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AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Objectives: • ESTIMATE areas (proportions of values) in a Normal distribution. • FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. • FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. • DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence. AP Stats ~ 2D: Standard Normal Distributions, and assessing normality All Normal distributions are the same if we measure in units of size σ from the mean µ as center. The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. N(0, 1) When we find z-scores (standardize a value), we are transforming our data so that it fits on a standard normal distribution. Remember that this allows us to compare distributions of different sizes, or different measures, etc. AP Stats ~ 2D: Standard Normal Distributions, and assessing normality TABLE A at the back of your book (pages T1 and T2 is called the STANDARD NORMAL TABLE. This is a table of areas under the standard normal curve. The entry in the table for each value of z gives you the area under the curve to the LEFT of z. (Remember percentiles are also the area to the left of the given value.) Loading... #⇐Ef!f ¥ AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A: Z - scores are the on outside edge Area inside Is on the . . AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Loading... AP Stats ~ 2D: Standard Normal Distributions, and assessing normality be * each number should listed and Labeled if you use normal cdf . AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Example: Suppose we have a Normal distribution, find the following proportions using Table A, then check them on your calculator. a) b) Find the proportion of observations to the left of z=-1.41. .org#q II. Table : Area .org = Normal cdf ( mints Max , = -1.41 , no , at ) = Find the proportion of observations less than 3.56. Table :D Normal cdf ( .998 Coftthetabkateotuhenatopggo , mm = - a , Max =3 56 . ,µ=o , 0=1 ) = xD . 99981 AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Example: Suppose we have a Normal distribution, find the following proportions using Table A, then check them on your calculator. c) The proportion of the distribution to the right of -2.13. ¥-4 d) Area =L area - Normal to the cdf ( min Left = - I = 2.13 , 0166 - . Max = a .ci# = ,µ=l , 0=0 ) = [email protected] The proportion of the distribution greater than .08. ' Area 88k€ = de Normal - It cdf ( = OR minMax = A ,µ=o , 0=1 ) = [email protected] AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Example: Suppose we have a Normal distribution, find the following proportions using Table A, then check them on your calculator. e) Find the proportion of observations between z=-.03 and 0<3*99-53) f) Area -_ 9953 . - . 3821=-61320 z=2.6. Find the proportion of observations between z=.23 and z=3.45. t.nl#ENormalcdf(mm=-.3,max=2.6,m=o,o=l)[email protected][email protected]*MgysN0rma1cdf(mM=.23,max=3.4s,n=o,[email protected] AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Example: The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are normally distributed with r=100 and =15. A school district classified children as “gifted” if their WISC score exceeds 135. There are 1300 sixth graders in the school district. About how many of them are gifted? o =X Z die = 135,4 - or = mm = 135 # of gifted Kids 0099113001=12.87 12 . T gifted students are = 1- . 9901 0099 = . - Normal cdf ( = Area 2.33 considered , = Max ,µ= a 100,0=15) = . 0098 AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Working Backwards: Sometimes we're given a percentile and we need to find the z-score. Loading... AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Examples: (double check your answers using the Normalcdf or invNorm functions on your calculator) a) Find the number z such that the proportion of observations that are less than z in a standard normal distribution is 0.98. mvNorm(area=98,µ=O,o=1 ) or .tn?afo3tabkatz=# -9M¥ Ynoektdrsast 2=2.0540 . b) Find the number z such that 22% of all observations from a standard distribution are greater than z. <¥ IM normal [email protected][email protected] orz=mvNorm( Look for -78 on the area -78 ,M= 0,0=11 AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Assessing the Normality of a Distribution: The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. A Normal probability plot provides a good assessment of whether a data set follows a Normal distribution. AP Stats ~ 2D: Standard Normal Distributions, and assessing normality METHOD 1: Use all of the methods from Chapter 1 in combination. 1. Construct a stemplot or histogram. Is it bell shaped and symmetric around the mean? 2. Find the mean and standard deviation. 3. Draw a normal distribution graph and mark the values at the mean, then at 1 and 2 standard deviations away. 4. Check the 68-95-99.7 rule. You can do this two ways standardize the data and find areas, or compare the relative frequency of the data in each section. NOTE: the smaller the set of data, the less likely it is to follow the rule exactly. You may have to make a judgment call then proceed with caution. AP Stats ~ 2D: Standard Normal Distributions, and assessing normality METHOD 2: Construct a NORMAL PROBABILITY PLOT on your calculator. If the graph looks linear, it means that our distribution is normal. If it’s a concave curve, the distribution is skewed left. If it’s a convex curve, the distribution is skewed right. Outliers will appear as points that are far away from the overall pattern. Step 1: Enter your data in list 1 Step 2: Define your statplot as below, then use Zoom 9:Stat to see the plot AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Example: The measurements listed below describe the usable capacity (in cubic feet) of a sample of 36 side-by-side refrigerators. Are the data close to normal? Show using both methods. 12.9 15.2 16.0 16.6 13.7 15.3 16.0 16.8 14.1 15.3 16.2 17.0 14.2 15.3 16.2 17.0 14.5 15.3 16.3 17.2 14.5 15.5 16.4 17.4 14.6 15.6 16.5 17.4 14.7 15.6 16.6 17.9 15.1 15.8 16.6 18.4 method [email protected] ' . # * z ÷÷mdIy¥M¥I¥Eiiµwn±¥MY" I:* I= 15.82 S×= 1.21 within ±2sy : 34/36=94.4 't : . : . . Within'=3s× 36/36=1001 : I iii. . Pretty linear . Approximately Normal . AP Stats ~ 2D: Standard Normal Distributions, and assessing normality Homework: page 129: #47-53, 55-67 odds, 72-74