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Chapter 18- Complex Statistics

Correlation What is it? A relationship or association between two variables.

Strength

5 Minute CHeck Construct a scatter plot using the following data. Then state if it has correlation, if it is positive or negative, and if it is strong, weak, or negative. (MAKE THE SCALES ON BOTH AXES FROM 60-90) Student A B C D E F G H I Math Test 64 67 69 70 73 74 77 82 84 Science Test 68 75 78 86

NOTE: Correlation is NOT causation Consider this: Arm Length and Running Speed It was found that amongst a group of children, there was a positive correlation found. Does this mean that short arms cause a reduction in running speed? Or that a high running speed causes your arms to grow long? NO! It is just a relationship, not a CAUSE.

Measuring Correlation The strength of a relationship is best measured by the correlation coefficient (r). An r value of 0 suggests there is no correlation A value of 1 suggests a perfect positive correlation A value of -1 suggests a perfect negative correlation

Pearson’s correlation coefficient 𝑟= 𝑥 − 𝑥 (𝑦− 𝑦) (𝑥− 𝑥 ) 2 (𝑦− 𝑦 ) 2 𝑟= 𝑠 𝑥𝑦 𝑠 𝑥 𝑠 𝑦 𝑟= 𝑥𝑦 −𝑛 𝑥 𝑦 𝑥 2 −𝑛 𝑥 2 𝑦 2 −𝑛 𝑦 2

Put data in List 1 and List 2 Stat- Tests- E: LinRegTTest- Enter Technolgoy Put data in List 1 and List 2 Stat- Tests- E: LinRegTTest- Enter X List: List 1, Y List: List 2, Calculate Scroll down to r

Coefficient of Determination Value Strength of Association r2 = 0 No correlation 0 < r2 < .25 Very weak correlation .25 ≤ r2 < .50 Weak correlation .50 ≤ r2 < .75 Moderate correlation .75 ≤ r2 < .90 Strong correlation .90 ≤ r2 < 1 Very strong correlation r2 = 1 Perfect correlation

Practice Problem 1 A chemical fertilizer company wishes to determine the extent of correlation between ‘quantity of compound X used’ and ‘lawn growth’ per day. Find Pearson’s correlation coefficient between the two variables. Lawn Compound X (g) Lawn Growth (mm) A 1 3 B 2 C 4 6 D 5 8

x y xy x2 Y2 1 3 2 4 6 5 8 12 20

Number of surviving lawn beetles Practice Problem 2 Wydox have been trying out a new chemical to control the number of lawn beetles in the soil. Determine the extent of the correlation between the quantity of chemical used and the number of surviving lawn beetles per square meter of lawn. Lawn Amount of chemical (g) Number of surviving lawn beetles A 2 11 B 5 6 C 4 D 3 E 9

Find the correlation if.. Sx = 14.7, sy = 19.2 and sxy = 136.8 The standard deviation of the x distribution is 8.71, the standard deviation of y is 13.23 and the covariance of x and y is -9.26 ∑x = 65, ∑y = 141, ∑xy = 1165, ∑x2 = 505, ∑y2 = 2745 , n = 11