Chapter 7 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 on or aboveIs always on or above the horizontal axis equal to oneHas an area exactly equal to one underneath it Often describes an overall distribution proportionsDescribe what proportions of the observations fall within each range of values

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

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

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

P(X < 2) & P(X < 2) 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) = Shape is a trapezoid – How long are the bases?.5( )(1)= ( )(2) =.5 b 2 =.375 b 1 =.5 h = 1

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

P(0.5 < X < 1.5) = ( )(.5) = (.5)(.25) =.125

Special Continuous Distributions

Uniform Distribution

Is a continuous distribution that is evenly (or uniformly) distributed Has a density curve in the shape of a rectangle Probabilities are calculated by finding the area under the curve Where: a & b are the endpoints of the uniform distribution How do you find the area of a rectangle?

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 the uniform distribution above. How long is this rectangle? What is the height of this rectangle? What shape does a uniform distribution have? 1/.12

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

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

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

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

c) What is the mean and standard deviation of this distribution?  = (5 + 40)/2 = 22.5   = (40 - 5) 2 /12 =  =

Normal Distributions Symmetrical bell-shaped (unimodal) density curve AboveAbove the horizontal axis N( ,  ) The transition points occur at  +  area under the curveProbability is calculated by finding the area under the curve increasesAs  increases, the curve flattens & spreads out decreasesAs  decreases, the curve gets taller and thinner How is this done mathematically?

Normal Distributions

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

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

Empirical Rule 68%Approximately 68% of the observations fall within  of  95%Approximately 95% of the observations fall within 2  of  99.7%Approximately 99.7% of the observations fall within 3  of 

Suppose that the height of male students at SHS 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? P(X > 73.5) = % =.32

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

Standard Normal Distribution A normal distribution with mean 0 and standard deviation 1, is called the standard (or standardized) normal distribution.

Normal Tables

Using the Normal Tables Find P(z < 0.46) Row labeled 0.4 Column labeled 0.06 P(z < 0.46) =

Using the Normal Tables Find P(z < -2.74) Row labeled -2.7 Column labeled 0.04 P(z < -2.74) =

Sample Calculations Using the Standard Normal Distribution Using the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfy each of the following: (a)P(z < 1.83) = (b)P(z > 1.83) = 1 – P(z < 1.83) = 1 – =

Using the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies each of the following: c)P(z < -1.83) = (d) P(z > -1.83) = 1 – P(z < -1.83) = 1 – =

Symmetry Property Notice from the preceding examples it becomes obvious that P(z > z*) = P(z < -z*) P(z > -2.18) = P(z < 2.18) =

Using the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies < z < 2.34, that is find P(-1.37 < z < 2.34). P(Z<2.34)= P(-1.37 < z < 2.34)= P(Z<-1.37)= =

P(0.54 < z < 1.61)= P(Z<1.61)= Using the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies 0.54 < z < 1.61, that is find P(0.54 < z < 1.61). P(Z<.54)= =

P(Z<-0.93)= P(-1.42 < z < -0.93)= P(Z<-1.42)= = Using the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfy < z < -0.93, that is find P(-1.42 < z < -0.93).

Using the standard normal tables, in each of the following, find the z values that satisfy : The closest entry in the table to is corresponding to a z value of 2.05 (a)The point z with 98% of the observations falling below it.

Using the standard normal tables, in each of the following, find the z values that satisfy : (b)The point z with 90% of the observations falling above it. The closest entry in the table to is corresponding to a z value of -1.28

Standard Normal Distribution Revisited This is called the standard normal distribution. If a variable X has a normal distribution with mean  and standard deviation , then the standardized variable has the normal distribution with mean 0 and standard deviation 1.

What is your z-score on a test that you scored an 86% on that had a mean score of 80% with a standard deviation of 5?

Strategies for finding probabilities or proportions in normal distributions 1.State the probability statement 2.Draw a picture 3.Calculate the z-score 4.Look up the probability (proportion) 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 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

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) = =.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%? P(X > ?) = Look up in table 0.95 to find z- score 1.645

The heights of the female students at SHS 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) = What is the z- score for the 63? -0.9

The heights of female teachers at SHS 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  = 4.5 &  =

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? ONLYNormalpdf – 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

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

Consider a random sample with n = 5. To find the appropriate z-scores for a sample of size 5, divide the standard normal curve into 5 equal-area regions. Why are these regions not the same width?

Consider a random sample with n = 5. Next – find the median z-score for each region Why is the median not in the “middle” of each region? These would be the z-scores (from the standard normal curve) that we would use to plot our data against.

Let’s construct a normal probability plot. The values of the normal scores depend on the sample size n. The normal scores when n = 10 are below: 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?

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?