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1 2.4 (cont.) Using the Mean and Standard Deviation Together 68-95-99.7 rule z-scores

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2 68-95-99.7 rule Mean and Standard Deviation (numerical) Histogram (graphical) 68-95-99.7 rule

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3 The 68-95-99.7 rule; applies only to mound-shaped data

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4 68-95-99.7 rule: 68% within 1 stan. dev. of the mean 68% 34% y-s y y+s

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5 68-95-99.7 rule: 95% within 2 stan. dev. of the mean 95% 47.5% y-2s y y+2s

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6 Example: textbook costs 286291307308315316327 328340342346347348348 349354355355360361364 367369371373377380381 382385385387390390397 398409409410418422424 425426428433434437440 480

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7 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

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8 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

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9 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

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10 The best estimate of the standard deviation of the mens weights displayed in this dotplot is 1. 10 2. 15 3. 20 4. 40

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12 Z-scores: Standardized Data Values Measures the distance of a number from the mean in units of the standard deviation

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13 z-score corresponding to y

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14 n Exam 1: y 1 = 88, s 1 = 6; exam 1 score: 91 Exam 2: y 2 = 88, s 2 = 10; exam 2 score: 92 Which score is better?

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15 Comparing SAT and ACT Scores n SAT Math: Eleanors score 680 SAT mean =500 sd=100 n ACT Math: Geralds score 27 ACT mean=18 sd=6 n Eleanors z-score: z=(680-500)/100=1.8 n Geralds z-score: z=(27-18)/6=1.5 n Eleanors score is better.

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16 Z-scores add to zero Student/Institutional Support to Athletic Depts For the 9 Public ACC Schools: 2013 ($ millions) SchoolSupporty - ybarZ-score Maryland15.56.41.79 UVA13.14.01.12 Louisville10.91.80.50 UNC9.20.10.03 VaTech7.9-1.2-0.34 FSU7.9-1.2-0.34 GaTech7.1-2.0-0.56 NCSU6.5-2.6-0.73 Clemson3.8-5.3-1.47 Mean=9.1000, s=3.5697 Sum = 0

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17 In 2007-08 the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the mean tuition was $4320. What is NCs z-score? 1. 1.03 2. -1.03 3. 2.39 4. 1865 5. -1865

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18 End of Section 2.4