AP Statistics Tuesday, 12 January 2016 OBJECTIVE TSW investigate normal distributions. QUIZ: Continuous & Uniform Distributions is graded. TODAY’S ASSIGNMENT (due tomorrow) − WS Normal Approximations to Binomial Distributions QUIZ: Normal Distributions & Approximations is tomorrow.

WS Normal Distributions 1)a) 18.14% b) points 2) years 3) ounces 4) minutes 5)a) 42 minutes b) c) )a) normal,  = $220,  = $46.10 b) c)

WS Continuous Distributions Review 1) or )  = $104.56,  = $ ) ) inches 5)a) 0.25 b) c) ) F 7)a) 0.45 b) 0.5

4 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

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

6 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

7 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: Think of selecting sample after sample of size 10 from a standard normal distribution. Then 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: 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?

8 Are these approximately normally distributed? 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?

AP Statistics Wednesday, 13 January 2016 OBJECTIVE TSW investigate normal distributions. QUIZ: Continuous & Uniform Distributions is graded. TODAY’S ASSIGNMENT − WS Normal Approximations to Binomial Distributions Due by end of period. –WS AP Review Continuous Distributions #1 –WS AP Review Free Response Questions – Distributions Both due on Friday, 01/15/16. QUIZ: Normal Distributions & Approximations is after the lesson TODAY.

Normal Approximation to the Binomial Before widespread use of technology, binomial probability calculations were very tedious. Let’s see how statisticians estimated these calculations in the past!

2) What is the mean and standard deviation of the above distribution? Premature babies are those born more than 3 weeks early. Newsweek (May 16, 1988) reported that 10% of the live births in the U.S. are premature. Suppose that 250 live births are randomly selected and that the number X of the “preemies” is determined. What is the probability that there are between 15 and 30 preemies, inclusive? (POD, p. 422) 1) Find this probability using the binomial distribution. P(15<X<30) = binomialcdf(250,0.1,30) – binomialcdf(250,0.1,14) = 0.866

3) If we were to graph a histogram for the above binomial distribution, what shape do you think it will have? 5) What do you notice about the shape? Since the probability is only 10%, we would expect the histogram to be strongly skewed right. Let’s graph this distribution – Put the numbers 0-45 in L1 seq(x,x,0,45) In L2, use binomialpdf to find the probabilities. binompdf(250,0.1,L1) Overlay a normal curve on your histogram: In Y1 = normalpdf(X, ,  )

13 Normal distributions can be used to estimate probabilities for binomial distributions when: 1) the probability of success is close to 0.5or 2) n is sufficiently large Rule: if n is large enough, then np > 10 & n(1 –p) > 10 Why 10?

14 Normal distributions extend infinitely in both directions; however, binomial distributions are between 0 and n. If we use a normal distribution to estimate a binomial distribution, we must cut off the tails of the normal distribution. This is OK if the mean of the normal distribution (which we use the mean of the binomial) is at least three standard deviations (3  ) from 0 and from n. (BVD, p. 334)

15 We require: Or As binomial: Square: Simplify: Since (1 - p) < 1: And p < 1: Therefore, we say the np should be at least 10 and n (1 – p) should be at least 10.

16 Normal distributions can be used to estimate probabilities for binomial distributions when: 1) the probability of success is close to 0.5or 2) n is sufficiently large Rule: if n is large enough, then np > 10 & n(1 –p) > 10 Since a continuous distribution is used to estimate the probabilities of a discrete distribution, a continuity correction is used to make the discrete values similar to continuous values.(+0.5 to discrete values) Why? Think about how discrete histograms are made. Each bar is centered over the discrete values. The bar for “1” actually goes from 0.5 to 1.5 & the bar for “2” goes from 1.5 to 2.5. Therefore, by adding or subtracting.5 from the discrete values, you find the actually width of the bars that you need to estimate with the normal curve.

The answer in question 1 is slightly larger. 17 (Back to our example) Since P(preemie) = 0.1 which is not close to 0.5, is n large enough? 6) Use a normal distribution with the binomial mean and standard deviation above to estimate the probability that between 15 & 30 preemies, inclusive, are born in the 250 randomly selected babies. Binomialwritten as Normal (w/cont. correction) P(15 < X < 30) 7) How does the answer in question 6 compare to the answer in question 1 (Binomial answer =0.866)? Normalcdf(14.5,30.5,25,4.743) = np = 250(0.1) = 25 & n(1-p) = 250(0.9) = 225 Yes, OK to use normal to approximate binomial  P(14.5 < X < 30.5) =

Assignment Finish WS Normal Approximations to Binomial Distributions. –Due tomorrow, Thursday, 14 January QUIZ: Normal Distributions and Approximations –You may use your charts! WS AP Review: Continuous Distributions #1 –Due on Friday, 15 January WS AP Review: Free Response Questions – Distributions –Due on Friday, 15 January