The distribution of heights of adult The distribution of heights of adult American men is approximately normal American men is approximately normal with mean 69 inches and standard with mean 69 inches and standard deviation 2.5 inches. Use the 68-95-deviation 2.5 inches. Use the 68-95-99.7 rule to answer the following 99.7 rule to answer the following questions:questions: What percent of men are taller than 74 What percent of men are taller than 74

inches?inches? Between what heights do the middle 95% Between what heights do the middle 95%

of men fall?of men fall? What percent of men are shorter than 66.5 What percent of men are shorter than 66.5

inches?inches? A height of 71.5 inches corresponds to what A height of 71.5 inches corresponds to what

percentile of adult male American heights?percentile of adult male American heights?

2.22.2More on Normal DistributionsMore on Normal Distributions

andandStandard Normal CalculationsStandard Normal Calculations

StandardizingStandardizing The standardized value The standardized value

is called a z-score. x is is called a z-score. x is the given value.the given value.

This tells you how many This tells you how many standard deviations you standard deviations you are from the mean.are from the mean.

This also allows you to This also allows you to find the percent of data find the percent of data under a given part of under a given part of the curve.the curve.

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Who’s Taller? (relatively Who’s Taller? (relatively speaking)speaking)

Verne is 67” tall. Assume the heights Verne is 67” tall. Assume the heights of women her age are normally of women her age are normally distributed with a mean distributed with a mean μμ = 64 inches = 64 inches and standard deviation and standard deviation σσ = 2.5 inches. = 2.5 inches.

Hank is 72” tall. Assume the heights of Hank is 72” tall. Assume the heights of men his age are normally distributed men his age are normally distributed with a mean with a mean μμ = 69.5 inches and = 69.5 inches and standard deviation standard deviation σσ = 2.25 inches. = 2.25 inches.

ExampleExample

The score for each student on a quiz The score for each student on a quiz is determined and a histogram is is determined and a histogram is created from the data. It is bell-created from the data. It is bell-shaped and symmetric with a mean shaped and symmetric with a mean of 80 and standard deviation of 10. of 80 and standard deviation of 10. Interpret a z-score for a student who Interpret a z-score for a student who scored a 87 on the quiz. scored a 87 on the quiz.

Standardized Normal CurvesStandardized Normal Curves Recall our formula for standardizing normal Recall our formula for standardizing normal

curves.curves.

Since any normal curve can be standardized, Since any normal curve can be standardized, we can find areas under the curve using one we can find areas under the curve using one table, Table A. This table is found in the table, Table A. This table is found in the front of your book or in your folder.front of your book or in your folder.

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Table ATable A

It is very important to remember that It is very important to remember that Table A gives the area under the Table A gives the area under the curve to the LEFT!!!curve to the LEFT!!!

Also, standardized normal curves Also, standardized normal curves have a mean of 0 and a standard have a mean of 0 and a standard deviation of 1.deviation of 1.

Reading Table AReading Table A

Use Table A to find the proportion of Use Table A to find the proportion of observations that have a z-score less observations that have a z-score less than 1.4 (this is 1.4 standard than 1.4 (this is 1.4 standard deviations from the mean).deviations from the mean).

Find the ones and the tenths digits in this column.

Find the hundredths digit across the top of the table. In this case, the hundredths digit is 0.

The answer is the intersection: .9192

P(Z<1.4)=.9192

Reading Table A:Reading Table A:“Greater Than” Problems“Greater Than” Problems

Use Table A to find the proportion of Use Table A to find the proportion of observations observations greater thangreater than a z-score a z-score of -2.15.of -2.15.

Table A gives us .0158 for the area to the LEFT. We want the area to the RIGHT (greater than -2.15), so subtract from 1.

P(Z>-2.15) = 1-.0158 = .9842

Steps in Finding Normal Steps in Finding Normal ProportionsProportions

Step 1: Step 1: DrawDraw a picture of the distribution and a picture of the distribution and shade the area of interest . shade the area of interest . Label Label the curve the curve with the values given (center and important with the values given (center and important points).points).

Step 2: Standardize x by using the formula.Step 2: Standardize x by using the formula.

Label your picture with the standardized values.Label your picture with the standardized values. Step 3: Use Table A to find the area under the Step 3: Use Table A to find the area under the

curve.curve. Step 4: State your conclusion Step 4: State your conclusion in wordsin words in the in the

context of the problem.context of the problem.

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Now to the actual Now to the actual problems…problems…

A commonly used IQ “cut-off” score A commonly used IQ “cut-off” score for AIG identification is 125. IQ for AIG identification is 125. IQ scores on the WISC-IV are normally scores on the WISC-IV are normally distributed with a mean = 100 and a distributed with a mean = 100 and a standard deviation = 15. Find the standard deviation = 15. Find the proportion of people whose IQ score proportion of people whose IQ score is at least 125.is at least 125.

““Between” ProblemsBetween” Problems

IQs between 140 and 170 are IQs between 140 and 170 are commonly referred to as commonly referred to as “moderately profoundly gifted.” “moderately profoundly gifted.” What proportion of the population What proportion of the population have IQ scores between 140 and have IQ scores between 140 and 170?170?

Working BackwardsWorking Backwards

Scores on the SAT Verbal Scores on the SAT Verbal approximately follow the approximately follow the NN(505,110) (505,110) distribution. How high must a distribution. How high must a student score to be in the top 10% of student score to be in the top 10% of all students taking the SAT?all students taking the SAT?

Caution about Test ItemsCaution about Test Items

Many test items Many test items ask students to ask students to distinguish distinguish between types between types of density of density curves. Once curves. Once the hear the the hear the word, students word, students have a have a tendency to call tendency to call everything everything “normal.” Be “normal.” Be careful!careful!

Density Curves

Symmetric

Unimodal

Bell Shaped

Normal Curves

68-95-99.7 Rule

Skewed

Bimodal – Two peaks… Looks

kinda like a camel

Just because a curve is

symmetric, has one peak and is bell shaped does not mean it is a

NORMAL curve!!!

What if they don’t tell me What if they don’t tell me whether the data are from a whether the data are from a

normal population?normal population?If you’re given the data, you have If you’re given the data, you have

several ways to assess normality.several ways to assess normality.

Start by looking at a histogram, Start by looking at a histogram, stemplot, dotplot, or box-and-whisker stemplot, dotplot, or box-and-whisker plot. Does the data appear plot. Does the data appear symmetrical, with most of the data symmetrical, with most of the data being near the center?being near the center?

And of course there is always And of course there is always the Empirical Rule…the Empirical Rule…

Another method is to check the 68-95-Another method is to check the 68-95-99.7 rule. First, find the mean and 99.7 rule. First, find the mean and standard deviation. Then count what standard deviation. Then count what percent of the observations fall percent of the observations fall within one standard deviation of the within one standard deviation of the mean. Is it close to 68%? Repeat for mean. Is it close to 68%? Repeat for 2 and 3 standard deviations away 2 and 3 standard deviations away from the mean.from the mean.

Normal Probability PlotsNormal Probability Plots

Another (and easier ) method is to Another (and easier ) method is to construct a normal probability plot construct a normal probability plot using your calculator.using your calculator.

If the plot is If the plot is approximatelyapproximately linear, it linear, it is safe to assume the data are from a is safe to assume the data are from a normal distribution.normal distribution.

Constructing Normal Prob. Constructing Normal Prob. PlotsPlots

Type your data in your calculator (it Type your data in your calculator (it is probably already there, because I is probably already there, because I know you have looked at your know you have looked at your histogram or box-and-whisker plot!).histogram or box-and-whisker plot!).

Go to StatPlot. Choose the last graph Go to StatPlot. Choose the last graph option. This represents Normal option. This represents Normal Probability.Probability.

Lets see what we can come up Lets see what we can come up with!with!

Let’s look at page 133 #64.Let’s look at page 133 #64.

HomeworkHomework

Chapter 2 #53-56, 60Chapter 2 #53-56, 60