2 Linear transformation A linear transformation changes the original variable x into a new variable given by an equation of the formAdding the constant a shifts all values of x upward or downward by the same amountMultiplying by the positive constant b changes the size of the unit of measurement
3 Effect of a linear transformation Multiplying each observation by a positive number b multiplies both measures of center (mean and median) and measures of spread (standard deviation and IQR) by bAdding the same number a (either positive or negative) to each observation adds a to measures of center and to quartiles but does not change measures of spread
4 How much will the mean salary increase? The median salary? Page A school system employs teachers at salaries between $30000 and $ The teacher’s union and the school board are negotiating the form of next year’s increase in the salary schedule. Suppose that every teacher is given a flat $1000 raise.How much will the mean salary increase? The median salary?The mean and the median will both increase by 1000Will a flat $1000 raise increase the spread as measured by the distance between the quartiles?No. Each Quartile will increase by 1000 therefore the difference will remain the sameWill a flat $1000 raise increase the spread as measured by the standard deviation of the salaries?No, the standard deviation remains unchanged when the same amount is added to each data value
5 Page 56 #1.46 Suppose that the teachers in the previous exercise each receive a 5% raise. The amount of the raise will vary from $1500 to $3000, depending on present salary. Will a 5% raise across-the-board increase the spread of the distribution as measured by the distance between the quartiles? Do you think it will increase the standard deviation?A 5% across-the –board raise will cause both the IQR and s to increase. The transformation being applied here is, where x = the old salary and = the new salary. Both IQR and s will increase by a factor of 1.05
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