Continuous Distributions Chapter 6 Continuous Distributions

Continuous random variables Are numerical variables whose values fall within a range or interval Are measurements Can be described by density curves

Density curves Is always on or above the horizontal axis Has an area exactly equal to one underneath it Often describes an overall distribution Describe what proportions of the observations fall within each range of values

Unusual density curves Can be any shape Are generic continuous distributions Probabilities are calculated by finding the area under the curve

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

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

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

P(X > 3) = P(1 < X < 3) = .5(.375+.5)(1)=.4375 Shape is a trapezoid – How long are the bases? b1 = .5 b2 = .375 h = 1 P(X > 3) = P(1 < X < 3) = .5(.375+.5)(1)=.4375 .5(.125+.375)(2) =.5

P(X > 1) = .75 .5(2)(.25) = .25 (2)(.25) = .5

P(0.5 < X < 1.5) = .28125 .5(.25+.375)(.5) = .15625 (.5)(.25) = .125

Special Continuous Distributions

How is this done mathematically? Normal Distributions Symmetrical bell-shaped (unimodal) density curve Above the horizontal axis N(m, s) The transition points occur at m + s Probability is calculated by finding the area under the curve As s increases, the curve flattens & spreads out As s decreases, the curve gets taller and thinner How is this done mathematically?

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

A B 6 s s 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? YES B A

Empirical Rule Approximately 68% of the observations fall within s of m Approximately 95% of the observations fall within 2s of m Approximately 99.7% of the observations fall within 3s of m

Suppose that the height of male students at PWSH 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? 71 1 - .68 = .32 P(X > 73.5) = 0.16 68%

Standard Normal Density Curves Always has m = 0 & s = 1 To standardize: Must have this memorized!

State the probability statement Draw a picture Calculate the z-score Strategies for finding probabilities or proportions in normal distributions State the probability statement Draw a picture Calculate the z-score Look up the probability (proportion) in the table

Write the probability statement Look up z-score in 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 can be expected to last less than 220 hours? Write the probability statement Draw & shade the curve 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 in table 0.95 to find z- score P(X > ?) = .05 .95 .05 1.645

What is the z-score for the 63? The heights of the female students at PWSH 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? What is the z-score for the 63? P(X < 63) = .185 -0.9 63

The heights of female teachers at PWSH 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 m = 4.5 & s = 3.3634

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

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 z-score for probability

Ways to Assess Normality Use graphs (dotplots, boxplots, or histograms) Normal probability (quantile) plot

What should happen if our data set is normally distributed? 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 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 . . . 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?

Normal Probability (Quantile) plots The observation (x) is plotted against known normal z-scores If the points on the quantile plot lie close to a straight line, then the data is normally distributed Deviations on the quantile plot indicate nonnormal data Points far away from the plot indicate outliers Vertical stacks of points (repeated observations of the same number) is called granularity

Are these approximately normally distributed? 50 48 54 47 51 52 46 53 52 51 48 48 54 55 57 45 53 50 47 49 50 56 53 52 What is this called? 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.

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!

1) Find this probability using the binomial 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. 2) What is the mean and standard deviation of the above distribution? P(15<X<30) = binomialcdf(250,.1,30) – binomialcdf(250,.1,14) =.866 m = 25 & s = 4.743

4) What do you notice about the shape? 3) If we were to graph a histogram for the above binomial distribution, what shape do you think it will have? 4) What do you notice about the shape? Let’s graph this distribution – Put the numbers 1-45 in L1 In L2, use binomialpdf to find the probabilities. Since the probability is only 10%, we would expect the histogram to be strongly skewed right. Set window: xmin:-0.5 xmax: 45 xscl: 1 ymin:0 ymax:0.2 yscl:1 Input the following in your calculator. Graph: histogram L1: seq(x,x,0,45) xlist: L1 L2: binomialpdf(250,.1,L1) freq: L2 Overlay a normal curve on your histogram: In Y1 = normalpdf(X,m,s)

Normal distributions can be used to estimate probabilities for binomial distributions when: 1) the probability of success is close to .5 or 2) n is sufficiently large Rule: if n is large enough, then np > 10 & n(1 –p) > 10 Why 10?

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 (3s) from 0 and from n. (BVD, p. 334)

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.

Think about how discrete histograms are made 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 .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.(+.5 to discrete values) Why?

(Back to our example) Since P(preemie) =. 1 which is not close to (Back to our example) Since P(preemie) = .1 which is not close to .5, is n large enough? 5) 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) 6) How does the answer in question 6 compare to the answer in question 1 (Binomial answer =0.866)? np = 250(.1) = 25 & n(1-p) = 250(.9) = 225 Yes, Ok to use normal to approximate binomial  P(14.5 < X < 30.5) = Normalcdf(14.5,30.5,25,4.743) = .8635