Continuous Random Variables

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?

Special Continuous Distributions

How do you find the area of a rectangle? 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 How do you find the area of a rectangle? Where: a & b are the endpoints of the uniform distribution

What shape does a uniform distribution have? 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. Construct the uniform distribution above. What shape does a uniform distribution have? What is the height of this rectangle? How long is this rectangle? 1/.12 4.98 5.04 4.92

What is the length of the shaded region? What is the probability that a randomly selected bag will weigh more than 4.97 pounds? P(X > 4.97) = .07(1/.12) = .5833 What is the length of the shaded region? 4.98 5.04 4.92 1/.12

What is the length of the shaded region? Find the probability that a randomly selected bag weighs between 4.93 and 5.03 pounds. What is the length of the shaded region? P(4.93<X<5.03) = .1(1/.12) = .8333 4.98 5.04 4.92 1/.12

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

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 or use calculator.

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

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) Use the Empirical Rule 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