PRED 354 TEACH PROBILITY STATIS FOR PRIMARY MATH

PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 14 Correlation & Regression

Correlation It is a statistical method that is used to measure and describe a relationship between two variables. Two variables are observed. No attempt to control or manipulate the variables.

Correlation Its characteristics: a. The direction of relationship 1. Positive correlation (Two variables tend to move in the same direction) (+) 2. Negative correlation (Two variables tend to move in the opposite direction) (-)

. . . Temperature Amount of coffee sold Amount of beer sold Correlation. . . Temperature

Correlation Its characteristics: b. The form of the relationship The relationship tend to have a linear form. (parabolic, in other words; not a straight-line relationship) The degree of the relationship A correlation measures how well the data fit the specific form being considered. A perfect correlation : +1 No fit: 0 c.

Correlation Where and why correlations are used: 1. Prediction: If two variables are known to be related in some systematic way, it possible to use one of the variables to make accurate predictions about the other. EX 2. Validity: Suppose that you develop a new test for a specific purpose. How could you show that this test truly is measuring what it claims? EX

Correlation Where and why correlations are used: 3. Reliability: A measurement procedure is considered reliable to the extent that it produces stable consistent measurement. EX 4. Theory Verification: Many theories make some specific prediction about the relationship between two variables EX

Correlation The Pearson Correlation (Interval o ratio scale)

Pearson Correlation NEW: The sum of products of deviations (SP): for measuring the amount of covariability between two variables. SP: a parallel procedure for measuring the amount covariability between two variables SS: to measure the amount variability for a single variable

Pearson Correlation EX: X 0 10 4 Y 1 3 1 8 8 2 3

Pearson Correlation: Interpretations 1. 2. 3. 4. It describes a relationship between two variables. It does not explain why the two variables are related. (Not proof of a cause-and-effect relationship between two variables) (EX: ) The value of correlation can be affected by the range of scores. (EX) One of two extreme points (outriders) can have dramatic effect on the value of a correlation (EX) How good? There is no proportion…(Coefficient of determination)

Coefficient of determination The value of r 2 is called by coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with other variables. EX: r = . 80 and r 2 =. 64. It means that 64 % of the variability in the Y scores can be predicted from the relationship with X.

Hypothesis testing H 0: ρ = 0 (There is no correlation). df = n-2 Table B. 6 (independent of sign) To be significant, the magnitude of the sample correlation must equal or exceed the value in the table. EX: A researcher obtains a correlation of r= -. 41 for a sample of n=25 individuals. Does this sample provide sufficient evidence to conclude there is significant, non-zero correlation in the population?

Regression Equation Amount of beer sold The regression equation for Y is the linear equation . . . Regresson line Temparature Where the constants b and a are determined by

Regression line 1. 2. 3. It makes relationship easier to see. It identifies the center or central tendency of the relationship. It can be used for prediction

Example EX: The following table presents students’ anxiety rating (15 minutes before the exam) and exam scores. a. Compute the correlation between Student Anxiety rating AR and ES. A 5 b. Is the sample correlation B 2 significant at the. 05 level? c. Find the regression line equation C 7 d. Predict a student’s reaction time D 7 whose AR score is 10. E 4 F 5 Exam Score 80 88 80 79 86 85