# Chapter 7 Continuous Distributions Continuous Random Variables Values

• Slides: 46

Chapter 7 Continuous Distributions

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

Density Curve • Always on or above the x-axis • Area underneath it equals 1 • Shows what proportion of data falls within an interval

Unusual Density Curves • Generic continuous distributions • Can be any shape • Probability = area under the curve

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

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

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

P(X > 3) =. 4375 P(1 < X < 3) =. 5

P(0. 5 < X < 1. 5) =. 28125

Special Continuous Distributions

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

Why 12? , where

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. What a mean of 4. 98 and a range shape doespounds a uniform What is the height of this of. 12 pounds. distribution have? rectangle? a) Construct this uniform How distribution. long is this rectangle? 1/. 12 4. 98 5. 04

b) What is the probability that a randomly selected bag will weigh more than 4. 97 pounds? What is the length of the shaded region? P(X > 4. 97) =. 07(1/. 12) =. 5833 1/. 12 4. 98 5. 04

c) Find the probability that a randomly selected bag weighs between 4. 93 and 5. 03 pounds. What is the length of the P(4. 93 < X < 5. 03) =. 1(1/. 12)shaded =. 8333 region? 1/. 12 4. 98 5. 04

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. is the height of the a) Draw the. What distribution. rectangle? Where should the rectangle end? 1/35 5 40

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

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

Normal Distribution • • • How is bell-shaped this done Symmetrical, density curve mathematically? Parameters: μ, Probability = area under the curve As increases, increases curve flattens & spreads out As decreases, decreases curve gets taller and thinner

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

A B 6 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

Empirical Rule • Approx. 68% of the data fall within 1 of μ • Approx. 95% of the data fall within 2 of μ • Approx. 99. 7% of the data fall within 3 of μ

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? 1 –. 68 =. 32 P(X > 73. 5) = 0. 16 68% 71

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

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

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 Write can bethe expected to last less & than 220 the hours? Draw shade curve probability statement P(X < 220) =. 9082 Look up z-score in table Calculate z-score

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

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%? Look up 0. 95 in table to find z-score P(X > ? ) =. 05. 95 . 05 1. 645

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 Whatstudents is the z-score are shorter than 63 inches? for 63? P(X < 63) =. 185 -0. 9 63

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

What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? P(X < 0) =. 0901 4. 5

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

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

Normal Scores Suppose we have the following observations To construct a normal probability plot, we Sketch a scatterplot by pairing the smallest Think of selecting sample after sample of size 10 What should of of contact windows inscores. integrated canwidths use quantities called normal The normal score with the smallest data value, from a standard normal distribution. Then happen if our nd nd data value, circuit chips: values of the normal scores depend on the 2 normal score with 2 and -1. 539 is the average of the smallest value from data set is so on sample size n. The normal scores when n = 10 each sample, -1. 001 is the average of the next normally smallest value from each sample, etc. aredistributed? below: 3. 21 2. 49 2. 94 4. 38 4. 02 3. 62 3. 30 2. 85 3. 34 3. 81 -1. 539 -1. 001 -0. 656 -0. 376 -0. 123 0. 376 0. 656 1. 001 1. 539 Widths of Contact Windows

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

Are these approximately normally distributed? 50 52 53 48 51 50 54 48 47 47 48 49 51 54 50 The histogram/boxplot is approx. symmetrical, so the data are approx. normal. 52 55 56 46 57 53 53 45 52 The normal probability plot is approx. linear, so the data are approx. normal.

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. P(15 < X < 30) = binomcdf(250, . 1, 30) – binomcdf(250, . 1, 14) =. 866 2) What is the mean and standard deviation of this distribution? μ = 25, = 4. 743

Premature babiesthis are distribution: those born more than 3 Let’s graph weeks early. Newsweek (May 16, 1988) reported that • 10% of the live L 1: seq(X, X, births 0, 45) in the U. S. are premature. Suppose 250 live births are randomly selected = the number of “preemies” is • L 2: and use. Xbinompdf to find the binomial determined. probabilities 3) If • we were graph a histogram xmin = to -0. 5, xmax = 45, xsclfor = 1 the above binomial distribution, what shape would it have? • ymin = 0, ymax = 0. 2, yscl = 1 p is only 10% should be skewed right Overlay a normal curve on your 4) What do you notice about the shape? histogram: • Y 1 = normalpdf(X, 25, 4. 743)

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

• 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

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.

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

5) Since P(preemie) =. 1 which is not close to. 5, is n large enough? np = 250(. 1) = 25 > 10 n(1 – p) = 250(. 9) = 225 > 10 We can use normal to approximate binomial 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 P(15 < X < 30) Normal (w/ cont. correction) P(14. 5 < X < 30. 5)

P(14. 5 < X < 30. 5) = normalcdf(14. 5, 30. 5, 25, 4. 743) =. 8634 7) How does the normal answer compare to the binomial answer? Pretty darn close!

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?