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AP Statistics Tuesday, 14 January 2014

AP Statistics Tuesday, 14 January 2014. OBJECTIVE TSW investigate normal distributions. QUIZ: Continuous & Uniform Distributions is not graded. ASSIGNMENT DUE WS Continuous Distributions Review  wire basket QUIZ: Normal Distributions & Approximations tomorrow.

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AP Statistics Tuesday, 14 January 2014

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  1. AP StatisticsTuesday, 14 January 2014 • OBJECTIVETSW investigate normal distributions. • QUIZ: Continuous & Uniform Distributionsis not graded. • ASSIGNMENT DUE • WS Continuous Distributions Review wire basket • QUIZ: Normal Distributions & Approximations tomorrow.

  2. WS Normal Distributions 1)a) 0.1814 b) 52.593 2) 3.191 yrs 3) 79.052 oz. 4) 9.652 min. 5) a) 42 min b) 0.0668 c) 0.8664 6) a) normal, = $220,  = $46.10 b) 0.2578 c) 0.7198

  3. 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

  4. 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!

  5. 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 2) What is the mean and standard deviation of the above distribution?

  6. 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) 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. • Overlay a normal curve on your histogram: • In Y1 = normalpdf(X,m,s)

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

  8. Normaldistributions 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 mustcut offthe tails of the normal distribution. This is OK if the mean of the normal distribution (which we use the mean of the binomial) isat least three standard deviations(3s) from 0 and from n. (BVD, p. 334)

  9. 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.

  10. 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. Normal distributions can be used to estimate probabilities for binomial distributions when: 1) the probability of success is close to 0.5 or 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?

  11. np = 250(0.1) = 25 & n(1-p) = 250(0.9) = 225 Yes, OK to use normal to approximate binomial (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. Binomial written 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)?  P(14.5 < X < 30.5) = Normalcdf(14.5,30.5,25,4.743) = 0.8635

  12. Assignment • WS Normal Approximations to Binomial Distributions • Due Thursday, 16 January 2014.

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