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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 68-95-99.7 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