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Chapter 6 The Normal Distribution and Other Continuous Distributions 6.1: Continuous Probability Distributions • Continuous Random Variables – If X is a continuous RV, then P(X=a) = 0, where “a” is any individual unique value – Because X has individual unique values – P(a X b) = “something nonzero” where “a” to “b” represents an interval • Normal is most important continuous probability distribution. 6.2: Normal Distribution • Also known as “Gaussian Distribution” • Works close enough for a lot of continuous RVs. • Works close enough for a few discrete RVs. • Necessary for our inferential statistics. • Bell-shaped and symmetric. • All measures of central tendency are equal. • In theory, X is continuous and unbounded. Normal RV • Probabilities for discrete RV were given by a probability distribution function. • Probabilities for continuous RV are given by a probability DENSITY function (pdf). • Normal pdf requires you to know two parameters to find probabilities: and . Finding Normal Probabilities • Equation 6.1: fun but not useful. • Like to have a table for each combination of and . – Can’t. • Generate 1 table that can be used by everyone. – Get everyone to convert or transform data so that it works with that one table! – Transform X into Z 6.3: Evaluating Normality • The assumption of Normality is made all the time: sometimes correctly so, and sometimes incorrectly so. • Said another way: not all continuous random variables are normally distributed. Checking Normality • Text discusses two ways in this section (other ways discussed in Stat 2!) 1 Compare what you know about the data to what you know about the normal distribution. 2 Construct a normal probability plot. Comparing actual data to theory • Central tendency: actual data mean, median, and mode should be similar. • Variability: – Is the interquartile range about equal to 1.33*the standard deviation? – Is the range about equal to 6 times the standard deviation? Comparing actual data to theory • Shape: – plot the data and check for symmetry. – check to determine if the Empirical Rule applies. • Sometimes samples are small--is the data non-normal or do you have a non-representative sample? Normal Probability Plot • Best left to software. • The straighter the line, the better the sample approximates a normal distribution. • Systematic deviation from a straight line indicates non-normality. Plot Construction • Order the data • Use inverse normal scores transformation to find the standardized normal quantile for each data point. ° P(Z < Oi) = i/(n+1) ° i.e. solve for Oi for the 1st data point and the second data point, etc. Plot Construction (cont.) • Plot the data points: – actual values on the Y axis – Standardized Normal Quantiles on the X axis • A straight line demonstrates normality. • A non-straight line demonstrates nonnormality.