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

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

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

c) What is the mean and standard deviation of this distribution?

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 CHS 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 CHS 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 CHS 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 Y=normalpdf(x,µ,σ) Normalcdf – will find probability of area from lower bound to upper bound (if starting from left, lower bound = -1x1099 ) Normalcdf(lower,upper,µ,σ) Invnorm (inverse normal) – will find z-score from probability number Z-score = Invnorm(probability)

Ways to Assess Normality Use graphs (dotplots, boxplots, or histograms) Use the Empirical Rule Normal probability (quartile) plot on calculator

Normal Probability (Quartile) plots The observation (x) is plotted against known normal z-scores If the points on the quartile plot lie close to a straight line, then the data is normally distributed Deviations from a straight line on the quartile 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 Both the histogram & boxplot are approximately symmetrical, so these data are approximately normal.

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? To create a Normal Probability plot, enter the data into L1 and then choose the 6th graph type in STATPLOT. L1 is x-axis and y-axis is theoretical z-scores for normal distribution The normal probability plot is approximately linear, so these data are approximately normal.