1. AP Statistics Wednesday, 06 January 2016 OBJECTIVE TSW investigate normal distributions. You need to have the following out: 1.Blue chart (Table A) 2.Calculator ASSIGNMENT DUE BY END OF PERIOD –WS Uniform Distributions  black tray ASSIGNMENT DUE ON FRIDAY, 01/08/16 –WS Normal Distributions

2. WS Unusual Density Curves 1)0.375 2)0.375 3)0.34375 4)0.5 5)0.35 6)0.3 7)0.55 8)0.8125 9)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%

3. Normal Distributions 3

4. 4 Normal Distribution 1. Symmetrical bell-shaped (unimodal) density curve Above 2. Above the horizontal axis 3. N( ,  ) 4. The transition points occur at  +  (Points of inflection) area under the curve 5. Probability is calculated by finding the area under the curve increases 6. As  increases, the curve flattens & spreads out decreases 7. As  decreases, the curve gets taller and thinner How is this done mathematically?

5. 5 Normal distributions occur frequently. Length of newborns Height Weight ACT or SAT scores Intelligence Number of typing errors Chemical processes

6. 6 A B Do these two normal curves have the same mean? If so, what is it? Which normal curve has a standard deviation of 3? Which normal curve has a standard deviation of 1? 6 YES B   A

7. 7 Empirical Rule 68%Approximately 68% of the observations fall within  of  95%Approximately 95% of the observations fall within 2  of  99.7%Approximately 99.7% of the observations fall within 3  of 

8. 8 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? P(X > 73.5) = 0.16 71 68% 1 - 0.68 = 0.32

9. 9 Standard Normal Density Curves Always has  = 0 &  = 1 To standardize: Must have this memorized!

10. 10 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

11. 11 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 less than 220 hours? P(X < 220) =0.9082 Write the probability statement Draw & shade the curve Calculate z-score Look up z- score in table

12. AP Statistics Friday, 08 January 2016 OBJECTIVE TSW investigate normal distributions. You need to have the following out: 1.Blue chart (Table A) 2.Calculator QUIZ: Continuous & Uniform Distributions on Monday, 11 January 2016. ASSIGNMENT DUE DATES –WS Uniform Distributions  if you haven’t already turned it in (I will collect) –WS Normal Distributions  end of period today (wire basket) –WS Continuous Distributions Review  Monday, 01/11/2016

13. 13 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

14. 14 The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. How long must a battery last to be in the top 5%? P(X > ?) = 0.05.95.05 Look up in table 0.95 to find z- score 1.645 224.675 hours  Label units when given

15. 15 The heights of the female students at JVHS are normally distributed with a mean of 65 inches. What is the standard deviation of this distribution if 18.5% of the female students are shorter than 63 inches? P(X < 63) = 0.185 63 What is the z- score for the 63? -0.9

16. 16 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. Normal distribution with  = 4.5 inches &  = 3.3634 inches

17. 17 What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? 4.5 P(X<0) =0.0901

18. 18 Will my calculator do any of this normal stuff? ONLYNormalpdf – 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

19. 19 Ways to Assess Normality 1.Use graphs (dotplots, boxplots, or histograms) 2.Use the Empirical Rule 3.Normal probability (quantile) plot

20. 20 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 granularity

21. 21 To construct a normal probability plot, you can use quantities called normal score. The values of the normal scores depend on the sample size n. The normal scores when n = 10 are below: -1.539 -1.001 -0.656 -0.376 -0.123 0.123 0.376 0.656 1.001 1.539 Think of selecting sample after sample of size 10 from a standard normal distribution. Then -1.539 is the average of the smallest observation from each sample & so on... Suppose we have the following observations of widths of contact windows in integrated circuit chips: 3.21 2.49 2.94 4.38 4.02 3.62 3.30 2.85 3.34 3.81 Sketch a scatterplot by pairing the smallest normal score with the smallest observation from the data set & so on Normal Scores Widths of Contact Windows What should happen if our data set is normally distributed?

22. 22 Are these approximately normally distributed? 5048544751524653 5251484854555745 5350474950565352 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. What is this called?

23. Assignments WS Normal Distributions –Due by end of period today (wire basket) WS Continuous Distributions Review –Due on Monday, 11 January 2016. QUIZ: Continuous & Uniform Distributions on Monday, 11 January 2016. 23