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LINEAR TRANSFORMATION RULE When adding a constant to a random variable, the mean changes but not the standard deviation. When multiplying a constant to a random variable, the mean and the standard deviation changes. An appliance repair shop charges a $30 service call to go to a home for a repair. It also charges $25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor? 30 25(1.25) $61.25 1 25 $8.33 3 RULES FOR COMBINING TWO VARIABLES To find the mean for the sum (or difference), add (or subtract) the two means To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root. Formulas: a b a b a b a b 2 a a b 2 b If variables are independent Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times? Phase Mean SD Unpacking Assembly Tuning 3.5 21.8 12.3 0.7 2.4 2.7 T 3.5 21.8 12.3 37.6 minutes T 0.7 2 2.42 2.7 2 3.680 minutes