A Day of Classy Review

Shifting and Scaling SAT/ACT Max SAT: 1600 (old school) Max ACT: 36 SAT = 40 x ACT + 150 ACT Summary Stats: Lowest = 19 Mean = 27 SD = 3 Q3 = 30 Median = 28 IQR = 6 Find equivalent SAT scores

SAT and ACT ACT Summary Stats: Lowest = 19 Mean = 27 SD = 3 Q3 = 30 Median = 28 IQR = 6 SAT = 40 x ACT + 150 𝐿𝑜𝑤𝑒𝑠𝑡 𝑆𝐴𝑇 =40 ∗19+150=910 𝑀𝑒𝑎𝑛 𝑆𝐴𝑇 =40 ∗27+150=1,230 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑆𝐴𝑇 =40∗3=120 𝑄3 𝑆𝐴𝑇 =40 ∗30+150=1,350 𝑀𝑒𝑑𝑖𝑎𝑛 𝑆𝐴𝑇 =40∗28+150=1,270 𝐼𝑄𝑅 𝑆𝐴𝑇 =40 ∗6=240

NormalCDF The NormalCDF( function finds the percent of the total area of the distribution that falls between two z-scores. For example, what would the NormalCDF(-1,1) be? (Hint: 68-95-99.7 rule)

NormalCDF Procedure First, determine the type of question being asked Once you’ve determined it is appropriate to use the NormalCDF function, convert given values into z-scores 𝑧= 𝑦− 𝑦 𝑆𝐷

NormalCDF Questions: Answers: What percent of the distribution falls between X and Y? What percent of the distribution is greater than Y? What percent of the distribution is less than X? Answers: NormalCDF(X,Y) NormalCDF(Y,99) NormalCDF(-99,X)

invNormal( Going the Other Way The invNormal( function finds what z-score would cut off that percent of the data Example: What z-score cuts off the top 10% in a Normal model? The bottom 20%? The trick here is figuring out what percent (as a decimal) to enter in to the function. Imagine invNormal calculates the area starting at -99. So to find the z-score that cuts off the top 10% we want the z-score that includes 90% invNormal (.9) = 1.28

invNormal( Example Based on the model N(1152, 84) describing angus steer weights, what are the cut-off values for A) highest 10% B) lowest 20% C) middle 40% A) invNormal(.9) = 1.28 B) invNormal(.2) = -.842 C) invNormal(.3) = -.524 invNormal(.7) = .524 1,259lbs 1,081lbs 1,108 – 1196lbs

New Topic – Normal Probability Plots How to decide when the normal model (unimodal and symmetric) is appropriate: Draw a picture (Histogram) Draw a picture! (Normal Probability Plot) A Normal Probability Plot is a plot of Normal Scores (z-scores) on the x-axis vs the units you were measuring on the y-axis (weights, miles per gallon, etc.) A unimodal and symmetric distribution will create a straight line

What it Looks Like

What it Looks Like

How It Works A Normal probability plot takes each data value and plots it against the z-score you would expect that point to have if the distribution were perfectly normal These are best done on a calculator or with some other piece of technology because it can be tricky to find what values to “expect”

Homework Page 136-141 # 3, 5, 12, 17, 28, 30