Scatter Diagram X-axis = independent variable. Y-axis = dependent variable. Each pair of data A point (x, y) X Y 2 3 (2, 3)
X Purpose of Drawing Scatter Diagram Is there a linear relationship between the two variables X and Y? Linear relationship = Scatter points (roughly at least) form the shape of a straight line. Y X Y Linear relationshipNo linear relationship
Measuring Strength of Linear Relationship Pearsons coefficient of correlation r Formula (2) (Not used in exam. Just for knowledge) Calculator Work For Casio 350MS Switch the calculator on. 1.Set calculator in LR (Linear Regression) mode: Press Mode. Press 3 for Reg (Regression). Press 1 for Linear. Check n. (Checking whether there are old data): Press Shift 1, next 3, and then =.
Calculator Work for r 3.Enter Data in Pairs: x-value, y-value M+ 4.Check n again: see step 2 above. 5.Press shift 2, then move by arrow to the right, press 3 for r, and then press =. Now you see the value of r.
Interpretation of r (Direct linear relationship) 1.If r is 1 or – 1, then all scatter points are on a straight line. 2.If r is 1, all points are on a straight line with a positive slope. 3.If r is -1, all points are on a straight line with a negative slope. 4.If a straight line has a positive slope, it rises up to the right. 5.If a straight line has a positive slope, if x increases, then y increases for the points (x, y) on it. (small x, small y) (large x, large y) 6.In this situation, we say that the two variables X and Y are directly or positively correlated.
Interpretation of r (Inverse linear relationship) 1.If r is -1, all points are on a straight line with a negative slope. 2.If a straight line has a negative slope, if x increases, then y decreases for the points (x, y) on it. (small x, large y) (large x, small y) 6.In this situation, we say that the two variables X and Y are inversely or negatively correlated.
Interpretation of r (strength) 1.If r is not exactly 1 or – 1, but it is.9 or -.9, then the points are around a straight line. They are close to a straight-line shape. 2.If r is.8 or -.8, then the points are close to a straight-line shape, but not so well as in case of.9 or Thus, the closer r is to 1 or – 1, the closer are the points to a straight-line shape. 4.Thus, the closer r is to 0, the farther are the points from a straight-line shape. 5.In r-values, 0.9 are stronger than 0.8, and 0.8 are weaker than 0.9.
Interpretation of r (strength) Values of r 0 No linear relationship 0.5 Weak linear relationship Weak linear relationship 1 Strong Perfect -1 Strong Perfect
Testing Linear Relationship 1.Pearson invented a formula to measure the strength and direction of a linear relationship between two variables. 2.The number given by his formula is called correlation coefficient. We call it Pearsons coefficient of correlation. 3.We write r for this value in a sample, and we write for this value in a population. 4.Testing whether the correlation is significant is scientific guessing whether there should be a correlation, in the population, between the two variables under consideration.
Null and Alternate Hypothesis 1.Test correlation: H 0 : = 0 and H a : 0 2.Test direct correlation: H 0 : 0 and H a : > 0 3.Test inverse correlation: H 0 : 0 and H a : < 0 4.Test positive correlation: H 0 : 0 and H a : > 0 5.Test inverse correlation: H 0 : 0 and H a : < 0
Three types of test 1.H 0 : = 0 and H a : 0 Two-tailed test 2.H 0 : 0 and H a : < 0 Left-tailed test 3.H 0 : 0 and H a : > 0 Right-tailed test
Critical value 1.Read t table. 2.Degrees of freedom (Df) = n n = number of pairs of data 4.Right-tailed test Positive sign 5.Left-tailed test Negative sign 6.Two-tailed test Both positive and negative sign
Test Statistic 1.Test statistic = Strength of evidence supporting alternate hypothesis H a 2.Original test statistic to test is r. 3.Convert r to t by Formula (10). 4.Learn to compute t by your calculator correctly.
Rejection region 1 For a two tailed-test, the rejection region is on the right of positive critical value and on the left of negative critical value. Real number line for t values 0Positive Critical Value Negative Critical Value Total area = Level of significance = Probability = α Rejection region T curve
Rejection region 2 For a left-tailed test, the rejection region is on the left of (negative) critical value. Real number line for t values 0 (Negative) Critical Value α = Area = Level of significance = Probability Rejection region t curve
Rejection region 3 For a right-tailed test, the rejection region is on the right of the (positive) critical value. Real number line for t values 0 (Positive) Critical Value Area = Level of significance = Probability = α Rejection region t curve
Decision Rule If the test statistic (TS) is in the rejection region, then reject H 0. Reject H 0 = H 0 is false, and hence H a is true. Fail to reject H 0 = H 0 is true, and hence H a is false.
Conclusion Conclusion = Decision Decision is the last step of statistical procedure. Conclusion is the report to the one who asked the original question.