26.1 Normal DistributionsMany continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables.The theoretical curve, called the bell curve or the Gaussian distribution, can be used to study many variables that are not normally distributed but are approximately normal.Bluman, Chapter 6
3Normal DistributionsThe mathematical equation for the a normal distribution is:Bluman, Chapter 6
4Normal DistributionsThe shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation.Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation.Bluman, Chapter 6
5Normal Distributions - different ones for different values of 𝜇 and 𝜎 Bluman, Chapter 6
6Normal Distribution Properties The normal distribution curve is bell-shaped.The mean, median, and mode are equal and located at the center of the distribution.The normal distribution curve is unimodal (i.e., it has only one mode).The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.Bluman, Chapter 6
7Normal Distribution Properties The curve is continuous—i.e., there are no gaps or holes. For each value of X, here is a corresponding value of Y.The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.Bluman, Chapter 6
8Normal Distribution Properties They said “it never meets the x axis—but it gets increasingly closer.”Example: for the standard normal distribution where 𝜇=0,𝜎=1, when 𝑥=5,𝑦=And when 𝑥=10,𝑦=7.7× 10 −23
9Normal Distribution Properties The total area under the normal distribution curve is equal to 1.00 or 100%.The area under the normal curve that lies withinone standard deviation of the mean is approximately 0.68 (68%).two standard deviations of the mean is approximately 0.95 (95%).three standard deviations of the mean is approximately ( 99.7%).Bluman, Chapter 6
10Normal Distribution Properties Bluman, Chapter 6
12Standard Normal Distribution Since each normally distributed variable has its own mean and standard deviation, the shape and location of these curves will vary. In practical applications, one would have to have a table of areas under the curve for each variable. To simplify this, statisticians use the standard normal distribution.The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.Bluman, Chapter 6
13The Standard Normal Distribution is our favorite This one is the most special of all of themThe mean: 𝜇=0The standard deviation: 𝜎=1Horizontal 𝑧=…,−3,−2,−1, 0, 1, 2, 3,⋯Total area between curve and axis = 1.000
14z value (Standard Value) The z value is the number of standard deviations that a particular X value is away from the mean. The formula for finding the z value is:Bluman, Chapter 6
15x value Going the other way: If you know the z value and you need to find the x value,𝑥=𝑧∙𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛+𝑚𝑒𝑎𝑛𝑥=𝑧∙𝜎+𝜇6
16Area Problems KEY CONNECTION: PROBABILITY is AREA AREA is PROBABILITY HOW THIS AFFECTS YOUR LIFE:You want to answer Probability questionsYou find the answers by finding Areas of regions underneath the Normal Curve.
17Three kinds of area problems “What is the area to the left of some 𝑧 ?”“What is the area to the right of some 𝑧 ?”“What is the area between 𝑧 1 and 𝑧 2 ?”
18How do you find these areas? By using a printed table that gives areas to the left of 𝑧=−#.## and 𝑧=#.##Or by using the TI-84 normalcdf(z1,z2) function.Or by using Excel’s =NORM.S.DIST(z,TRUE) function
19Area under the Standard Normal Distribution Curve 1. To the left of any z value:Look up the z value in the table and use the area given.Bluman, Chapter 6
20Area under the Standard Normal Distribution Curve 2. To the right of any z value:Look up the z value and subtract the area from 1.Bluman, Chapter 6
21Area under the Standard Normal Distribution Curve 3. Between two z values:Look up both z values and subtract the corresponding areas.Bluman, Chapter 6
22TI-84 Methods To get normalcdf() Area between two z vals. 2ND DISTR It’s on the VARS key2:normalcdf(Beware! You do NOT want to use 1:normalpdf( in most cases in this course.Normalcdf(zLow, zHigh)Example: Area between 𝑧=−1 and 𝑧=+1Agrees with the Empirical Rule (68/95/99.7) !
23TI-84 Methods Area to the left of z Area to the right of z normalcdf(-99,z)Example: Area to the left of z=1.23normalcdf(z,99)Example: Area to the right of z=1.23Observe: area to left + area to right =Not a coincidence!!!!
24Example 6-1: Area under the Curve Find the area to the left of z = 1.99.The value in the 1.9 row and the .09 column of Table E is The area isBluman, Chapter 6
25Example 6-1: Area under the Curve Find the area to the left of z = 1.99.They got using the printed table.
26Example 6-2: Area under the Curve Find the area to right of z =The value in the -1.1 row and the .06 column of Table E is The area is =Bluman, Chapter 6
27Example 6-2: Area under the Curve Find the area to right of z =They used the printed table, which only gives areas to the left, so they had to subtract,=The TI-84 normalcdf() was more direct.
28Example 6-3: Area under the Curve Find the area between z = 1.68 and z =The values for z = 1.68 is and for z = is The area is =Bluman, Chapter 6
29Example 6-3: Area under the Curve Find the area between z = 1.68 and z =With the printed tables, they had to do two lookups and subtract the results to get .8682The TI-84 normalcdf() was more direct.Bluman, Chapter 6
30Problems that work backwards They give you the areaYou have to work backwards to find the z score.
31Example of a backwards problem What z value divides the area under the standard normal curve so that the area to the left of that z is ? And what is the area to the right of that z score?
32How to solve it using the printed table The area to the left is Then look for that value inside Table E.The z value is 0.56.Bluman, Chapter 6
33Backwards problem using TI-84 invNorm( is the toolExample: area2ND DISTR again3:invNorm(Stands for “Inverse Normal”Tell him area to the leftHe responds with the z score.
34If they give you area to the right Example: Find z so area to the right of z isHow to solve itIn table:Lookup in table.If it’s not there, take the closest value.Read out to find z again.With TI-84 invNorm(But Tables & TI-84 deal with areas to the left.COMPLEMENT: If area to the right is , are to the left is 1 –So we seek the z that has to its left.
35Backwards Area-Between “Find the z scores that delimit the middle 80% of the area” DRAW A PICTURE!!!is in the middleSo – = in two tails÷ 2 = in each tail
36Backwards Area-Between Using Printed TableUsing TI-84Look deep in table for closest match toRead out to find the negative z on the left.Because of symmetry, the positive z on the right is the opposite of that value.invNorm(.1000)Because of symmetry, the positive z on the right is the opposite of that value.Answers: 𝑧 = −1.28 and 𝑧 = +1.28
37Confirming this with TI-84 Using normalcdf()Using DRAWnormalcdf(-1.28,1.28) should give about area they asked forMore decimals for more precision2ND DRAW 1:ClrDraw2ND DISTRRight arrow to DRAW1:ShadeNorm(z1,z2)
38Confirming this with TI-84 WINDOW settingsUsing DRAWIt probably won’t turn out well on its own. You may need to do some thinking.What we did for this one:2ND DRAW 1:ClrDraw2ND DISTRRight arrow to DRAW1:ShadeNorm(z1,z2)