Correlation Section 10 2 Objectives 1 Determine whethere

Correlation Section 10. 2

Objectives 1) Determine whethere is a linear correlation between two variables. 2) Find the correlation coefficient r. 3) Perform statistic test for correlation coefficient r.

Learning Outcomes After the lesson, the students will be able to 1) Using the data graph to determine whethere is a linear correlation between two variables. 2) Using calculator to find the correlation coefficient r. 3) Conducting hypotheses test for correlation coefficient r.

The Basics • A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable. • What correlations can you think of in your life?

Variety is the Spice of Life • There are many different types of correlations. POSITIVE CORRELATION NEGATIVE CORRELATION

Variety is the Spice of Life • There are many different types of correlations. NO CORRELATION NON-LINEAR CORRELATION

How Strong Is That? • The linear correlation coefficient r measures the strength of the linear correlation between the paired quantitative x and y values in a sample. • Note the “paired” data means that the each pair of values has some relationship and comes from two different objects. It does not mean the same object is measured twice at two different times.

Properties of r 1. The value of r is always between -1 and 1 inclusive. • -1 ≤ r ≤ 1 2. r measures the strength of a linear relationship 3. r is very sensitive to outliers. CAUTION This section ONLY applies to linear correlations. If you conclude there does not appear to be a correlation, know that it is possible that there might be some other association that is not linear.

Finding r • Formula 10 -1 Known as the “shortcut” formula.

Finding r Input the following data n 2 4 5 7 9 11 12 s 6 10 16 18 21 25 28 n - Number of times the dart hit on the target s - The score you will get in the game

Finding r • Calculator Steps – The step-up • 2 nd, “ 0” key • Scroll down to Diagnostic. On • Hit enter twice

Finding r • Plot Steps • “ 2 nd ” and then “y=” • Turn Plot to “on” • Choose “Type” to be the first icon • In the “Xlist”, point to the first data set • In the “Ylist”, point to the second data set • Then “Zoom” and “ 9”

Finding r • Calculator Steps – The actual process • Stat • Edit, Option 1: Edit … • Enter values in L 1 and in L 2 • Then … • Stat • Calc, Lin. Reg(ax+b) • See r = for correlation coefficient

Finding r Input the following data n 2 4 5 7 9 11 12 s 6 10 16 18 21 25 28 n - Number of times the dart hit on the target s - The score you will get in the game r = 0. 9847

Before we start … • Requirement Check – The data are a simple random sample of quantitative data. – The plotted points approximate a straight line. – There are no outliers. (correlation coefficient r is very sensitive to outliers in the sense that a single outlier can dramatically affect its value. )

Practice Round • Eric Bram, a NY teenager, noticed that the cost of a slice of cheese pizza was typically the same as the cost of a subway ride. Over the years, he noticed that as one increased, so did the other. When the cost of a slice of pizza increased, he told the New York Times that the cost of subway fares would rise as well.

Practice Round • Here is some of his data: Cost of $0. 15 Pizza $0. 35 $1. 00 $1. 25 $1. 75 $2. 00 Subway $0. 15 Fare $0. 35 $1. 00 $1. 35 $1. 50 $2. 00 • Check the requirements, if met then … • Find the correlation coefficient. r = 0. 9878

Interpreting r • So r = 0. 9878 … great, now what? • We can use Table A-6 to make a meaningful conclusion… USING TABLE A-6 TO INTERPRET r If r exceeds the value in Table A-6, conclude that there is a linear correlation. Otherwise, there is not sufficient evidence to support the conclusion of a linear correlation.

Formal Hypothesis Test • Using the methods of Section 10. 2, we can conduct claim tests to determine whether or not correlations actually exist.

The Process - Notation • n = number of pairs of sample data • r = linear correlation coefficient for a sample of paired data. • ρ = linear correlation coefficient for a population of paired data.

The Process – Hypotheses • Null Hypotheses – Ho: ρ=0 (There is no linear correlation). • Alternative Hypotheses – H 1: ρ≠ 0 (There is a linear correlation).

The Process - Conclusion • If r exceeds the value in Table A-6, conclude that there is sufficient evidence to support a linear correlation. • If r does not exceeds the value in Table A-6, conclude that there is not sufficient evidence to support a linear correlation.

Practice Round • Here is some of his data: Cost of $0. 15 Pizza $0. 35 $1. 00 $1. 25 $1. 75 $2. 00 Subway $0. 15 Fare $0. 35 $1. 00 $1. 35 $1. 50 $2. 00 • Check the requirements, if met then … • Find the correlation coefficient. r = 0. 9878

Common Errors 1. A common error is to conclude that correlation implies causality. There is a correlation between the costs of pizza and subway fares, but we cannot conclude that increases in pizza cost (massive cheese price spike) causes subways to increase their rates. Both costs might be affected by some other variable lurking (creepily) in the background.

Common Errors 2. Another error arises with data based on averages. Averages suppress individual variations and may inflate the correlation coefficient. One study produced a 0. 4 linear correlation coefficient for paired data relating income and education among individuals, but the linear correlation coefficient became 0. 7 when regional averages were used.

Common Errors 3. A third error involves the property of linearity. Remember, this section only deals with linear correlations. Just because we find that there is no correlation between two data sets supplied in this section, that does not mean there is no correlation PERIOD. There might be a non-linear relationship.

Class Activity • Collect data from each student consisting of the number of credit cards and the number of keys that the student has in his or her possession. Is there a correlation? • Try to identify at least one reasonable explanation for the presence or absence of a correlation.

Homework • Pg. 532 -533 #16, 18, 20, 23

Homework • 10. 2 Worksheet