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Chapter 16 Part 3 MORE ABOUT MEANS AND VARIANCES Adding a constant to data (adding the same amount to each value) shifts the mean but the variance and standard deviation do not change. πΈ π±π =πΈ π ±π πππ π ± π = πππ π ππ· π ± π = ππ· π Example: Couples at the Quiet Nook can expect discounts averaging $5.83 with a standard deviation of $8.62. Suppose that for several weeks the restaurant has also been distributing coupons worth $5 off any one meal per table. If every couple there brings a coupon, what will be the mean and standard deviation of the total discounts theyβll receive? E(x) = 5.83 + 5 = $10.83 SD(x) = $8.62 (unchanged) Multiplying each value of a random variable by a constant multiplies the mean and the standard deviation by that constant. πΈ ππ = π β πΈ(π) ππ· ππ = π β ππ·(π) Example: (continued) Couples at the Quiet Nook can expect discounts averaging $5.83 with a standard deviation of $8.62. When two couples dine together on a single check, the restaurant doubles the discount offer - $40 for the ace of hearts on the 1st draw or $20 for ace of hearts on 2nd draw. What are the mean and standard deviation? E(x) = 2(5.83) = $11.66 StandardDev(x) = 2(8.62) = $17.24 Example: Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: Standard 2Y + 20 Mean Deviation Mean = E(2Y+20) = 2E(Y)+20 = 2(12)+20 = 44 X 80 12 SD = SD(2Y+20) = 2SD(Y) = 2(3) = 6 Y 12 3 Adding/Subtracting Random Variables: πΈ π ± π = πΈ(π) ± πΈ(π) If X and Y are independent, then πππ π ± π = πππ π + πππ π ππ· π ± π = ππ·(π)2 + ππ·(π)2 Example: Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: Standard X+Y Mean Deviation Mean = E(X+Y) = E(X)+E(Y)= 80+12= 92 X 80 12 SD = SD(X+Y) = 122 + 32 = 12.37 Y 12 3 Practice: Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: Standard a) 3X b) 0.25X+Y c) X β 5Y d) π1 +π2 + π3 Mean Deviation X 80 12 Y 12 3 Answers: a)240; 36 b)32; 4.24 c)20; 19.21 d)240; 20.78 Todayβs Assignment: ο±Add to HW #9 p. 384 #24-26, 33-36