M31- Dist of X-bars 1  Department of ISM, University of Alabama, Sampling Distributions Our Goal? To make a decisions. What are they? Why do we care?

M31- Dist of X-bars 2  Department of ISM, University of Alabama, Lesson Objectives  Know what is meant by the “sampling distribution” of a statistic, and the “population of all possible X-bar values.”  Know when the population of all possible X-bar values IS normal.  Know when the population of all possible X-bar values IS NOT normal.

M31- Dist of X-bars 3  Department of ISM, University of Alabama, Descriptive NumericalGraphical Statistics

M31- Dist of X-bars 4  Department of ISM, University of Alabama, Statistical Inference Generalizing from a sample to a population, by using a statistic to estimate a parameter. Goal:.

Population Census True Parameter Sample Statistic a guess

Parameter Statistic Mean: Standard deviation: Proportion: s X    estimates p

M31- Dist of X-bars 7  Department of ISM, University of Alabama, A statistic is a. Before we can make decisions about parameters and control the degree of risk, we must know:  the of the statistic  and its values. Objective of this section:

Original Population: 300 ST 260 students. X = Exam 2 score  = population mean (unknown)  = population std deviation (unknown) Calculate: n = 4 x = mean s = std dev Example 1: Population Sample

M31- Dist of X-bars 9  Department of ISM, University of Alabama, Think of X as a random variable. Fact: Different samples of size “n” will produce different values of the sample mean. The population mean is fixed as long as the population does not change.

M31- Dist of X-bars 10  Department of ISM, University of Alabama, Shape? (Skewed? Symmetric?) Center? (Mean? Median?) Spread? (Std. Deviation? IQR?) Is it one of our “special” distributions? (Normal? Exponential?) For samples of size n, what is the distribution of the statistic X?

M31- Dist of X-bars 11  Department of ISM, University of Alabama, For samples of size n, what is the distribution of p, i.e, a sample proportion? Shape? (Skewed? Symmetric?) Center? (Mean? Median?) Spread? (Std. Deviation? IQR?) Possible values? (0/n, 1/n, 2/n, …, n/n) Is it one of our “special” distributions? (normal, exponential, binomial, Poisson) ^

M31- Dist of X-bars 12  Department of ISM, University of Alabama, From pop. of all ST 260 students, randomly select n = 1 student. Record exam 2 grade: Sampled value: 76 X = 76.0 How close can we expect this estimate to be to the true mean  ? Example 1, continued:

M31- Dist of X-bars 13  Department of ISM, University of Alabama, Sampled values: 64, 78, 94, 46 X = ( ) / 4 = 70.5 From pop. of all ST 260 students, randomly select n = 4 students. Record exam 2 grades: How close can we expect this estimate to be to the true mean  ? Example 1, continued:

M31- Dist of X-bars 14  Department of ISM, University of Alabama, x’s from samples tend to be to the true mean,  than x ’s from smaller samples. Fact:

M31- Dist of X-bars 15  Department of ISM, University of Alabama, Sampling Distribution of X is the distribution of all possible sample means calculated from all possible samples of size n. Also called “the population of all possible x-bars”. Also called “the population of all possible x-bars”.

M31- Dist of X-bars 16  Department of ISM, University of Alabama, And so on,..., until we collect every possible sample of size n = 4. How many samples of size 4 are there from a population of 300 members?

M31- Dist of X-bars 17  Department of ISM, University of Alabama, Sampling Distribution of x for n = 4 Based on all samples of size n = 4  x  x x-axis And the shape looks like a Normal dist.

M31- Dist of X-bars 18  Department of ISM, University of Alabama,  x  x Definitions, from previous slide: The subscripts identify the population to which the parameter refers. = average of all possible X’s (center of the sampling dist.) = std. deviation of all pos. X’s (spread of the sampling dist.)

M31- Dist of X-bars 19  Department of ISM, University of Alabama, Compare parameters of the original population of all scores and the parameters of the sampling dist. of all possible x’s mean =  & std. dev. =  Original population:  x  x   (same mean as individual values)   n (different std. dev., but related!)

M31- Dist of X-bars 20  Department of ISM, University of Alabama, If  = 75 and  = 10, Original Population: 300 ST 260 students. X = Exam 2 score. then the population of all possible X-values for n = 4 will have   x  x   n 

M31- Dist of X-bars 21  Department of ISM, University of Alabama, Questions What is the probability that one randomly selected Exam2 score will be within 10 points of the population mean, 75? X: 65 to 85 Z:  =, What is the probability that a sample mean of n = 4 randomly selected Exam 2 scores will be within 10 points of the pop. mean?  =, X X: 65 to 85 Z:

M31- Dist of X-bars 22  Department of ISM, University of Alabama, Investing in Stocks Individual stocks vs. Diversified portfolios: The same expected earnings, but different risks!   x Diversified Individual  n  DiversifiedIndividual <

M31- Dist of X-bars 23  Department of ISM, University of Alabama, We now know the parameters of the population of all possible x-bar values. What is the distribution ? Look back at the plot exam 2 grades.

M31- Dist of X-bars 24  Department of ISM, University of Alabama, If X ~ N (  ), then for samples of size n, X ~ N (, ). If original population has a Normal dist., then the distribution of X values is Normal also.   n

M31- Dist of X-bars 25  Department of ISM, University of Alabama, Original Population : Normal (  = 50,  = 18)  = 18.00

M31- Dist of X-bars 26  Department of ISM, University of Alabama, Original Population : Normal (  = 50,  = 18) n = 36 n = 16 n = 4 n = 2  = 9.00 x  = x  = 4.50 x  = 3.00 x  = 18.00

M31- Dist of X-bars 27  Department of ISM, University of Alabama, Bottle filling machine for soft drink. Bottles should contain ounces; assume actual contents follow a normal distribution with a mean of oz. and a standard deviation of 0.12 oz. X = contents of one randomly selected bottle X ~ N(  = 20.18,  = 0.12) Example 2: This is the original population.

M31- Dist of X-bars 28  Department of ISM, University of Alabama, P( X < 20.00) = 0 = P( Z < ) a. Find the proportion of individual bottles contain less than oz? Z = X = content of one bottle. X ~ N(  = 20.18,  = ) of the bottles will contain less than ounces. Is this a problem? = Z-axis X-axis = == =

M31- Dist of X-bars 29  Department of ISM, University of Alabama, = P( Z < ) a. Find the proportion of six-packs whose mean content is less than oz? Z = Only ________% of the six-packs will contain an average less than ounces. = Z-axis X-axis ==== Is population of x-bars Normal? Yes; because original pop. is Normal. X = mean of six-pack. X ~ N(  = 20.18,  = ) X x x  = ) P( X < 20.00) = Got this from Excel

M31- Dist of X-bars 30  Department of ISM, University of Alabama,

M31- Dist of X-bars 31  Department of ISM, University of Alabama, New situation Such as an... Exponential Distribution? But what if the original population is not normally distributed?

M31- Dist of X-bars 32  Department of ISM, University of Alabama, Demonstration of the Central Limit Theorem Page 289

Original Population: Exponential (  = 4) n = 1  = 4 4

Original Population: Exponential (  = 4) n = 30 n = 15 n = 5 n = 2  = 4 Sampling distribution for X  = x  = x  = x  = x 4

M31- Dist of X-bars 35  Department of ISM, University of Alabama, If X ~ NOT Normal, then for large samples of size n, X ~ N (, ), approximately. If original population does NOT have a Normal dist.,   n Central Limit Theorem Page 289 the X values are approximately Normal IF n is large.

M31- Dist of X-bars 36  Department of ISM, University of Alabama, How big is BIG ? Bigger is better, but is enough! This same phenomena will happen for ANY non-normal distribution, IF “n” is BIG!

M31- Dist of X-bars 37  Department of ISM, University of Alabama, “Investment opportunity” Earnings: x P(X=x) P(player looses) =.40 Expected value:  = -80 (.40) + 0 (.10) + 60 (.50) Also,  = 66.0 = = Example 3 (C.L.T.) This is definitely NOT normal!

M31- Dist of X-bars 38  Department of ISM, University of Alabama, P( X < 0.0) = ? 0Z-axis After 36 plays, what is the probability that the average earnings is negative? -2.0 X = earning for one play X ~ NOT Normal  = x X ~ N(  = -2.0,  = ) x X = Avg. earnings, 36 plays X-axis X-bar pop. is Normal because n is BIG.

M31- Dist of X-bars 39  Department of ISM, University of Alabama, Z-axis After 6400 plays, what is the probability that the average earnings is negative? -2.0 Same as previous, BUT....  = x X ~ N(  = -2.0,  = ) x X-axis P( X < 0.0) = ?

M31- Dist of X-bars 40  Department of ISM, University of Alabama, Summary different values of”n” Number of plays P( you lose) Expected Total Amount of earnings ,800 12, ,000

M31- Dist of X-bars 41  Department of ISM, University of Alabama, The house never loses!

M31- Dist of X-bars 42  Department of ISM, University of Alabama, X = number of accidents in one week On-the-job accidents in a company. X ~ Poisson ( = 2.2 acc/wk ) Example 4 (C.L.T.) a.Find the probability of having two or fewer accidents in one randomly selected week. P(X < 2) =, from Table A.4. This is a Chapter 6 problem. The probability of being two or less is greater than.5, but the mean is 2.2! How is this possible?

M31- Dist of X-bars 43  Department of ISM, University of Alabama, Example 4 continued X = number of accidents in one week On-the-job accidents in a company. X ~ Poisson ( = 2.2 acc/wk )

M31- Dist of X-bars 44  Department of ISM, University of Alabama, Example 4 continued Ori. pop. is definitely NOT normal; BUT n is large! b.What is the probability that the average number of accidents for next 52 weeks will be 2.0 or less? X = mean for 52 weeks; n = 52. What is the sampling distribution? X = number of accidents in one week On-the-job accidents in a company. X ~ Poisson ( = 2.2 acc/wk )

M31- Dist of X-bars 45  Department of ISM, University of Alabama, What is the sampling distribution of X ? X X X ~ N (  =,  = ) / 52 By the C.L.T., it is approximately Normal. Recall: for Poisson the mean is, the standard deviation is the square root of. = Example 4 continued X = number of accidents in one week On-the-job accidents in a company. X ~ Poisson ( = 2.2 acc/wk )

M31- Dist of X-bars 46  Department of ISM, University of Alabama, b.What is the probability that the average number of accidents for next 52 weeks will be 2.0 or less? 2.2 It is much less likely that the average number of accidents per week will be two or less, than any one specific week. Z-axis X- axis X = mean of accidents. X ~ N(  = 2.2,  = _______) x x P( X < 2.0) = Example 4 cont.

M31- Dist of X-bars 47  Department of ISM, University of Alabama,  Anytime the original pop. is Normal (true for any n).  Anytime the original pop. is not Normal, but n is BIG (n > 30). Reminder When is the population of all possible X values Normal?

M31- Dist of X-bars 48  Department of ISM, University of Alabama,  Anytime the original population is not Normal AND n is NOT BIG. Remember When is the population of all possible X values NOT Normal?