1. Chapter 7 Continuous Distributions

2. Continuous Random Variables Values fall within an interval Measurements Described by density curves

3. Density Curve on or aboveAlways on or above the x-axis equals 1Area underneath it equals 1 proportionShows what proportion of data falls within an interval

4. Unusual Density Curves Generic continuous distributions Can be any shape area under the curveProbability = area under the curve

5. P(X < 2) = How do we find the area of a triangle?

6. P(X = 2) =0 P(X < 2) =.25 What is the area of a line segment?

7. P(X < 2) & P(X < 2) In continuous distributions, P(X < 2) & P(X < 2) are the same! Hmmmm… Is this different than discrete distributions?

8. P(X > 3) = P(1 < X < 3) =.4375.5

9. P(X > 1) =.75.5(2)(.25) =.25 (2)(.25) =.5

10. P(0.5 < X < 1.5) =.28125

11. Special Continuous Distributions

12. Uniform Distribution Evenly (uniformly) distributed  Every value has equal probability Density curve: rectangle Probability = area under the curve (a & b are the endpoints of the distribution) How do you find the area of a rectangle?

13. Why 12?, where

14. 4.985.044.92 The Citrus Sugar Company packs sugar in bags labeled 5 pounds. However, the packaging isn’t perfect and the actual weights are uniformly distributed with a mean of 4.98 pounds and a range of.12 pounds. a) Construct this uniform distribution. How long is this rectangle? What is the height of this rectangle? What shape does a uniform distribution have? 1/.12

15. b) What is the probability that a randomly selected bag will weigh more than 4.97 pounds? 4.985.044.92 1/.12 P(X > 4.97) =.07(1/.12) =.5833 What is the length of the shaded region?

16. c) Find the probability that a randomly selected bag weighs between 4.93 and 5.03 pounds. 4.985.044.92 1/.12 P(4.93 < X < 5.03) =.1(1/.12) =.8333 What is the length of the shaded region?

17. The time it takes for students to drive to school is evenly distributed with a minimum of 5 minutes and a range of 35 minutes. a) Draw the distribution. 5 Where should the rectangle end? 40 What is the height of the rectangle? 1/35

18. b) What is the probability that it takes less than 20 minutes to drive to school? 5 40 1/35 P(X < 20) =(15)(1/35) =.4286

19. c) What are the mean and standard deviation of this distribution? μ = (5 + 40)/2 = 22.5  2 = (40 – 5) 2 /12 = 102.083  = 10.104

20. Normal Distribution Symmetrical, bell-shaped density curve Parameters: μ,  area under the curveProbability = area under the curve increasesAs  increases, curve flattens & spreads out decreasesAs  decreases, curve gets taller and thinner How is this done mathematically?

21. Normal distributions occur frequently. Length of infants Height Weight ACT scores Intelligence Number of typing errors Velocities of ideal gas molecules Yearly rainfall Quantum harmonic oscillators Diffusion Thermal light Size of living tissue Blood pressure Compound interest Exchange rates Stock market indices

22. 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

23. Empirical Rule 68%Approx. 68% of the data fall within 1  of μ 95%Approx. 95% of the data fall within 2  of μ 99.7%Approx. 99.7% of the data fall within 3  of μ

24. Suppose the height of male GBHS students is normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. What is the probability that a randomly selected male student is taller than 73.5 inches? P(X > 73.5) = 0.16 71 68% 1 –.68 =.32

25. Standard Normal Density Curve μ = 0 &  = 1 To standardize any normal data: Make your life easier – memorize this!

26. To find normal probabilities/proportions: 1.Write the probability statement 2.Draw a picture 3.Calculate the z-score 4.Look up the probability in the table

27. 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) =.9082 Write the probability statement Draw & shade the curve Calculate z-score Look up z-score in table

28. 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 –.9082 =.0918

29. 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 > ?) =.05.95.05 Look up 0.95 in table to find z-score 1.645

30. The heights of the female GBHS students 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) =.185 63 What is the z-score for 63? -0.9

31. The heights of female GBHS teachers are normally distributed with mean 65.5 inches and standard deviation 2.25 inches. The heights of male GBHS teachers are normally distributed with mean 70 inches and standard deviation 2.5 inches. Describe the distribution of differences of teacher heights (male – female) Normal distribution with μ = 4.5,  = 3.3634

32. What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? 4.5 P(X < 0) =.0901

33. Will my calculator do any of this stuff? Normalpdf: Doesn't make sense  P(X = x) = 0! ONLY  Used for graphing ONLY Normalcdf: Calculates probability  normalcdf(lower bound, upper bound) Invnorm (inverse normal): Finds z-score for a probability to the left

34. Ways to Assess Normality Dotplots, boxplots, histograms Normal Probability (Quantile) Plot

35. To construct a normal probability plot, we can use quantities called normal scores. 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 value from each sample, -1.001 is the average of the next smallest value from each sample, etc. 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 data value, 2 nd normal score with 2 nd data value, and so on Normal Scores Widths of Contact Windows What should happen if our data set is normally distributed?

36. Normal Probability (Quantile) Plots Plot data against known normal z-scores Points form a straight line  data is normally distributed Stacks of points (repeat observations): granularity

37. Are these approximately normally distributed? 5048544751524653 5251484854555745 5350474950565352 The histogram/boxplot is approx. symmetrical, so the data are approx. normal. The normal probability plot is approx. linear, so the data are approx. normal.

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

39. 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 250 live births are randomly selected and X = the number of “preemies” is determined. What is the probability that there are between 15 and 30 preemies, inclusive? 1)Find this probability using the binomial distribution. 2) What is the mean and standard deviation of this distribution? P(15 < X < 30) = binomcdf(250,.1, 30) – binomcdf(250,.1, 14) =.866 μ = 25,  = 4.743

40. 3) If we were to graph a histogram for the above binomial distribution, what shape would it have? 4) What do you notice about the shape? Since p is only 10%, we'd expect it to be skewed right. Let’s graph this distribution: L1: seq(X, X, 0, 45) L2: use binompdf to find the binomial probabilities xmin = -0.5, xmax = 45, xscl = 1 ymin = 0, ymax = 0.2, yscl = 1 Overlay a normal curve on your histogram: Y1 = normalpdf(X, 25, 4.743 )

41. We can estimate binomial probabilities with the normal distribution IF… 1) p is close to.5or 2) n is sufficiently large  np > 10 & n(1 –p) > 10 Why 10?

42. Normal distributions extend infinitely in both directions Binomial distributions go from 0 to n If we use normal to estimate binomial, we have to cut off the tails of the normal distribution This is okay if the mean of the normal distribution (which we use the mean of the binomial for) is at least three standard deviations (3  ) from 0 and from n

43. We require: Or As binomial: Square it: Simplify: Since (1 – p) < 1: And since p < 1: Therefore, np should be at least 10 and n(1 – p) should be at least 10.

44. Continuity Correction Discrete histograms: Each bar is centered over a discrete value Bar for "1" actually goes from 0.5 to 1.5, bar for "2" goes from 1.5 to 2.5, etc. So if we want to estimate a discrete distribution with a continuous one…  Add/subtract 0.5 from each discrete value

45. 5) Since P(preemie) =.1 which is not close to.5, is n large enough? 6) Use a normal distribution to estimate the probability that between 15 and 30 preemies, inclusive, are born in the 250 randomly selected babies. Binomial Normal (w/ cont. correction) P(15 < X < 30) np = 250(.1) = 25 > 10 n(1 – p) = 250(.9) = 225 > 10  We can use normal to approximate binomial  P(14.5 < X < 30.5)

46. 7) How does the normal answer compare to the binomial answer? normalcdf(-2.21, 1.16) =.8634P(14.5 < X < 30.5) = Pretty darn close!

47. Estimate each probability using the normal distribution: a) What is the probability that less than 20 preemies are born out of the 250 babies? b) What is the probability that at least 30 preemies are born out of the 250 babies? c) What is the probability that less than 35 preemies but more than 20 preemies are born out of the 250 babies?