Linear Correlation and Regression Analysis
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1 Linear Correlation and Regression Analysis
2 Set Up the Calculator 2 nd CATALOG D arrow down DiagnosticOn ENTER ENTER
3 SCATTER DIAGRAM
4 Positive Linear Correlation
5 Positive Correlation Variables will tend to pair as high measurements for x with high measurements for y and low x with low y
6 Negative correlation
7 Negative Correlation Variables will tend to pair high x measurements with low measurements for y and low x with high y.
8 No correlation
9 r THE COEFFICIENT OF LINEAR CORRELATION Pearson s Correlation Coefficient, r, measures the strength of a linear relationship between two variables for a sample. n x n 2 xy x y 2 2 x n y y 2
10 Interpreting the Values of r r = 1 Perfect Positive Correlation (Positive Slope)
11 Interpreting the Values of r r = -1 Perfect Negative correlation (Negative Slope)
12 Interpreting the Values of r r = 0 no correlation
13 Possible Values of r 1 r 1 strong correlation weak correlation r = -1 r = 0 r = +1 l r = r = r = 0.07 r = 0.69
14 Calculate r
15 1-Var Stat TEMP1 and TEMP2
16 What is xy
17
18 TESTING THE SIGNIFICANCE OF THE CORRELATION COEFFICIENT Is there a significant correlation between: HS average and success in college? Hours of T.V. watched and IQ score? Age and blood pressure? Advertising expenditures and sales? Mothers shoe size and daughters IQ?
19 Caution #2: Correlation Doesn t Indicate a Cause-and-Effect Relationship The number of storks nesting in various European towns in the early 1900 s and the number of human babies born in the same towns had a very high correlation. However, we can t conclude that an increase in the number of storks will cause an increase in the number of babies.
20 TESTING THE SIGNIFICANCE OF THE CORRELATION COEFFICIENT The population correlation coefficient is symbolized by Greek letter rho: ρ The sample correlation coefficient r is an estimate of the population correlation coefficient.
21 Null Hypothesis Form Ho: The population correlation coefficient is equal to zero. Ho: ρ = 0 There is no linear correlation between the two variables. r 0
22 3 Forms of the Alternative Hypothesis Ha:
23 Interpretation of Ha: Form #1 Ha: ρ > 0 There is a positive correlation between the two variables.
24 General Solution (positive) 1TT > r 0 r r formula
25 Interpretation of Ha: Form #2 Ha: ρ < 0 There is a negative correlation between the two variables.
26 General Solution (negative) 1TT > r formula r 0 r
27 Interpretation of Ha: Form #3 Ha: ρ 0 There is a (some) correlation between the two variables.
28 General Solution (some) 2TT > rformula r r 0 r
29 Degrees of freedom for testing the correlation coefficient Subtract two from the number of pairs of data df = n 2
30 Coefficient of Determination r 2 is the influence that the variance in the independent variable has on the dependent variable. 1 r 2 is unexplained.
31 Determine the variables grades vs. study hours independent is dependent is
32 Determine the variables credits vs. age sun light vs. height of a plant bike vs. presidents gift vs. work
33 LINEAR REGRESSION ANALYSIS Regression Line Formula: y' = a + bx y' is the predicted value of y, the dependent variable, given the value of x, the independent variable.
34 Example A scientists wants to determine if there is a linear relationship between the amount of rainfall in May and the number of mosquitoes. For each of the selected years, data pairs have listed in the table.
35 YEAR Sample data MOSQUITO INDEX RAIN
36 Enter data
37 Procedure a) Construct a scatter diagram. b) Calculate the sample correlation coefficient, r. c) Determine if r is significant at α = 1%.
38 Procedure 2 d) Find r and interpret its meaning. e) Determine the regression equation, y'. f) Using the regression equation, predict the mosquito population index if we have 3.1 inches of rain in May.
39 Scatter Plot
40 Perform Linear Reg T Test CORRECT ORDER?
41 Select Ha; Set Y1 (once)
42 Calculate, arrow down Record : p-value, a, b, r 2 and r
43 Interpretations r = : close to r = 1 therefore a Strong Positive correlation. p-value is less then α = 1%, so reject Ho and accept Ha. There is a correlation between rainfall and the mosquito population index.
44 r Coefficient of determination: 87.22% of the dependent variable variation is attributed to variation in the independent variable; 12.78% is unexplained.
45 Regression Equation y' = a + b (x) y' = (x) Prediction for the mosquito index: Substitute rain = 3.1 inches y' =
46 Confidence in Prediction of population index Since r = , is very close to r = 1 (a perfect correlation), there is a high level of confidence in the accuracy of the prediction.
47 Interpolation vs. Extrapolation Use a rain prediction = 6739 Substitute Y1(6739) Result = How much confidence?
48 Using the Regression Line BLOOD PRESSURE Interpolate if maximum weight collected 256 lbs BODY WEIGHT
49 Using the Regression Line BLOOD PRESSURE Interpolate Extrapolate 256 lbs 9842 BODY WEIGHT
50 Redo Scatter Plot for Regression line
51
52
53 Calculate correlation coefficient, r for Rain & Mosquito Data r n x n 2 xy x y 2 2 x n y y 2 calculated r is the EV
54 Determine critical page 810 table V value from Pairs of mosquito & rain values, n= 6 r Subtract two from the number of pairs of data values to generate Degrees of freedom: df = n 2 = 4 We were given α = 1% 2 Tail
55 critical r 0.92 calculated r
56
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