M11-Normal Distribution 1 1  Department of ISM, University of Alabama, Lesson Objective  Learn the mechanics of using the table for the Normal Distribution.  Given a region for a variable that follows the Normal Distribution, find the probability that a randomly selected item will fall in this region.  Given a probability, find the region for a normally distributed variable that corresponds to this probability.

M12-Normal Distribution 2 2  Department of ISM, University of Alabama, The Normal Distributions a.k.a., “The Bell Shaped Curve” Describes the shape for some quantitative, continuous random variables.

M12-Normal Distribution 2 3  Department of ISM, University of Alabama,  = mean determines the location.  = standard deviation determines spread, variation, scatter. Normal Population Distribution has two parameters:

M12-Normal Distribution 2 4  Department of ISM, University of Alabama, X ~ N(  = 66,  = 9) or N(66, 9) Z = the number of standard deviations that an X - value is from the mean. Notation: X -   Z = Z ~ N(  = 0,  = 1 ) or N(0,1) Z follows the “Standard Normal Distribution”

M12-Normal Distribution 2 5  Department of ISM, University of Alabama, ____, ±1  ______, ±2  ______, ±3  Empirical Rule of the Normal Distribution Where does this come from?

M12-Normal Distribution 2 6  Department of ISM, University of Alabama, Recall The “area” under the curve within a range of X values is equal to proportion of the population within that range of X values. Question: How do we compute “areas”? Geometry formulas Calculus (integration) Tables Excel Minitab

M12-Normal Distribution 2 7  Department of ISM, University of Alabama, Reading the Standard Normal Table (finding areas under the normal curve) Step 1 for all problems: DTDP

M12-Normal Distribution 2 8  Department of ISM, University of Alabama, Table gives P(0 < z < ?) = Find P(0 < z < 1.72) = Up to the 1 st decimal place 2 nd decimal place.4573 Standard Normal Table

M12 Normal Distribution 2 9  Department of ISM, University of Alabama, P(-1.23 < Z < 2.05) = ? 0Z 2.05 What proportion of Z values are between –1.23 and +2.05? P(1.23 < Z < 2.05) = ? 0Z What proportion of Z values are between and +2.05? = = = = ?

M12 Normal Distribution 2 10  Department of ISM, University of Alabama, P( 10.0 < X < 15.72) = ? Z P( 0 < Z < 1.43) Weights of packages are normally distributed with mean of 10 lbs. and standard deviation of 4.0 lbs. Find the proportion of packages that weigh between 10 and lbs. 10X Z = – = Z = 10.0 – = X = weight of packages. X ~ N(  = 10,  = 4.0) =

M12 Normal Distribution 2 11  Department of ISM, University of Alabama, P( X > 15.72) = ? 0Z 1.43 P( Z > 1.43) Same situation. What proportion of packages weigh more than lbs? 10X Z = – = 1.43 X = weight of packages. X ~ N(  = 10,  = 4.0) = = ?

M12 Normal Distribution 2 12  Department of ISM, University of Alabama, P( X < 14.2) = ? 0Z P( Z < 1.05) Same situation. What proportion of packages weigh less than 14.2 lbs? 10X 14.2 Z = 14.2 – = 1.05 X = weight of packages. X ~ N(  = 10,  = 4.0) = =

M12-Normal Distribution 2 13  Department of ISM, University of Alabama, Same situation. What proportion of packages weigh between 5.08 and 18.2 lbs? X = weight of packages. X ~ N(  = 10,  = 4.0)

M12-Normal Distribution 2 14  Department of ISM, University of Alabama, Same situation. What proportion of packages weigh either less than 2.4 lbs or greater than 11.0 lbs? X = weight of packages. X ~ N(  = 10,  = 4.0) Homework

M12 Normal Distribution 2 15  Department of ISM, University of Alabama, P( X < ?) =.10 0Z P( Z < ) Same situation. Find the weight such that 10% of all packages weigh less than this weight. 10X ? X = weight of packages. X ~ N(  = 10,  = 4.0) = – 1.28 = ? – 10 4 ? = 10 – = 10 – 5.12 = 4.88 pounds This is a backwards problem! We are given the probability; we need to find the boundary. 10% weigh less than 4.88 pounds; 90% weigh more than 4.88 pounds. 10% weigh less than 4.88 pounds; 90% weigh more than 4.88 pounds.

M12 Normal Distribution 2 16  Department of ISM, University of Alabama, Table gives P(0 < z < ?) = Standard Normal Table Find P( __ < z < 0) = Find the Z value to cut off the top 10%.

M12 Normal Distribution 2 17  Department of ISM, University of Alabama, Table gives P(0 < z < ?) = Standard Normal Table.25 ? Find the Z values that define the middle 50%..25 ?

M12 Normal Distribution 2 18  Department of ISM, University of Alabama, Table gives P(0 < z < ?) = Standard Normal Table Find the Z values that define the middle 95%.

M12-Normal Distribution 2 19  Department of ISM, University of Alabama, Normal Functions in Excel NORMDIST – Used to compute areas under any normal curve. Can also compute height of curve (not useful except for drawing normal curves). NORMSDIST - Used to compute areas under a standard normal ( N(0,1) or Z curve ).

M12-Normal Distribution 2 20  Department of ISM, University of Alabama, Normal Functions in Excel NORMINV - Used to find the X value corresponding to a given cumulative probability for any normal distribution. NORMSINV - Used to find the Z value corresponding to a given cumulative probability for a standard normal distribution.

M12-Normal Distribution 2 21  Department of ISM, University of Alabama, P( Z < –1.92) = 2. P( Z < 2.56) = 3. P( Z > 0.80) = 4. P( Z = 1.42) = 5. P(.32 < Z < 2.48) = 6. P( < Z < 1.75) = 7. P( Z < 4.25) = 8. P( Z > 4.25) = 9. P(-.05 < Z <.05) = 10. Find Z such that only 12% are smaller. Practice problems. You MUST know how to work ALL of these problems and the following practice problems to pass this course.

M12-Normal Distribution 2 22  Department of ISM, University of Alabama, –.7881 = –.6255 = –.0401 = Practice problem answers

M12-Normal Distribution 2 23  Department of ISM, University of Alabama, Question: What do we do when we have a normal population distribution, but the mean is not “0” and/or the standard deviation is not “1”? Z = X –   Use the Universal Translator Example: Suppose X ~ N(120, 10). 11. Find P ( X > 150 ). 12. Find the quartiles of this distribution.

M12-Normal Distribution 2 24  Department of ISM, University of Alabama, P( 0 < Z < 1.43) = ? 0Z 1.43 What proportion of Z values are between 0 and 1.43?.4236 =.4236 P(-1.43 < Z < 0) = ? 0Z What proportion of Z values are between and 0?.4236 =.4236 ori

M12-Normal Distribution 2 25  Department of ISM, University of Alabama, P(Z > 1.43) = ? 0Z 1.43 What proportion of Z values are greater than 1.43?.4236 P(Z < 1.43) = ? 0Z What proportion of Z values are less than 1.43? = =.0764 = =

M12-Normal Distribution 2 26  Department of ISM, University of Alabama, P(-1.23 < Z < 2.05) = ? 0Z 2.05 What proportion of Z values are between –1.23 and 2.05?.3907 P(1.23 < Z < 2.05) = ? 0Z What proportion of Z values are between 1.23 and 2.05? = =.8705 = =

M12-Normal Distribution 2 27  Department of ISM, University of Alabama, P( 4.28 < X < 10.0) = ? 0Z = P( < Z < 0) Same situation. Find the proportion of packages that weigh between 4.28 and 10.0 lbs. 10X 4.28 Z = 4.28 – = Z = 10.0 – = 0 X = weight of packages. X ~ N(  = 10,  = 4.0) =.4236

M12-Normal Distribution 2 28  Department of ISM, University of Alabama, P( 13.0 < X < 17.84) = ? 0Z = P(.75 < Z < 1.96) Same situation. What proportion of the packages weigh between 13.0 and lbs? 10X Z = – = 1.96 X = weight of packages. X ~ N(  = 10,  = 4.0) = = Z = 13.0 – =.75 ?

M12-Normal Distribution 2 29  Department of ISM, University of Alabama, Z Same situation. Find the weight such that a. 16% weigh more less than this value. b. You have the boundaries of the middle 80%. c. The top 25% weigh more. d. You have the quartiles. X X = weight of packages. X ~ N(  = 10,  = 4.0)