Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Normal Distributions 1 Normal Distributions 1. 2. 3. 4. Symmetrical bell-shaped (unimodal) density curve How is this done Above the horizontal axis mathematically? N(m, s) The transition points occur at m + s (Points of inflection) 5. Probability is calculated by finding the area under the curve 6. As s increases, the curve flattens & spreads out 7. As s decreases, the curve gets 2 taller and thinner Normal distributions occur frequently. • • • • • • • Length of newborns Height Weight ACT or SAT scores Intelligence Number of typing errors Chemical processes 3 A 6 B s s Do these two normal curves have the same mean? If so, what is it? YES Which normal curve has a standard deviation of 3? B Which normal curve has a standard deviation of 1? A 4 Empirical Rule • Approximately 68% of the observations fall within s of m • Approximately 95% of the observations fall within 2s of m • Approximately 99.7% of the observations fall within 3s of m 5 Suppose that the height of male students at JVHS is normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. What is the probability that the height of a randomly selected male student is more than 73.5 inches? 1 - 0.68 = 0.32 P(X > 73.5) = 0.16 68% 71 6 Standard Normal Density Curves Always has m = 0 & s = 1 To standardize: x m z s Must have this memorized! 7 Strategies for finding probabilities or proportions in normal distributions 1. State the probability statement 2. Draw a picture 3. Calculate the z-score 4. Look up the probability (proportion) in the table 8 AP Statistics Friday, 09 January 2015 • OBJECTIVE TSW investigate normal distributions. • You need to have the following out: 1. 2. Blue chart (Table A) Calculator (sign up for a new/old number, if needed) • QUIZ: Continuous & Uniform Distributions on Monday, 12 January 2015. • ASSIGNMENT DUE DATES – – – – WS Normal Distributions Monday, 01/12/2015 WS Continuous Distributions Review Tuesday, 01/13/2015 Bookwork: 7.67, 7.69, 7.71 Monday, 01/12/2015 Bookwork: 7.73, 7.77, 7.80 Monday, 01/12/2015 WS Unusual Density Curves 1) 2) 3) 4) 5) 6) 7) 8) 9) 0.375 0.375 0.34375 0.5 0.35 0.3 0.55 0.8125 a) A = 0.05(0.4)(5) = 1 b) P(X < 0.20) = 50% P(X < 0.10) = 12.5% c) 37.5% d) 43.75% WS Uniform Distributions 1) a) b) μ = 2 min, σ = 1.15470 min 0.375 2) a) b) 10 min 0.05 3) a) b) c) 0.666 0.333 82.5 degrees 4) a) b) c) d) e) continuous μ = 7 oz., σ = 0.28868 oz. 0.75 0.25 0.57735 5) 0.00024414 The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standardDraw deviation of 15 & shade Write the hours. What proportion of these the curve probability batteries can be expected to last less statement than 220 hours? P(X < 220) = 0.9082 Look up z220 200 score in z 1.33 table 15 Calculate z-score 12 The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last more than 220 hours? P(X>220) = 1 - 0.9082 = 0.0918 220 200 z 1.33 15 13 The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 Look up in table 0.95 hours. How long must a battery last to be in the top 5%? to find z- score P(X > ?) = 0.05 x 200 1.645 15 x 224.675 224.675 hours .95 .05 1.645 Label units when given 14 The heights of the female students at JVHS are normally distributed with a What is the zmean of 65 inches. What is the for the standard deviation of this score distribution 63? if 18.5% of the female students are shorter than 63 inches? P(X < 63) = 0.185 -0.9 63 65 0.9 s 63 2 s 2.2222 inches 0.9 15 The heights of female teachers at JVHS are normally distributed with mean of 65.5 inches and standard deviation of 2.25 inches. The heights of male teachers are normally distributed with mean of 70 inches and standard deviation of 2.5 inches. • Describe the distribution of differences of heights (male – female) teachers. m = 70 - 65.5 m = 4.5 s = 2.252 + 2.52 s = 3.36340... Normal distribution with m = 4.5 inches & s = 3.3634 inches 16 • What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? P(X<0) = 0.0901 0 4.5 z 1.34 3.3634 4.5 17 Will my calculator do any of this normal stuff? • Normalpdf – use for graphing ONLY • Normalcdf – will find probability of area from lower bound to upper bound • Invnorm (inverse normal) – will find X-value for probability 18 Assignments • WS Normal Distributions – Due on Monday, 12 January 2015. • WS Continuous Distributions Review – Due on Tuesday, 13 January 2015. • Bookwork: 7.67, 7.69, 7.71, 7.73, 7.77, 7.80 – Due on Monday, 12 January 2015. • QUIZ: Continuous & Uniform Distributions on Monday, 12 January 2015. 19 AP Statistics Monday, 12 January 2015 • OBJECTIVE TSW quiz over continuous and uniform • READ • • distributions. – – – – Sec. 7.7: pp. 409-416 ASSIGNMENTS DUE WS Normal Distributions wire basket Bookwork: 7.67, 7.69, 7.71, 7.73, 7.77, 7.80 ASSIGNMENT DUE TOMORROW WS Continuous Distributions Review black tray Ways to Assess Normality 1. Use graphs (dotplots, boxplots, or histograms) 2. Use the Empirical Rule 3. Normal probability (quantile) plot 21 Normal Probability (Quantile) plots 1. The observation (x) is plotted against known normal z-scores 2. If the points on the quantile plot lie close to a straight line, then the data is normally distributed 3. Deviations on the quantile plot indicate nonnormal data 4. Points far away from the plot indicate outliers 5. Vertical stacks of points (repeated observations of the same number) is called 22 granularity Normal Scores Suppose we have the following To construct a normal probability plot, Sketch a scatterplot by pairing the Think of selecting sample after sample of observations of widths of contact you cansmallest use quantities called normal normal score with the size 10 from a standard normal windows in integrated circuit chips: What should score. The values of the normal scores smallest observation from the the 1distribution. Then -1.539 is data happen if sample size n. The normal depend on the set smallest & so on observation average of the our when data n = 10 are below: scores from each sample & so on . . . is 3.21 set2.49 2.94 4.38 4.02 2 33.34 4 3.81 5 3.62normally 3.301 2.85 distributed? -1 -1.539 -1.001 -0.656 -0.376 -0.123 Contact 1.001 Windows1.539 0.123 Widths 0.376of 0.656 23 Are these approximately normally distributed? 50 48 54 47 51 52 46 53 What 52 51 48 48 54 55 57is this 45 53 50 47 49 50 56 called? 53 52 Both the histogram & boxplot are approximately symmetrical, so these data are approximately normal. The normal probability plot is approximately linear, so these data are approximately normal. 24 QUIZ: Continuous and Uniform Distributions • Checkerboard, please. • ASSIGNMENT DUE TOMORROW – WS Continuous Distributions Review 25