# ch06 - STA 2023 Holbrook The Standard Deviation as a Ruler...

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Unformatted text preview: STA 2023 - Holbrook The Standard Deviation as a Ruler and the Normal Model Chapter 6 6-1 STA 2023 - Holbrook Interpreting the Standard Deviation 6-2 68­95­99.7 Rule STA 2023 - Holbrook • Rule of thumb that applies to data sets with Rule frequency distributions (think histograms) that are mound shaped or Normal. Normal 6-3 68% of the measurements will fall within 1 68% standard deviation of the mean. standard 95% of the measurements will fall within 2 95% standard deviations of the mean. standard 99.7% (virtually all) of the measurements will fall 99.7% within 3 standard deviations of the mean. within Example STA 2023 - Holbrook • Page 149, problem #28 6-4 Page 100, problem #12 – First edition STA 2023 - Holbrook • • Chebyshev’s Rule (optional) Applies to any data set, regardless of Applies the shape. the For k>1, at least 1 – (1/k)2 of the measurements will fall within k standard deviations of the mean. standard 6-5 Chebyshev’s Rule (optional) STA 2023 - Holbrook K (standard dev.) 2 1-(1/K)2 3/4 Percentage (at least) 75% 3 8/9 88.89% 4 15/16 93.75% 6-6 STA 2023 - Holbrook Numerical Measures of Relative Standing 6-7 Measures of Relative Standing STA 2023 - Holbrook • Z-Scores • Incorporate the mean, standard deviation, Incorporate and the 68-95-99.7 Rule and Percentile Ranks 6-8 Used by standardized tests (SAT, ACT, Used GRE) GRE) STA 2023 - Holbrook Measures of Relative Standing • Z-Score – the number of standard deviations a given measurement (y) is away from the mean. away • Percentile Rank – A number such that p % of the measurements fall below the pth percentile and (100-p)% fall above it. percentile 6-9 Z­Scores STA 2023 - Holbrook • • How many standard deviations away How from the mean an observation is. from Where an observation is, relative to the Where group. •For a population Z = (Y - µ) σ 6 - 10 10 •For a sample Z = (Y - Y) (Y s Example STA 2023 - Holbrook The mean price of a music cd in The Gainesville is \$12.39 with a standard deviation of \$0.80 while the mean price of a movie dvd is \$19.99 with a standard deviation of \$1.60. If you pay \$13.99 for a cd and \$22.39 for a dvd at Borders, which is relatively more expensive? relatively 6 - 11 11 Solution STA 2023 - Holbrook For the cd Zcd = 13.99 – 12.39 = 2.00 13.99 0.80 For the dvd Zdvd = 22.39 – 19.99 = 1.50 22.39 1.60 6 - 12 12 STA 2023 - Holbrook Normal Distribution 6 - 13 13 Importance of Normal Distribution STA 2023 - Holbrook 1. Describes Many Random Processes or Describes Continuous Phenomena Continuous • GPA of SFCC students • Height or weight in pounds • Pairs of jeans, CD’s or DVD’s owned • Units taken at SFCC • Length of fish in Newnans Lake Length 2. Basis for Classical Statistical Inference 2. 6 - 14 14 Normal Distribution STA 2023 - Holbrook 1. ‘‘Bell-Shaped’ & Bell-Shaped’ Symmetrical Symmetrical 2. f(X ) Mean, Median, Mean, Mode Are Equal Mode 3. 3. Random Variable Has Infinite Range Has 6 - 15 15 X Mean Mean Median Mode Mode Effect of Varying Parameters (µ & σ ) STA 2023 - Holbrook f(X) B A C X 6 - 16 16 Probability Density Function STA 2023 - Holbrook 1 f ( x) = e σ 2π f(x) σ π x µ 6 - 17 17 = = = = = µ2 1 x − − 2 σ Frequency of Random Variable x Frequency Population Standard Deviation 3.14159; e = 2.71828 Value of Random Variable (-∞ < x < ∞ ) Population Mean STA 2023 - Holbrook Normal Distribution Probability Probability is Probability area under curve! curve! d P(c ≤ x ≤ d ) = ∫ f ( x) dx c f(x ) c 6 - 18 18 d x ? STA 2023 - Holbrook Infinite Number of Tables Normal distributions differ by Normal mean & standard deviation. mean f(X) X 6 - 19 19 STA 2023 - Holbrook Infinite Number of Tables Normal distributions differ by Normal mean & standard deviation. mean Each distribution would Each require its own table. require f(X) X That’s an infinite number! That’s number! 6 - 20 20 STA 2023 - Holbrook 6 - 21 21 Standardize the Normal Distribution Standardize the Normal Distribution STA 2023 - Holbrook Normal Distribution σ µ 6 - 22 22 Xy Standardize the Normal Distribution STA 2023 - Holbrook Y −µ Z= σ Normal Distribution Standardized Normal Distribution σ σ =1 µ 6 - 23 23 Xy µ =0 One table! Z Standardizing Example STA 2023 - Holbrook • Psychology students at Wittenberg University Psychology completed the Dental Anxiety Scale questionnaire. Scores on the exam range from 0 (no anxiety) to 20 (extreme anxiety). Assume the distribution of scores on the questionnaire is normal with a mean score of 11 and a standard deviation of 3.5? standard a) b) c) 6 - 24 24 If you scored a 16, what is your z-score? Find the probability that someone scores between 10 and 15. Find the probability that someone scores above 17. Part A Solution STA 2023 - Holbrook For your score of 16: Z = 16 – 11 = 1.43 16 3.5 3.5 You scored 1.43 standard deviations above the mean. 6 - 25 25 Part B Solution STA 2023 - Holbrook Y − µ 10 − 11 Z= = = − 0.29 σ 3.5 Y − µ 15 − 11 Z= = = 1.14 Normal σ 3.5 Standardized Distribution Normal Distribution σ σ Xy = .4 7.8 6 - 26 26 σ =1 11 8 8.2 XY -.50 -0.29 0 .50 1.14 Z For P(Z < ­0.29) (from Z­Table) STA 2023 - Holbrook Y − µ 10 − 11 Z= = = − 0.29 σ 3.5 Y − µ 15 − 11 Z= = = 1.14 Normal σ 3.5 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 .3859 7.8 6 - 27 27 11 8 8.2 XY -.50 -0.29 0 .50 1.14 Z For P(Z < 1.14) (from Z­Table) STA 2023 - Holbrook Y − µ 10 − 11 Z= = = − 0.29 σ 3.5 Y − µ 15 − 11 Z= = = 1.14 Normal σ 3.5 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 .8729 7.8 6 - 28 28 11 8 8.2 XY -.50 -0.29 0 .50 1.14 Z For Area Between P(­0.29 < Z < 1.14) STA 2023 - Holbrook Y − µ 10 − 11 Z= = = − 0.29 σ 3.5 Y − µ 15 − 11 Z= = = 1.14 Normal σ 3.5 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 (.8729 - .3859) = .4870 7.8 6 - 29 29 11 8 8.2 XY -.50 -0.29 0 .50 1.14 Z Part C Solution STA 2023 - Holbrook Y − µ 17 − 11 Z= = = 1.71 σ 3 .5 Normal Distribution Standardized Normal Distribution σy p-Value 0 11 1.50 6 - 30 30 Wep-Value this want ZY 0 1.50 1.71 Z For P(Z < 1.71) (from Z­Table) STA 2023 - Holbrook Y − µ 17 − 11 Z= = = 1.71 σ 3 .5 Normal Distribution Standardized Normal Distribution σy p-Value Wep-Value this want .9564 0 11 1.50 6 - 31 31 ZY 0 1.50 1.71 Z For Area Above P(Z > 1.71) STA 2023 - Holbrook Y − µ 17 − 11 Z= = = 1.71 σ 3 .5 Normal Distribution Standardized Normal Distribution σy (1 - .9564) = .0436 p-Value p-Value 0 11 1.50 6 - 32 32 ZY 0 1.50 1.71 Z STA 2023 - Holbrook Normal Distribution Thinking Challenge You work in Quality Control for You GE. Light bulb life has a normal distribution with distribution µ = 2000 hours & σ = 200 hours. What’s the probability that a bulb will last will A. between 1800 & 2400 A. 1800 2400 hours? hours? B. less than 1470 hours? B. less 1470 6 - 33 33 Part A Solution STA 2023 - Holbrook Y − µ 1800 − 2000 Z= = = − 1.00 σ 200 Y − µ 2400 − 2000 Z= = = 2 .00 Normal σ 200 Standardized Distribution Normal Distribution σ σ Xy = .4 7.8 6 - 34 34 σ =1 2400 8.2 8 XY -.50 -1.00 0 .50 2.00 Z For P(Z < ­1.00) (from Z­Table) STA 2023 - Holbrook Y − µ 1800 − 2000 Z= = = − 1.00 σ 200 Y − µ 2400 − 2000 Z= = = 2 .00 Normal σ 200 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 .1587 7.8 6 - 35 35 2400 8.2 8 XY -.50 -1.00 0 .50 2.00 Z For P(Z < 2.00) (from Z­Table) STA 2023 - Holbrook Y − µ 1800 − 2000 Z= = = − 1.00 σ 200 Y − µ 2400 − 2000 Z= = = 2 .00 Normal σ 200 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 .9772 7.8 6 - 36 36 2400 8.2 8 XY -.50 -1.00 0 .50 2.00 Z STA 2023 - Holbrook For Area Between P(­1.00 < Z < 2.00) Y − µ 1800 − 2000 Z= = = − 1.00 σ 200 Y − µ 2400 − 2000 Z= = = 2 .00 Normal σ 200 Standardized Distribution Normal Distribution σ σ Xy = .4 σ =1 (.9772 - .1587) = .8185 7.8 6 - 37 37 2400 8.2 8 XY -.50 -1.00 0 .50 2.00 Z Part B Solution STA 2023 - Holbrook Y − µ 1470 − 2000 Z= = = −2.65 σ 200 Normal Distribution Standardized Normal Distribution σ 1470 6 - 38 38 σ =1 µ X Y -2.65 µ = 0 Z For P(Z < ­2.65) (from Z­Table) STA 2023 - Holbrook Y − µ 1470 − 2000 Z= = = −2.65 σ 200 Normal Distribution Standardized Normal Distribution σ σ =1 .0040 1470 6 - 39 39 µ X Y -2.65 µ = 0 Z Working Backwards? STA 2023 - Holbrook • (optional) Page 151, Problem #41 6 - 40 40 Page 101, Problem #22. STA 2023 - Holbrook Chapter 6 Skipping the following sections: ­ Rescaling Data, pg. 126­127 ­ Are You Normal, pg. 141­142 ­ Normal Probability Plot, pg. 143 6 - 41 41 End of Chapter Any blank slides that follow are blank intentionally. ...
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