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The Normal Model Ch. 6 “All models are wrong – but some are useful.” -- George Box Approximating with the Normal model Nearly Normal Condition • The shape of the data’s distribution is roughly unimodal and symmetric. • Check this by making a histogram (or Normal probability plot) • YOU MUST CHECK THE NEARLY NORMAL CONDITION!!! Statistics vs. Parameters Statistics • Summaries of actual data Parameters • Values that specify the model • 𝜇 -- mean • -- standard deviation • N(𝜇, )—Normal model with mean 𝜇 and standard deviation Rescale the Data • Standardize the data so that the: – average is 0 – standard deviation is 1 – Convert the data to z-scores • Each value is converted to a Standard Score (zscore) Standard Score (z-score) 𝑦−𝜇 𝑧= 𝜎 The Normal Model The Normal Model 3 -2 -1 + 1 + 2 + 3 The 68-95-99.7 Rule a.k.a. The Empirical Rule 68% 95% 99.7% 3 -2 -1 + 1 + 2 + 3 Three Rules of Working with the Normal model 1. Make a picture 2. Make a picture 3. Make a picture The Setup The distribution of fuel efficiency of a particular vehicle is roughly unimodal and symmetric with mean 24 mpg and standard deviation 6 mpg. 1. 2. 3. 4. 5. 6. 7. 8. Check the Nearly Normal Condition Sketch the Normal model What percent of all cars get less than 15 mpg? Describe the fuel efficiency of the worst 20% of all cars. What percent of all cars get between 20 mpg and 30 mpg? What percent of cars get more than 40 mpg? What gas mileage represents the third quartile? Describe the gas mileage of the most efficient 5% of all cars.