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 < O i ) = i/(n+1) °i.e. solve for O i 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 non- normality.