down over time; one increasing, while the other is decreasing over time). When data are observational, we cannot claim a causal relation exists between two variables. We can only claim causality when the data are collected through a designed experiment. © 2010 Pearson Prentice Hall. All rights reserved 4-23 23 Another way that two variables can be related even though there is not a causal relation is through a lurking variable. A lurking variable is related to both the explanatory and response variable. For example, ice cream sales and crime rates have a very high correlation. Does this mean that local governments should shut down all ice cream shops? No! The lurking variable is temperature. As air temperatures rise, both ice cream sales and crime rates rise. © 2010 Pearson Prentice Hall. All rights reserved 4-24 © 2010 Pearson Prentice Hall. All rights reserved 4-25 25 This study is a prospective cohort study, which is an observational study. Therefore, the researchers cannot claim that increased cola consumption causes a decrease in bone mineral density. Some lurking variables in the study that could confound the results are: • • • • • • body mass index height smoking alcohol consumption calcium intake physical activity © 2010 Pearson Prentice Hall. All rights reserved 4-26 Section 4.2 Least-squares Regression © 2010 Pearson Prentice Hall. All rights reserved 4-27 27 Using the following sample data: (a) Find a linear equation that relates x (the explanatory variable) and y (the response variable) by selecting two points and finding the equation of the line containing the points. Using (2, 5.7) and (6, 1.9): 5.7 1.9 26 0.95 m y y1 m x x1 y 5.7 0.95 x 2 y 5.7 0.95 x 1.9 y 0.95 x 7.6 © 2010 Pearson Prentice Hall. All rights reserved 4-28 28 (b) Graph the equation on the scatter diagram. 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 (c) Use the equation to predict y if x = 3. y 0.95 x 7.6 0.95(3) 7.6 4.75 © 2010 Pearson Prentice Hall. All rights reserved 4-29 © 2010 Pearson Prentice Hall. All rights reserved 4-30 The difference between the observed value of y and the predicted value of y is the error, or residual. Using the line from the last example, and the predicted value at x = 3: residual = observed y – predicted y = 5.2 – 4.75 = 0.45 7 6 (3, 5.2) y – predicted y } residual == observed 5.2 – 4.75 5 = 0.45 4 3 2 1 0 0 1 2 3 4 5 © 2010 Pearson Prentice Hall. All rights reserved 6 7 4-31 © 2010 Pearson Prentice Hall. All rights reserved 4-32 © 2010 Pearson Prentice Hall. All rights reserved 4-33 EXAMPLE Finding the Least-squares Regression Line Using the drilling data (a) Find the least-squares regression line. (b) Predict the drilling time if drilling starts at 130 feet. (c) Is the observed drilling time at 130 feet above, or below, average. (d) Draw the least-squares regression line on the scatter diagram of the data. © 2010 Pearson Prentice Hall. All rights reserved 4-34 (a) We agree to round the estimates of the slope and intercept to four decimal places. yˆ 0.0116 x 5.5273 (b) yˆ 0.0116 x 5.5273 0.0116(130) 5.5273 7.035 (c) The observed drilling time is 6.93 seconds. The predicted drilling time is 7.035 seconds. The drilling time of 6.93 seconds is below average. © 2010 Pearson Prentice Hall. All rights reserved 4-35 (d) 8.5 Time to Drill 5 Feet 8 7.5 7 6.5 6 5.5 0 20 40 60 80 100 120 140 160 180 200 Depth Drilling Begins © 2010 Pearson Prentice Hall. All rights reserved 4-36 © 2010 Pearson Prentice Hall. All rights reserved 4-37 Interpretation of Slope: The slope of the regression line is 0.0116. For each additional foot of depth we start drilling, the time to drill five feet increases by 0.0116 minutes, on average. Interpretation of the y-Intercept: The y-intercept of the regression line is 5.5273. To interpret the y-intercept, we must first ask two questions: 1. Is 0 a reasonable value for the explanatory variable? 2. Do any observations near x = 0 exist in the data set? A value of 0 is reasonable for the drilling data (this indicates that drilling begins at the surface of Earth. The smallest observation in the data set is x = 35 feet, which is reasonably close to 0. So, interpretation of the y-intercept is reasonable. The time to drill five feet when we begin drilling at the surface of Earth is 5.5273 minutes. © 2010 Pearson Prentice Hall. All rights reserved 4-38 If the least-squares regression line is used to make predictions based on values of the explanatory variable that are much larger or much smaller than the observed values, we say the researcher is working outside the scope of the model. Never use a least-squares regression line to make predictions outside the scope of the model because we can’t be sure the linear relation continues to exist. © 2010 Pearson Prentice Hall. All rights reserved 4-39 © 2010 Pearson Prentice Hall. All rights reserved 4-40 To illustrate the fact that the sum of squared residuals for a least-squares regression line is less than the sum of squared residuals for any other line, use the “regression by eye” applet. © 2010 Pearson Prentice Hall. All rights reserved 4-41 Section 4.3 Diagnostics on the Least-squares Regression Line © 2010 Pearson Prentice Hall. All rights reserved 4-42 42 © 2010 Pearson Prentice Hall. All rights reserved 4-43 43 The coefficient of determination, R2, measures the proportion of total variation in the response variable that is explained by the least-squares regression line. The coefficient of determination is a number between 0 and 1, inclusive. That is, 0 < R2 < 1. If R2 = 0 the line has no explanatory value If R2 = 1 means the line variable explains 100% of the variation in the response variable. © 2010 Pearson Prentice Hall. All rights reserved 4-44 44 The data to the right are based on a study for drilling rock. The researchers wanted to determine whether the time it takes to dry drill a distance of 5 feet in rock increases with the depth at which the drilling begins. So, depth at which drilling begins is the predictor variable, x, and time (in minutes) to drill five feet is the response variable, y. Source: Penner, R., and Watts, D.G. “Mining Information.” The American Statistician, Vol. 45, No. 1, Feb. 1991, p. 6. © 2010 Pearson Prentice Hall. All rights reserved 4-45 45 © 2010 Pearson Prentice Hall. All rights reserved 4-46 Sample Statistics Mean Standard Deviation Depth 126.2 52.2 Time 6.99 0.781 Correlation Between Depth and Time: 0.773 Regression Analysis The regression equation is Time = 5.53 + 0.0116 Depth © 2010 Pearson Prentice Hall. All rights reserved 4-47 Suppose we were asked to predict the time to drill an additional 5 feet, but we did not know the current depth of the drill. What would be our best “guess”? © 2010 Pearson Prentice Hall. All rights reserved 4-48 Suppose we were asked to predict the time to drill an additional 5 feet, but we did not know the current depth of the drill. What would be our best “guess”? © 2010 Pearson Prentice Hall. All rights reserved 4-49 Suppose we were asked to predict the time to drill an additional 5 feet, but we did not know the current depth of the drill. What would be our best “guess”? ANSWER: The mean time to drill an additional 5 feet: 6.99 minutes © 2010 Pearson Prentice Hall. All rights reserved 4-50 Now suppose that we are asked to predict the time to drill an additional 5 feet if the current depth of the drill is 160 feet? ANSWER: Our “guess” increased from 6.99 minutes to 7.39 minutes based on the knowledge that drill depth is positively associated with drill time. © 2010 Pearson Prentice Hall. All rights reserved 4-51 © 2010 Pearson Prentice Hall. All rights reserved 4-52 The difference between the observed value of the response variable and the mean value of the response variable is called the total deviation and is equal to © 2010 Pearson Prentice Hall. All rights reserved 4-53 The difference between the predicted value of the response variable and the mean value of the response variable is called the explained deviation and is equal to © 2010 Pearson Prentice Hall. All rights reserved 4-54 The difference between the observed value of the response variable and the predicted value of the response variable is called the unexplained deviation and is equal to © 2010 Pearson Prentice Hall. All rights reserved 4-55 © 2010 Pearson Prentice Hall. All rights reserved 4-56 Total Variation = Unexplained Variation + Explained Variation © 2010 Pearson Prentice Hall. All rights reserved 4-57 Total Variation = Unexplained Variation + Explained Variation Unexplained Variation 1= Explained Variation + Total Variation Total Variation Explained Variation =1– Total Variation Unexplained Variation Total Variation © 2010 Pearson Prentice Hall. All rights reserved 4-58 To determine R2 for the linear regression model simply square the value of the linear correlation coefficient. © 2010 Pearson Prentice Hall. All rights reserved 4-59 EXAMPLE Determining the Coefficient of Determination Find and interpret the coefficient of determination for the drilling data. Because the linear correlation coefficient, r, is 0.773, we have that R2 = 0.7732 = 0.5975 = 59.75%. So, 59.75% of the variability in drilling time is explained by the least-squares regression line. © 2010 Pearson Prentice Hall. All rights reserved 4-60 Draw a scatter diagram for each of these data sets. For each data set, the variance of y is 17.49. © 2010 Pearson Prentice Hall. All rights reserved 4-61 Data Set A Data Set B © 2010 Pearson Prentice Hall. All rights reserved Data Set C 4-62 © 2010 Pearson Prentice Hall. All rights reserved 4-63 Residuals play an important role in determining the adequacy of the linear model. In fact, residuals can be used for the following purposes: • To determine whether a linear model is appropriate to describe the relation between the predictor and response variables. • To determine whether the variance of the residuals is constant. • To check for outliers. © 2010 Pearson Prentice Hall. All rights reserved 4-64 If a plot of the residuals against the predictor variable shows a discernable pattern, such as curved, then the response and predictor variable may not be linearly related. © 2010 Pearson Prentice Hall. All rights reserved 4-65 © 2010 Pearson Prentice Hall. All rights reserved 4-66 A chemist as a 1000-gram sample of a radioactive material. She records the amount of radioactive material remaining in the sample every day for a week and obtains the following data. Day Weight (in grams) 0 1 2 3 4 5 6 7 1000.0 897.1 802.5 719.8 651.1 583.4 521.7 468.3 © 2010 Pearson Prentice Hall. All rights reserved 4-67 Linear correlation coefficient: -0.994 © 2010 Pearson Prentice Hall. All rights reserved 4-68 © 2010 Pearson Prentice Hall. All rights reserved 4-69 Linear model not appropriate © 2010 Pearson Prentice Hall. All rights reserved 4-70 If a plot of the residuals against the explanatory variable shows the spread of the residuals increasing or decreasing as the explanatory variable increases, then a strict requirement of the linear model is violated. This requirement is called constant error variance. The statistical term for constant error variance is homoscedasticity © 2010 Pearson Prentice Hall. All rights reserved 4-71 © 2010 Pearson Prentice Hall. All rights reserved 4-72 A plot of residuals against the explanatory variable may also reveal outliers. These values will be easy to identify because the residual will lie far from the rest of the plot. © 2010 Pearson Prentice Hall. All rights reserved 4-73 © 2010 Pearson Prentice Hall. All rights reserved 4-74 EXAMPLE Residual Analysis Draw a residual plot of the drilling time data. Comment on the appropriateness of the linear least-squares regression model. © 2010 Pearson Prentice Hall. All rights reserved 4-75 © 2010 Pearson Prentice Hall. All rights reserved 4-76 Boxplot of Residuals for the Drilling Data © 2010 Pearson Prentice Hall. All rights reserved 4-77 © 2010 Pearson Prentice Hall. All rights reserved 4-78 An influential observation is one that has a disproportionate affect on the value of the slope and y-intercept in the least-squares regression equation. © 2010 Pearson Prentice Hall. All rights reserved 4-79 Explanatory, x Influential observations typically exist when the point is large relative to its X value. So, Case 3 is likely influential. © 2010 Pearson Prentice Hall. All rights reserved 4-80 EXAMPLE Influential Observations Suppose an additional data point is added to the drilling data. At a depth of 300 feet, it took 12.49 minutes to drill 5 feet. Is this point influential? © 2010 Pearson Prentice Hall. All rights reserved 4-81 © 2010 Pearson Prentice Hall. All rights reserved 4-82 © 2010 Pearson Prentice Hall. All rights reserved 4-83 With influential Without influential © 2010 Pearson Prentice Hall. All rights reserved 4-84 As with outliers, influential observations should be removed only if there is justification to do so. When an influential observation occurs in a data set and its removal is not warranted, there are two courses of action: (1) Collect more data so that additional points near the influential observation are obtained, or (2) Use techniques that reduce the influence of the influential observation (such as a transformation or different method of estimation e.g. minimize absolute deviations). © 2010 Pearson Prentice Hall. All rights reserved 4-85 Section 4.4 Contingency Tables and Association © 2010 Pearson Prentice Hall. All rights reserved 4-86 86 A professor at a community college in New Mexico conducted a study to assess the effectiveness of delivering an introductory statistics course via traditional lecture-based method, online delivery (no classroom instruction), and hybrid instruction (online course with weekly meetings) methods, the grades students received in each of the courses were tallied. The table is referred to as a contingency table, or two-way table, because it relates two categories of data. The row variable is grade, because each row in the table describes the grade received for each group. The column variable is delivery method. Each box inside the table is referred to as a cell. © 2010 Pearson Prentice Hall. All rights reserved 87 4-87 © 2010 Pearson Prentice Hall. All rights reserved 4-88 88 A marginal distribution of a variable is a frequency or relative frequency distribution of either the row or column variable in the contingency table. © 2010 Pearson Prentice Hall. All rights reserved 4-89 89 EXAMPLE Determining Frequency Marginal Distributions A professor at a community college in New Mexico conducted a study to assess the effectiveness of delivering an introductory statistics course via traditional lecture-based method, online delivery (no classroom instruction), and hybrid instruction (online course with weekly meetings) methods, the grades students received in each of the courses were tallied. Find the frequency marginal distributions for course grade and delivery method. © 2010 Pearson Prentice Hall. All rights reserved 4-90 90 EXAMPLE Determining Relative Frequency Marginal Distributions Determine the relative frequency marginal distribution for course grade and delivery method. © 2010 Pearson Prentice Hall. All rights reserved 4-91 91 © 2010 Pearson Prentice Hall. All rights reserved 4-92 92 A conditional distribution lists the relative frequency of each category of a variable given a specific value of the other variable in the contingency table. © 2010 Pearson Prentice Hall. All rights reserved 4-93 93 EXAMPLE Determining a Conditional Distribution Construct a conditional distribution of course grade by method of delivery. Comment on any type of association that may exist between course grade and delivery method. It appears that students in the hybrid course are more likely to pass (A, B, or C) than the other two methods. © 2010 Pearson Prentice Hall. All rights reserved 4-94 94 EXAMPLE Drawing a Bar Graph of a Conditional Distribution Using the results of the previous example, draw a bar graph that represents the conditional distribution of grade earned by method of delivery. © 2010 Pearson Prentice Hall. All rights reserved 4-95 95 The following contingency table shows the survival status and demographics of passengers on the ill-fated Titanic. Draw a conditional bar graph of survival status by demographic characteristic. Survival Status on the Titanic 0.9 0.8 Relative Frequeny 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Men Women Boys Girls Survived 0.19716647 0.753554502 0.453125 0.6 Died 0.80283353 0.246445498 0.546875 0.4 © 2010 Pearson Prentice Hall. All rights reserved 4-96 96 © 2010 Pearson Prentice Hall. All rights reserved 4-97 97 EXAMPLE Illustrating Simpson’s Paradox Insulin dependent (or Type 1) diabetes is a disease that results in the permanent destruction of insulin-producing beta cells of the pancreas. Type 1 diabetes is lethal unless treatment with insulin injections replaces the missing hormone. Individuals with insulin independent (or Type 2) diabetes can produce insulin internally. The data shown in the table below represent the survival status of 902 patients with diabetes by type over a 5-year period. Type 1 Type 2 Total Survived 253 326 579 Died 105 218 323 358 544 902 From the table, the proportion of patients with Type 1 diabetes who died was 105/358 = 0.29; the proportion of patients with Type 2 diabetes who died was 218/544 = 0.40. Based on this, we might conclude that Type 2 diabetes is more lethal than Type 1 diabetes. © 2010 Pearson Prentice Hall. All rights reserved 4-98 98 However, Type 2 diabetes is usually contracted after the age of 40. If we account for the variable age and divide our patients into two groups (those 40 or younger and those over 40), we obtain the data in the table below. Type 1 Survived Died Type 2 Total < 40 > 40 < 40 > 40 129 124 15 311 579 1 104 0 218 323 130 228 15 529 902 Of the diabetics 40 years of age or younger, the proportion of those with Type 1 diabetes who died is 1/130 = 0.008; the proportion of those with Type 2 diabetes who died is 0/15 = 0. Of the diabetics over 40 years of age, the proportion of those with Type 1 diabetes who died is 104/228 = 0.456; the proportion of those with Type 2 diabetes who died is 218/529 = 0.412. The lurking variable age led us to believe that Type 2 diabetes is the more dangerous type of diabetes. © 2010 Pearson Prentice Hall. All rights reserved 4-99 99 Simpson’s Paradox represents a situation in which an association between two variables inverts or goes away when a third variable is introduced to the analysis. © 2010 Pearson Prentice Hall. All rights reserved 100 Section 4.5 Nonlinear Regression © 2010 Pearson Prentice Hall. All rights reserved 101 Exponential Model: y abx Power Model: y axb © 2010 Pearson Prentice Hall. All rights reserved 104 EXAMPLE Using the Definition of a Logarithm Rewrite the logarithmic expressions to an equivalent expression involving an exponent. Rewrite the exponential expressions to an equivalent logarithmic expression. (a) log315 = a (b) 45 = z © 2010 Pearson Prentice Hall. All rights reserved 107 In the following properties, M, N, and a are positive real numbers, with a 1, and r is any real number. loga (MN) = loga M + loga N loga Mr = r loga M EXAMPLE Simplifying Logarithms Write the following logarithms as the sum of logarithms. Express exponents as factors. (a) log2 x4 (b) log5(a4b) If a = 10 in the expression y = logax, the resulting logarithm, y = log10x is called the common logarithm. It is common practice to omit the base, a, when it is equal to 10 and write the common logarithm as y = log x EXAMPLE Evaluating Exponential and Logarithmic Expressions Evaluate the following expressions. Round your answers to three decimal places. (a) log 23 (b) 102.6 © 2010 Pearson Prentice Hall. All rights reserved 112 y = abx Exponential Model log y = log (abx) Take the common logarithm of both sides log y = log a + log bx log y = log a + x log b Y=A+Bx b = 10B where a = 10A EXAMPLE 4 Finding the Curve of Best Fit to an Exponential Model A chemist as a 1000gram sample of a radioactive material. She records the amount of radioactive material remaining in the sample every day for a week and obtains the following data. Day Weight (in grams) 0 1 2 3 4 5 6 7 1000.0 897.1 802.5 719.8 651.1 583.4 521.7 468.3 (a) Draw a scatter diagram of the data treating the day, x, as the predictor variable. (b) Determine Y = log y and draw a scatter diagram treating the day, x, as the predictor variable and Y = log y as the response variable. Comment on the shape of the scatter diagram. (c) Find the least-squares regression line of the transformed data. (d) Determine the exponential equation of best fit and graph it on the scatter diagram obtained in part (a). (e) Use the exponential equation of best fit to predict the amount of radioactive material is left after 8 days. © 2010 Pearson Prentice Hall. All rights reserved 118 y = axb Power Model log y = log (axb) Take the common logarithm of both sides log y = log a + log xb log y = log a + b log x Y=A+bX where a = 10A EXAMPLE Finding the Curve of Best Fit to a Power Model Distance Cathy wishes to measure 1.0 1.1 the relation between a light bulb’s intensity and 1.2 the distance from some 1.3 light source. She 1.4 measures a 40-watt light 1.5 bulb’s intensity 1 meter 1.6 from the bulb and at 0.1- 1.7 meter intervals up to 2 1.8 meters from the bulb and 1.9 obtains the following data. 2.0 Intensity 0.0972 0.0804 0.0674 0.0572 0.0495 0.0433 0.0384 0.0339 0.0294 0.0268 0.0224 (a) Draw a scatter diagram of the data treating the distance, x, as the predictor variable. (b) Determine X = log x and Y = log y and draw a scatter diagram treating the day, X = log x, as the predictor variable and Y = log y as the response variable. Comment on the shape of the scatter diagram. (c) Find the least-squares regression line of the transformed data. (d) Determine the power equation of best fit and graph it on the scatter diagram obtained in part (a). (e) Use the power equation of best fit to predict the intensity of the light if you stand 2.3 meters away from the bulb. Modeling is not only a science but also an art form. Selecting an appropriate model requires experience and skill in the field in which you are modeling. For example, knowledge of economics is imperative when trying to determine a model to predict unemployment. The main reason for this is that there are theories in the field that can help the modeler to select appropriate relations and variables.