3. Shallow men believe in luck. Strong men believe in cause and effect… Ralph Waldo Emerson American Poet The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning… Stephen Jay Gould American Paleonthologyst and Biologist

4. A study of the correlation and regression of two variables x and y, begins with a graph of the pairs ( x, y)

5. A study of the correlation and regression of two variables x and y, begins with a graph of the pairs (x, y)

6. § Scatter Diagrams A graph in which pairs of points, ( x, y), are plotted with x on the horizontal axis and y on the vertical axis. The explanatory variable is x. The response variable is y. One goal of plotting paired data is to determine if there is a linear relationship between x and y Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation

7. § Scatter Diagrams Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation

8. § Why a Scatter Diagram? Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation One goal of data analysis is to find a mathematical equation that best represents the data. For our purposes, best means the line that comes closest to each point on the scatter diagram. Find the best fitting line

9. § Fitting a Line to a Scatter Diagram Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation

10. § Fitting a Line to a Scatter Diagram Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation But not all relationships are linear

11. § Correlation Strength Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Correlation is a measure of the linear relationship between two quantitative variables. If the points lie exactly on a straight line, we say there is perfect correlation. As the strength of the relationship decreases, the correlation moves from perfect to strong to moderate to weak.

12. § Correlation Strength Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Moderate Linear Correlation Perfect Linear Correlation

13. § Correlation Measurement Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation The mathematical measurement that describes the correlation is called the sample correlation coefficient r

14. § Correlation Coefficient Features Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation −r−r has no units. −− 1 ≤ r ≤ 1 −P−Positive values of r indicate a positive relationship between x and y (likewise for negative values). −r−r = 0 indicates no linear relationship. −S−Switching the explanatory variable and response variable does not change r. −C−Changing the units of the variables does not change r.

15. § Correlation Coefficient Features Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation r = 0 There is no linear relationship between the variables

16. § Correlation Coefficient Features Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation r = 1 / r =  1 There is a perfect linear relationship between the variables. Points lie on the line

17. § Correlation Coefficient Features Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation 0 < r < 1 Positive correlation. Large x values corresponds to large y values, and small x values corresponds to small y values

18. § Correlation Coefficient Features Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation  1 < r < 0 Negative correlation. Large x values corresponds to small y values, and small x values corresponds to large y values

19. § Developing a Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Negative Correlation Positive Correlation Little or No Correlation

20. § Developing a Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation −F−For both x and for y, define “high” to be an observation above its mean. −T−Then, the relationship on the previous page can be represented as follows:

21. § Developing a Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Positive Correlation

22. § Developing a Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Negative Correlation

23. § Developing a Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation −T−To calculate r, we can sum over all points and divide each summation by its standard deviation (to make it unit-free). −F−Finally, divide by n 11 to take the average

24. § Computational Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation −N−No Means / Standard Deviations involved

25. § Computational Formula for r Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Obtain a random sample of n data pairs ( x, y). The data pairs should have a bivariate normal distribution, i.e. that for a fixed value of x, the y values should have a normal distribution (or at least mound shaped and symmetrical), and for a fixed y, the x values should have their own (approximately) normal distribution −T−To use the Computational Formula…

26. Braser–Braser Chapter 10.1 pp 596 AP Statistics Scatter Diagrams & Linear Correlation

27. Braser–Braser Chapter 10.1 pp 597 AP Statistics Scatter Diagrams & Linear Correlation

28. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 c m/sec ) and the drift rate of sand dunes y ( 100 g /cm/sec )?

29. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 cm/sec ) and the drift rate of sand dunes y ( 100 g/cm/sec )?

30. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 cm/sec ) and the drift rate of sand dunes y ( 100 g/cm/sec )? It is required to calculate:  x,  y  x 2,  y 2,  xy

31. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 cm/sec ) and the drift rate of sand dunes y ( 100 g/cm/sec )?

32. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 cm/sec ) and the drift rate of sand dunes y ( 100 g/cm/sec )?

33. § Computational Formula for r. Example Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation In the desert, is there a linear relationship between wind velocity x ( 10 cm/sec ) and the drift rate of sand dunes y ( 100 g/cm/sec )? Since r = 0.949, there is a strong, positive, linear relationship between wind velocity and sand dune drift rates.

34. § Population Correlation vs. Sample Correlation Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Sample Correlation Coefficient r : computed from a random sample of ( x, y) data pairs Population Correlation Coefficient :: computed from all data pairs ( x, y)

35. Braser–Braser Chapter 10.1 pp 597 AP Statistics Scatter Diagrams & Linear Correlation

36. Braser–Braser Chapter 10.1 pp 598 AP Statistics Scatter Diagrams & Linear Correlation

37. § Correlation and Causation Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Correlation does not mean Causation!! High correlations do not imply a cause and effect relationship between the explanatory variable x and response variable y !! Beware:

38. § Correlation and Causation Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Correlation does not mean Causation!! The fact that two variables tend to increase or decrease together does not mean a change in one is causing a change in the other A strong correlation is sometimes due to other(s) (known or unknown) variable(s). Such variable(s) is/are called Lurking Variable(s)

39. § Correlation and Causation Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Correlation does not mean Causation!! A lurking variable is neither the explanatory variable nor the response variable. Lurking variables may be responsible for the evident changes in x and y. Ex. Over a period of time, the population of a certain town increased. It was observed that during this period the correlation between x, the number of people attending church, and y, the number of people in the city jail was r = 0.90. Does going to church cause people to go to jail?

40. § Correlations Between Averages Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation If the two variables of interest are averages rather than raw data values, the correlation between the averages will tend to be higher! The use of averages reduces the variation existing between individual measurements A high correlation based on two variables consisting of averages does not necessarily imply a high correlation between to variables consisting of individual measurements

41. Braser–Braser Chapter 10.1 pp 596 AP Statistics Scatter Diagrams & Linear Correlation

42. Braser–Braser Chapter 10.1 pp 596 AP Statistics Scatter Diagrams & Linear Correlation

43. Braser–Braser Chapter 10.1 pp 596 AP Statistics Scatter Diagrams & Linear Correlation

44. Braser–Braser Chapter 10.1 AP Statistics Scatter Diagrams & Linear Correlation Practice Textbook Section 10.1 Problems: pp. 303 – 308 Checking for Understanding HM STAT Space (10.1)

45. Love is the energizing elixir of the universe, the cause and effect of all harmonies … It is common error to infer that things which are consecutive in order of time have necessarily the relation of cause and effect… Jacob Bigelow American Botanist

46. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Linear Regression 1971-2010 2001-2010 r = 0.9637 r = 0.7459

47. When we are in the paired-data setting: −D−Do the data points have a linear relationship? −C−Can we find an equation for the best fitting line? −C−Can we predict the value of the response variable for a new value of the predictor variable? −W−What fractional part of the variability in y is associated with the variability in x ? Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Linear Regression

48. The line that we fit to the data will be such that the sum of the squared distances from the line will be minimized. Let d = the vertical distance from any data point to the regression line. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Least-Squares Criterion

49. a : y -intercept. b : slope Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Least-Squares Criterion Least Square Line:

50. b : slope Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation Least Square Line: a : y -intercept. : Sample Means x, y : Data pairs with bivariate normal distribution

51. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example Let x = the size of the caribou population ( 100 s) and let y = the size of the wolf population in Denali National Park Use x as the explanatory variable and determine the least-squares regression equation

52. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example

53. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example

54. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example

55. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example The least-squares regression equation for the caribou population ( x ) and the wolf population ( y ) in Denali National Park is:

56. Braser–Braser Chapter 10.2 pp 614 AP Statistics Linear Regression & Coefficient r

57. Braser–Braser Chapter 10.2 pp 614 AP Statistics Linear Regression & Coefficient r

58. Braser–Braser Chapter 10.2 pp 615 AP Statistics Linear Regression & Coefficient r

59. Braser–Braser Chapter 10.2 pp 615 AP Statistics Linear Regression & Coefficient r

60. Braser–Braser Chapter 10.2 pp 615 AP Statistics Linear Regression & Coefficient r

61. Braser–Braser Chapter 10.2 pp 615 AP Statistics Linear Regression & Coefficient r

62. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Using the Regression Equation The point Also, the slope tells us that if we increase the explanatory variable by one unit, the response variable will change by the slope (increase or decrease, depending on the sign of the slope). is always on the least-squares line.

63. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Making Predictions We can simply plug in x values into the regression equation to calculate y values. Interpolation : Predicting ŷ values for x values that are between observed x values in the data set Extrapolation : Predicting ŷ values for x values that are beyond observed x values in the data set Extrapolation can result in unreliable predictions

64. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Making Predictions

65. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example Predict the wolf population when the caribou population is 21 (hundred).

66. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Finding the Regression Equation. Example Explain why predicting the wolf population when the caribou population is 12 (hundred) might be unreliable

67. Braser–Braser Chapter 10.2 pp 616 AP Statistics Linear Regression & Coefficient r

68. Braser–Braser Chapter 10.2 pp 616 AP Statistics Linear Regression & Coefficient r

69. Braser–Braser Chapter 10.2 pp 616 AP Statistics Linear Regression & Coefficient r

70. Braser–Braser Chapter 10.2 pp 616 AP Statistics Linear Regression & Coefficient r

71. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination Another way to gauge the fit of the regression equation is to calculate the coefficient of determination, r 2 We can view the mean of y as a baseline for all the y values

72. Chapter 10.2 Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination Total Deviation: Explained Deviation: Unexplained Deviation:

73. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination Total Deviation = Explained Deviation – Unexplained Deviation

74. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination Total Deviation = Explained Deviation – Unexplained Deviation

75. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination The ratio r 2 is the measure of the total variability in y that is explained by the corresponding variation on x, when the least squares lines is used. Is called the coefficient of variation, and it is a measure of how good is the least square line as an instrument of regression

76. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Coefficient of Determination The ratio −r−r 2 is the measure of the total variability in y that is explained by the corresponding variation on x −1−1 – r 2 is then the fractional amount of total variation in y that is due to random chance or possible lurking variables that influence y is called the Coefficient of Determination, and it is a measure of how good is the least square line as an instrument of regression

77. Braser–Braser Chapter 10.2 pp 618 AP Statistics Linear Regression & Coefficient r

78. Braser–Braser Chapter 10.2 pp 618 AP Statistics Linear Regression & Coefficient r

79. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots The Least Squares line usually does not go through all the sample data points ( x, y). In fact, for a specified x value from a data pair, there is usually a difference between the predicted value ŷ and the y value paired with x. This difference is called residual. residual

80. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots Definition. The residual i s the difference between the y value in a specified data pair ( x, y) and the value ŷ = a + bx predicted by the least squares line for the same x : Residual: y  ŷ Standard Residual

81. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots. E xamples If the least squares line provides a reasonable model for the data, the pattern points in the plot will seem random and un- structured around the horizontal line x = 0

82. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots. Examples If a point on the residual plot seems far outside the pattern of the other points, it may reflect an unusual data point or outlier. Such points have quite an influence on the least squares model. Influential Observation

83. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots. Examples If a point on the residual plot seems far above or below the line x = 0, such point is called a large residual. This kind of points are not necessarily influential points

84. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots. Examples If the residual plot seems to follow a pattern, or curve, then the linear regression is not appropriate. Sometimes a non- linear pattern indicates a biased sample

85. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r § Residual Plots. Examples If in the residual plot points spread uniformly, this indicates a variable standard deviation for the data

86. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r A new medication’s effectiveness is tested in the lab. Different doses are used and a count of bacteria population taken. The results are summarized in the adjoin table. 150015 190014 210013 320012 360011 380010 56009 60008 104007 106006 142005 166004 197003 211002 355001 Bacteria (x100)Dosage (mg) (a) Construct a scatter plot the given data, and find the linear regression line (b) Obtain the corresponding residuals plot. Is the linear regression model appropriate to model this problem? Explain your answer

87. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r The authors of the article Age, Spacing and Growth Rate of Tamarix as an indication of Lake Boundary Fluctuations at Sebkhet Kelbia Tunisia (Journal of Arid Environments [1982]) use a simple linear regression model to describe the relationship between y = vigor (average width in centimeters of the last two annual rings), and x = stem density (stems/cm 2 ). The estimated model was based on the following data: 0.400.450.350.000.550.650.600.551.200.75x (stems/cm 2 ) 22211915 149654y (cm) (a). Construct a scatter plot the given data, and find the linear regression line (b). Obtain the corresponding residuals plot. Is the linear regression model appropriate to model this problem? Explain your answer

88. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r The article Effects of Gamma Radiation on Juvenile and Mature Cuttings of Quaking Aspen (Forest Science [1967]) reported the following data on x = exposure to radiation (kR/6 hrs) and y = dry weight of roots (mg): x (kR/6 hrs)02468 y (mg)1101231198662 (a). Construct a scatter plot the given data, and find the linear regression line (b). Obtain the corresponding residuals plot. Is the linear regression model appropriate to model this problem? Explain your answer

89. Braser–Braser Chapter 10.2 AP Statistics Linear Regression & Coefficient r Practice Textbook Section 10.2 Problems: pp. 595 – 602 Checking for Understanding HM STAT Space (10.2)

90. Life is a perpetual instruction in cause and effect… R alph Waldo Emerson American Poet God created the law of free will, and God created the law of cause and effect. And he himself will not violate the law. We need to be thinking less in terms of what God did and more in terms of whether or not we are following those laws… Marianne Williamson American Writer

91. From a population ( x, y) data pairs, it is possible to compute the population correlation coefficient, ρ, and the population regression line slope, β Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Inferences on the Regression Equation Sample Statistics Population Parameter

92. To make inferences about a population from a sample (x, y) of data pairs: Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Inferences on the Regression Equation −T−The set of ordered pairs ( x, y) constitutes a random sample from all ordered pairs in the population. −F−For each fixed value of x, y has a normal distribution. −A−All distributions for all y values have equal variance and a mean that lies on the regression equation.

93. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Statistical Test for  Obtain a random sample of n > 2 data pairs ( x, y). Find the value of the slope b and y –intercept a for the least squares line: Find the value of the standard error of estimate:

94. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Statistical Test for  1.State the hypothesis: Null Hypothesis H o :  = 0 Alternate Hypothesis H 1 :  or  or  2.Set the level of significance  (Usually 0.01 or 0.05 ) 3.Compute the sample test statistic t :

95. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Statistical Test for  4. Use the Student’s t distribution and type of test: one- tailed (  or  ), two-tailed (  ), to find the P-value corresponding for the sample test statistic 5.Conclude the test: If P -value  t hen reject   if P -value  t hen do not reject H 0 6.State your conclusion in the context of the application

96. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § S§ Statistical Test for  EE xample Plate tectonics and the spread of the ocean floor are very important in modern studies of earthquakes and earth science in general. A random sample of islands in the Indian Ocean gave the following information: x : Age of volcanic island (x 1 0 6 years) y : Distance from center of the midoceanic ridge (x 1 00 Km) - Test the claim that  at a 5 % level of significance

97. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Confidence Interval for  1.Set a confidence level c, ( 0 < c < 1 ) 2.Find the critical value t c for the set confidence level using the Student’s T distribution with d.f. = n – 2 3.Compute the maximal margin of error E : : 4.Find the interval’s lower an upper limits:

98. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § C§ Confidence Interval EE xample Plate tectonics and the spread of the ocean floor are very important in modern studies of earthquakes and earth science in general. A random sample of islands in the Indian Ocean gave the following information: x : Age of volcanic island (x 1 0 6 years) y : Distance from center of the midoceanic ridge (x 1 00 Km) - Find a 7 5 % confidence interval for 

99. Braser–Braser Chapter 10.3 pp 636 AP Statistics Inferences for Correlation & Regression

100. Braser–Braser Chapter 10.3 pp 637 AP Statistics Inferences for Correlation & Regression

101. Braser–Braser Chapter 10.3 pp 637 AP Statistics Inferences for Correlation & Regression

102. Is there any statistical significance of the sample correlation coefficient r ? To prove this it will be necessary a statistical test of ,, the population correlation coefficient Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Testing the Correlation Coefficient

103. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Testing the Correlation Coefficient. Procedure Let r be the sample correlation coefficient computed using data pairs ( x, y ). 1.State the level of significance  f or the test, and take as null hypothesis that x and y have no linear correlation H o :  = 0 2. State in the context of the problem the alternate hypothesis H 1 :  > 0, or  < 0, or   0

104. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Testing the Correlation Coefficient. Procedure 3.Obtain a random sample of pairs ( x, y) of size n  3. 4. Compute the sample test statistics with the formula d.f. = n – 2 5. Use the Student’s T distribution to estimate the P -value corresponding to the test statistic

105. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Testing the Correlation Coefficient. Procedure 6.Conclude the test: If P -value  , then reject H 0 If P -value  t hen do not reject H 0 7. State the conclusion in the context of the problem

106. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § E§ Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. x9.911.48.114.78.512.6 y37.143.033.447.526.540.2 x = % of population 2 5 y.o. or older with at least 4 years of college y = % growth in per capita income over the last 7 years

107. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. For these data: r = 0.887 and n = 6, then state the test hypothesis: H 0 :  = 0 (No Linear Correlation) H 1 :   0 (Positive Linear Correlation)

108. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. Computing the test statistics with d.f. = 4:

109. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. Computing the probability for the computed t - value : Correlation Coefficient r = 0.887 Sample Test Statistics t = 3.84

110. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. Computing the probability for the computed t- value : 0.005 < P Value < 0.01 P -Value is between:

111. Braser–Braser Chapter 10.3 AP Statistics Inferences for Correlation & Regression § Example: Testing the Correlation Coefficient Suppose we have conducted a study investigating the relationship between years in school and income. Test for significance of the correlation coefficient in the positive direction at  = 0.01. Since 0.005 < P Value < 0.01 is less than our level of significance  = 0.01, we can conclude that the correlation coefficient between x and y is positive.

112. Custom Shows