Ch 2 – The Normal Distribution YMS – 2.1 Density Curves and the Normal Distributions

Vocabulary Mathematical Model Density Curve An idealized description of a distribution Density Curve Is always on or above the horizontal axis Has area = 1 underneath it Can roughly locate the mean, median and quartiles, but not standard deviation Mean is “balance” point while median is “equal areas” point.

Reminder: Exploring Data on a Single Quantitative Variable Always plot your data Identify socs Calculate a numerical summary to briefly describe center and spread Describe overall shape with a smooth curve Label any outliers

Next 2 classes – Fathom Activity and Sketching WS Greek Notation Population mean is μ and population standard deviation is σ These are for idealized distributions (population vs. sample) Classwork p83 #2.1 to 2.5 Next 2 classes – Fathom Activity and Sketching WS

Activity: Beauty and the Geek More Vocabulary Normal Curves Are symmetric, single-peaked and bell-shaped They describe normal distributions Inflection point Point where change of curvature takes place Could use this to estimate standard deviation

3 Reasons for Using Normal Distributions 1. They are good descriptions for some distributions of real data. 2. They are good approximations to the results of many kinds of chance outcomes. 3. Many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions.

The 68-95-99.7 Rule In N(μ, σ), rule gives percent of data that falls within 1, 2, and 3 standard deviations, respectively. AKA Empirical Rule Classwork p89 #2.6-2.9 Homework p90 #2.12, 2.14, 2.18 and 2.2 Reading Blueprint

Sketch a bell curve for each of the following: p(x < a ) = 0.5 p(x > b) = 0.5 p(x < c) = 0.8 p(x < d) = 0.2 p(x > e) = 0.05 p(x > f) = .95

Standard Normal Calculations YMS – 2.2 Standard Normal Calculations

Standards Standard Normal Distribution Standardized value of x (z-score) Data point minus mean divided by standard deviation Gives you the number of standard deviations the data point is from the mean

Table A Left Column has ones.tenths digit Top Row has 0.0hundreths digit LEFT COLUMN + TOP ROW = Z-SCORE Area is always to the LEFT of the z-score

TI-83 Plus Keystrokes 2nd DISTR 1: normalpdf Finds height of density curve at designated point We won’t be using this 2: normalcdf(lower limit, upper limit, mean, st. dev.) Gives area under the curve to left or right of a point 3:invNorm(area, mean, standard deviation) *When you don’t enter a mean or standard deviation, it assumes it is the Normal Distribution (0, 1)

In Class Exercises p95 #2.19-2.20 Homework p103 #2.21-2.25

Activity: Grading Curves WS Normal Probability Plots (NPP) Is a plot of z-scores vs. data values Use the calculator! If it’s a straight line, the data is normally distributed. How else do we assess normality?

In Class Exercises (Next 3 days) Shape of Distributions WS p108 #2 In Class Exercises (Next 3 days) Shape of Distributions WS p108 #2.27 p113 #2.41-2.42, 2.46-2.47, 2.51-2.52, 2.54 AP Practice Packet