The Normal Model Chapter 6 Density Curves and Normal Distributions

A Density Curve: Is always on or above the x axis Has an area of exactly 1 between the curve and the x axis Describes the overall pattern of a distribution The area under the curve above any range of values is the proportion of all the observations that fall in that range.

Mean vs Median The median of a density curve is the equal area point that divides the area under the curve in half The mean of a density function is the center of mass, the point where curve would balance if it were made of solid material

Normal Curves Bell shaped, Symmetric,Single-peaked Mean = µ Standard deviation = Notation N(µ, ) One standard deviation on either side of µ is the inflection points of the curve

Rule 68% of the data in a normal curve at least is within one standard deviation of the mean 95% of the data in a normal curve at least is within two standard deviations of the mean 99.7% of the data in a normal curve at least is within three standard deviations of the mean

Why are Normal Distributions Important? Good descriptions for many distributions of real data Good approximation to the results of many chance outcomes Many statistical inference procedures are based on normal distributions work well for other roughly symmetric distributions

Standard Normal Curve Chapter 6

Standardizing (z-score) If x is from a normal population with mean equal to µ and standard deviation,  then the standardized value z is the number of standard deviations x is from the mean Z = (x - µ)/  The unit on z is standard deviations

Standard Normal Distribution A normal distribution with µ = 0 and  = 1, N(0,1) is called a Standard Normal distribution Z-scores are standard normal where z=(x-µ)/ 

Standard Normal Tables Table A in the front of your book has the areas to the left of given cut-off points Find the 1st 2 digits of the z value in the left column and move over to the column of the third digit and read off the area. To find the cut-off point given the area, find the closest value to the area ‘inside’ the chart. The row gives the first 2 digits and the column give the last digit

Solving a Normal Proportion State the problem in terms of an x variable in the context of the problem Draw a picture and locate the required area Standardize the variable using z =(x-µ)/  Use the calculator/table and the fact that the total area under the curve = 1 to find the desired area

Finding a Cutoff Given the Area State the problem in terms of x and area Draw a picture and shade the area Use the table to find the z value with the desired area Unstandardize the z value using z =(x-µ)/  and solving for x

Assessing Normality In order to use the previous techniques the population must be normal Method 1 for assessing normality :  Construct a stem plot or histogram and see if the curve is bell shaped and symmetric around the mean

Assessing Normality: Revisited Normality Probably Plot on calculator Enter data into a list Stat Plot select graph type number 6 Specify the correct list and the data axis as x Zoom Stat If the graph is nearly linear the distribution is nearly normal

TI–83/84 Commands Normal Curves Chapter 6

Graphing Normal Curves Setting the window is NOT automatic. The user (you) must set an appropriate window Window settings suggestions  X[mean – 4 (Std dev), mean + 4(Std dev)]  Xscale = Std dev  Y[–.01,.02]  Yscale =.01

ShadeNorm(xmin,xmax,mean,sd ) Draws the graph and returns the proportion of the data in a normal distribution that is between xmin and xmax Xmin is the smallest x value in the range  Xmin = –10 for ‘less than’, ‘no more than’ Xmax is the largest x value in the range  Xmax = 10 for ‘at least’, ‘greater than’ Mean = mean of the distribution Sd = standard deviation of distribution

Normalcdf(xmin,xmax) Returns the proportion of the data in a Standard Normal distribution that is between xmin and xmax Xmin is the smallest x value in the range  Xmin = –10 for ‘less than’, ‘no more than’ Xmax is the largest x value in the range  Xmax = 10 for ‘at least’, ‘greater than’

invNorm(prob) Returns, in a standard normal distribution, the cutoff point that has an area of prob to the left. Prob = the proportion to the left of the cutoff where 0≤prob≤1 If prob <.5 then the cutoff returned will be negative