Ch 2 – The Normal DistributionCh 2 – The Normal Distribution

YMS – 2.1YMS – 2.1

Density Curves and Density Curves and

the Normal Distributionsthe Normal Distributions

VocabularyVocabulary Mathematical Model Mathematical Model

An idealized description of a distributionAn idealized description of a distribution Density CurveDensity Curve

Is always on or above the horizontal axis Is always on or above the horizontal axis Has area = 1 underneath itHas area = 1 underneath it Can roughly locate the mean, median and Can roughly locate the mean, median and

quartiles, but not standard deviationquartiles, but not standard deviation Mean is “balance” point while median is Mean is “balance” point while median is

“equal areas” point.“equal areas” point.

Reminder: Exploring Data on a Reminder: Exploring Data on a Single Quantitative Variable Single Quantitative Variable

Always plot your dataAlways plot your data Identify socs Identify socs

Calculate a numerical summary to Calculate a numerical summary to briefly describe center and spreadbriefly describe center and spread

Describe overall shape with a smooth Describe overall shape with a smooth curvecurve

Label any outliers Label any outliers

Greek NotationGreek Notation Population mean is μ and population Population mean is μ and population

standard deviation is σstandard deviation is σ These are for idealized distributions These are for idealized distributions

(population vs. sample) (population vs. sample)

Classwork p83 #2.1 to 2.5Classwork p83 #2.1 to 2.5

Next 2 classes – Fathom Activity and Next 2 classes – Fathom Activity and Sketching WSSketching WS

Activity: Beauty and the Activity: Beauty and the GeekGeek

More VocabularyMore Vocabulary Normal Curves Normal Curves

Are symmetric, single-peaked and bell-Are symmetric, single-peaked and bell-shapedshaped

They describe normal distributionsThey describe normal distributions Inflection point Inflection point

Point where change of curvature takes place Point where change of curvature takes place Could use this to estimate standard deviationCould use this to estimate standard deviation

3 Reasons for Using3 Reasons for UsingNormal Distributions Normal Distributions

1. They are good descriptions for some 1. They are good descriptions for some distributions of distributions of real datareal data..

2. They are good approximations to the 2. They are good approximations to the results of many kinds of results of many kinds of chance chance outcomes.outcomes.

3. Many 3. Many statistical inferencestatistical inference procedures procedures based on normal distributions work well based on normal distributions work well for other roughly symmetric distributions. for other roughly symmetric distributions.

The 68-95-99.7 Rule The 68-95-99.7 Rule

In N(μ, σ), rule gives percent of data that In N(μ, σ), rule gives percent of data that falls within 1, 2, and 3 standard falls within 1, 2, and 3 standard deviations, respectively.deviations, respectively.

AKA Empirical RuleAKA Empirical Rule

Classwork p89 #2.6-2.9Classwork p89 #2.6-2.9

Homework p90 #2.12, 2.14, 2.18 Homework p90 #2.12, 2.14, 2.18

and 2.2 Reading Blueprintand 2.2 Reading Blueprint

Sketch a bell curve for each Sketch a bell curve for each of the following:of the following:

p(x < a ) = 0.5p(x < a ) = 0.5 p(x > b) = 0.5p(x > b) = 0.5 p(x < c) = 0.8p(x < c) = 0.8 p(x < d) = 0.2p(x < d) = 0.2 p(x > e) = 0.05p(x > e) = 0.05 p(x > f) = .95p(x > f) = .95

YMS – 2.2YMS – 2.2

Standard Normal CalculationsStandard Normal Calculations

StandardsStandards

Standard Normal DistributionStandard Normal Distribution N(0, 1)N(0, 1)

Standardized value of x (z-score)Standardized value of x (z-score) Data point minus mean divided by Data point minus mean divided by

standard deviation standard deviation Gives you the number of standard Gives you the number of standard

deviations the data point is from the deviations the data point is from the meanmean

Table A Table A

Left Column has Left Column has ones.tenthsones.tenths digit digit Top Row has Top Row has 0.0hundreths 0.0hundreths digitdigit LEFT COLUMN + TOP ROW = Z-LEFT COLUMN + TOP ROW = Z-

SCORESCORE

Area is always to the LEFT of the z-Area is always to the LEFT of the z-scorescore

TI-83 Plus TI-83 Plus KeystrokesKeystrokes

22ndnd

DISTRDISTR 1: normalpdf1: normalpdf

Finds height of density curve at designated pointFinds height of density curve at designated point We won’t be using thisWe won’t be using this

2: normalcdf(lower limit, upper limit, mean, st. 2: normalcdf(lower limit, upper limit, mean, st. dev.)dev.)

Gives area under the curve to left or right of a pointGives area under the curve to left or right of a point 3:invNorm(area, mean, standard deviation)3:invNorm(area, mean, standard deviation)

*When you don’t enter a mean or standard *When you don’t enter a mean or standard deviation, it assumes it is the Normal Distribution deviation, it assumes it is the Normal Distribution

(0, 1)(0, 1)

In Class ExercisesIn Class Exercises

p95 #2.19-2.20p95 #2.19-2.20

HomeworkHomework

p103 #2.21-2.25p103 #2.21-2.25

Activity: Grading Curves WSActivity: Grading Curves WS

Normal Probability Plots Normal Probability Plots (NPP) (NPP)

Is a plot of z-scores vs. data valuesIs a plot of z-scores vs. data values Use the calculator!Use the calculator!

If it’s a straight line, the data is If it’s a straight line, the data is normally distributed.normally distributed. How else do we assess normality?How else do we assess normality?

In Class ExercisesIn Class Exercises(Next 3 days)(Next 3 days)

Shape of Distributions WSShape of Distributions WSp108 #2.27p108 #2.27

p113 #2.41-2.42, 2.46-2.47,p113 #2.41-2.42, 2.46-2.47,2.51-2.52, 2.54 2.51-2.52, 2.54

AP Practice PacketAP Practice Packet