If r equals 1 or 1 then all points fall directly on a

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If r equals −1 or 1 then all points fall directly on a straight line in a scatter plot. Values of r close to −1 or 1 indicate a strong linear relationship between X and Y . The closer r is to 0, the weaker the linear relationship between X and Y . If X and Y are interchanged, the value of r does not change. r and the slope of the line in a scatter plot will have the same sign (positive or negative). EXAMPLE: (Adapted from Question 13, page 675 in Business Statistics , 2 nd Canadian Ed. (2014). Sharpe, N.R., DeVeaux, R.D., Velleman, P.F., Wright, D. Pearson Toronto) Data on the number of sales associates working, and the number of sales (in \$1000s), were recorded for 10 randomly selected small book stores. The objective of the study was to determine if there was a linear relationship between the number of sales associates on the floor and the amount of business done in sales. A partial table of the data is presented below. Number of Salespeople Sales (in \$1000s) 2 10 3 11 7 13 ... ... 20 26 Since we have two quantitative variables, we can use a scatter plot to get a quick visual representation: Here, the summary statistics are provided (since they are tedious to calculate by hand!): SS XX = 286 . 4, SS YY = 256 . 4, SS XY = 261 . 6 We can now calculate the correlation as: 24
3.8 Linear Transformations Linear transformations occur when our data is transformed from one set of units/measurements to another. Common examples of linear transformations include: Currency exchanges (i.e. Canadian to US dollars) Temperature conversions (i.e. degrees Celsius to degrees Fahrenheit) Weight changes (i.e. pounds to kilograms) Calculating a Z-score! We can transform our data by adding a constant to each observation, and/or multiplying each observation by a constant. Adding a Constant Consider the following data set: 2, 4, 6, 8, 10. The mean, median, variance, and standard deviation of this data set is: Consider what happens when we add a constant (say, 3) to each observation, such that our new data set is: 5, 7, 9, 11, 13. The mean, median, variance, and standard deviation of the new data set is: In general, adding a constant to every observation in the data set will change the Measures of Centre, but not the Measures of Variability. Multiplying by a Constant Consider our first data set again: 2, 4, 6, 8, 10 Now, consider what happens when we multiply each observation in the data set by a constant (say, 2), such that our new data set is: 4, 8, 12, 16, 20. The mean, median, variance, and standard deviation of the new data set is: 25
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