Class 15. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 15 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 17 by D.B. Rowe 1

Agenda: Recap Chapter 7.1 7. Lecture Chapter 7. Discussion of Chapters Problem Solving Session. Go Over Eam 3.

Recap Chapter 7.1 7. 3

7. The Sampling Distribution of Sample Means When we take a random sample 1,, n from a population, one of the things that we do is compute the sample mean. The value of is not μ. Each time we take a random sample of size n, we get a different set of values 1,, n and a different value for. There is a distribution of possible s. 4

7. The Sampling Distribution of Sample Means Sampling Distribution of a sample statistic: The distribution of values for a sample statistic obtained from repeated samples, all of the same size and all drawn from the same population. If we take a random sample of size n and compute sample mean,. The collection of all possible means is called the sampling distribution of the sample mean (SDSM). 5

7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values:,, 4, 6, 8. 4 6 8 P( ) 1/ 5 1/ 5 4 1/ 5 6 1/ 5 8 1/ 5 P()..16.1.8.4 5 possible values 4 6 8 [ P( )] 4 [( ) P( )] 8 8 6

7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n= with replacement. Population data values: 4 6 8 8 4 6 4 6 8,, 4, 6, 8. (,) (,) (4,) (6,) (8,) 4 6 8 4 6 8 4 6 8 4 6 8 (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) 5 possible samples 7

5 possible samples. 7. The Sampling Distribution of Sample Means (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 =.4 1 3 4 1 3 4 5 3 4 5 6 3 4 5 6 7 4 5 6 7 8 8

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 =.4 1 3 4 1 3 4 5 3 4 5 6 3 4 5 6 7 4 5 6 7 8, one time 1, two times, three times 3, four times 4, five times 5, four times 6, three times 7, two times 8, one time 5 possible samples. (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) 9

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 =.4 1 3 4 1 3 4 5 3 4 5 6 3 4 5 6 7 4 5 6 7 8, one time 1, two times, three times 3, four times 4, five times 5, four times 6, three times 7, two times 8, one time 5 possible samples. (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 1

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 Represent this distribution function with a histogram. note that intermediate values of 1,3,5,7 are now possible. Figure from Johnson & Kuby, 1. 11

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n=1 or (with replacement). P() from n=1 distribution from n= distribution..16.1.8.4 4 6 8 Figure from Johnson & Kuby, 1. 1

7. The Sampling Distribution of Sample Means Sample distribution of sample means (SDSM): If all possible random samples, each of size n, are taken from any population with mean μ and standard deviation σ, then the sampling distribution of sample means will have the following: 1. A mean equal to μ. A standard deviation equal to n Furthermore, if the sampled population has a normal distribution, then the sampling distribution of will also be normal for all samples of all sizes. Discuss Later: What if the sampled population does not have a normal distribution? 13

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P( ) (1/ 5) 1( / 5) (3 / 5) 3(4 / 5) 4(5 / 5) 5(4 / 5) 6(3 / 5) 7( / 5) 8(1/ 5) 4 Eactly as from formula! 4 P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 14

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). ( 4) (1/ 5) (1 4) ( / 5) 4 ( 4) (3 / 5) (3 4) (4 / 5) (4 4) (5 / 5) (5 4) (4 / 5) ( ) P( ) (6 4) (3 / 5) (7 4) ( / 5) (8 4) (1/ 5) Eactly as from formula! n P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 15

7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n=1 or (with replacement). P(). from n=1 distribution 4, 8 from n= distribution 4, n.16.1.8.4 4 6 8 Figure from Johnson & Kuby, 1. 16

Questions? Homework: Chapter 7 # 6, 1, 3, 9, 33, 35 17

Lecture Chapter 7. continued 18

7. The Sampling Distribution of Sample Means Sample distribution of sample means (SDSM): If all possible random samples, each of size n, are taken from any population with mean μ and standard deviation σ, then the sampling distribution of sample means will have the following: 1. A mean equal to μ. A standard deviation equal to n Furthermore, if the sampled population has a normal distribution, then the sampling distribution of will also be normal for all samples of all sizes. Discuss Later: What if the sampled population does not have a normal distribution? 19

7. The Sampling Distribution of Sample Means P()..16 from n=1 distribution 4, 8 from n= distribution 4, 8.1.8.4 4 6 8? from n=3 distribution 4, 8 3 Looks like? Figure from Johnson & Kuby, 1. from n=4 distribution 4, 8 4? Looks like? n large? 4 8 Looks like? n

7. The Sampling Distribution of Sample Means We have a couple of definitions. Standard error of the mean ( ): The standard deviation of the sampling distribution of sample means. Central Limit Theorem (CLT): The sampling distribution of sample means will more closely resemble the normal distribution as the sample size increases. The CLT is etremely important in Statistics. 1

7. The Sampling Distribution of Sample Means The Central Limit Theorem: Assume that we have a population (arbitrary distribution) with mean μ and standard deviation σ. If we take random samples of size n (with replacement), then for large n, the distribution of the sample means, the s, is approimately normally distributed with, n where in general n 3 is sufficiently large, but can be as small as15 or as big as 5 depending upon the shape of distribution!

7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: For a discrete distribution P(), we found the mean μ and variance σ from P( ) and ( ) P( ). 3

7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: For a discrete distribution P(), we found the mean μ and variance σ from P( ) and ( ) P( ). For a continuous distribution f(), we find the mean μ and variance σ from f ( ) d and ( ) f ( ) d. 4

7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: Uniform(a,b) Normal(μ,σ ) f( ) 1 b a f( ) e 1 a μ b μ 5

7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: Uniform(a,b) Normal(μ,σ ) f( ) 1 b a f( ) e 1 b b a f ( ) d a b ( b a) ( ) f ( ) d 1 a a μ b f ( ) d ( ) f ( ) d μ 6

7. The Sampling Distribution of Sample Means b a ( b a) 1 Eample: The random observation comes from either a Uniform(,) population or Normal population μ=1,σ=57.7. Uniform(a=,b=) Normal(μ=1,σ = /1). Generate data,..., 1 n, for n=1,, 3, 4, 5, and 15. Calculate, and repeat one million times. 7

7. The Sampling Distribution of Sample Means Eample: The random observation comes from either a Uniform(,) population or Normal population μ=1,σ=57.7. Uniform(a=,b=) Normal(μ=1,σ = /1)....18 1 1 f( ) 1.16 b a.14 57.7 e f( ).1.18.16.14.1.1.1 b a ( b a) 1 1 57.7.8.6.4..8.6.4. 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 Generate data,..., 1 n, for n=1,, 3, 4, 5, and 15. Calculate, and repeat one million times. 8 limited range

7. The Sampling Distribution of Sample Means According to SDSM, if we had a sample of size n=1,, 3, 4, 5, and 15 from Uniform(a=,b=), then n Sample Size, Mean, SD, 1 1 57.735 1 4.848 3 1 33.3333 4 1 8.8675 5 1 5.8199 15 1 14.971,..., 1 n For Uniform a to b b a ( b a) 1 Theoretical values from the SDSM n 9

7. The Sampling Distribution of Sample Means According to SDSM, if we had a sample of size n=1,, 3, 4, 5, and 15 from Normal(μ=1,σ =(57.7) ), then n Sample Size, Mean, SD, 1 1 57.735 1 4.848 3 1 33.3333 4 1 8.8675 5 1 5.8199 15 1 14.971,..., 1 n Theoretical values from the SDSM For Normal b a ( b a) 1 n 3

7. The Sampling Distribution of Sample Means I wrote a computer program to generate 1 million random observations,..., 1 1 from the Uniform(a=,b=) and also from Normal(μ=1,σ =(57.7) ) distributions. Show histograms! 31

7. The Sampling Distribution of Sample Means Epanded program to generate n million random observations 1,..., n 1 6 from the Uniform(a=,b=) and also from the Normal(μ=1,σ =(57.7) ) distributions, for each of n=1,, 3, 4, 5, and 15. 6 data sets of Uniform and Normal random observations 3

7. The Sampling Distribution of Sample Means n=1 1 1 6 observations scale changes 1 14 9 8 7 1 57.73 1 14 9 8 7 1 57.73 6 5 observations are uniform 6 5 observations are normal 4 3 4 3 limited range 1 1 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 33

7. The Sampling Distribution of Sample Means n= 1 6 observations scale changes 1.8 1.6 1.4 15 1 57.73 1.8 1.6 1.4 15 1 57.73 1. 1 observations are uniform 1. 1 observations are normal.8.6.8.6 limited range.4.4.. 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 34

7. The Sampling Distribution of Sample Means n=3 3 1 6 observations scale changes.5 3 15 1 57.73.5 3 15 1 57.73 1.5 observations are uniform 1.5 observations are normal 1 1 limited range.5.5 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 35

7. The Sampling Distribution of Sample Means n=4 4 1 6 observations scale changes 3.5 4 15 3 1 57.73 3.5 4 15 3 1 57.73.5 observations are uniform.5 observations are normal 1.5 1.5 limited range 1 1.5.5 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 36

7. The Sampling Distribution of Sample Means n=5 5 1 6 observations scale changes 4.5 4 3.5 5 15 1 57.73 4.5 3.5 5 15 4 1 57.73 3.5 observations are uniform 3.5 observations are normal 1.5 1.5 limited range 1 1.5.5 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 37

7. The Sampling Distribution of Sample Means n=15 15 1 6 observations scale changes 15 15 1 1 57.73 observations are uniform 15 15 1 1 57.73 observations are normal 5 5 limited range 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 38

7. The Sampling Distribution of Sample Means Sample means and standard deviations from each of the n million observations from the Uniform(a=,b=) and Normal(μ=1,σ =(57.7) ) distributions. 6 data sets of Uniform and Normal 39

7. The Sampling Distribution of Sample Means Sample means and standard deviations from each of the n million observations from the Uniform(a=,b=) and Normal(μ=1,σ =(57.7) ) distributions. i.e. n=5, 5 1 6 Groups of n=5 Mean of groups 1 1,, 3, 4, 5 1 6 data sets of Uniform and Normal... 5 6, 7 8, 9, 1... 4999996,.,.,., 5... 1 6 s Histogram of 's 4

7. The Sampling Distribution of Sample Means Computed sample means and standard deviations from the one million means,..., for n=1,, 3, 4, 5, and 15. 1 1 6 Sample Size n Mean Mean U Mean N SD SD U SD N 1 1 1.64 1.77 57.735 57.771 57.7888 1 99.988 1.37 4.848 4.8418 4.86 3 1 99.999 99.967 33.3333 33.3418 33.984 4 1 99.9559 1.64 8.8675 8.8946 8.816 5 1 1.74 1.3 5.8199 5.7865 5.8397 15 1 1.134 99.9517 14.971 14.935 14.8918 s s 41

7. The Sampling Distribution of Sample Means n=1 1 1 6 means scale same 1 57.73.5 15.5 15 Histogram of means from uniform Histogram of means from normal 1.5 1.5 1 s 1 s limited range.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 4

7. The Sampling Distribution of Sample Means n= 1 1 6 means scale same 1 57.73.5 15.5 15 Histogram of means from uniform Histogram of means from normal 1.5 1.5 1 s 1 s limited range.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 43

7. The Sampling Distribution of Sample Means n=3 1 1 6 means scale same 1 57.73.5 15.5 15 Histogram of means from uniform Histogram of means from normal 1.5 1.5 1 s 1 s limited range.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 44

7. The Sampling Distribution of Sample Means n=4 1 1 6 means scale same 1 57.73.5 15 1.5 Histogram of means from uniform.5 15 1.5 Histogram of means from normal 1 1 s s.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 45

7. The Sampling Distribution of Sample Means n=5 1 1 6 means scale same 1 57.73.5 15 1.5 Histogram of means from uniform.5 15 1.5 Histogram of means from normal 1 1 s s.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 46

7. The Sampling Distribution of Sample Means n=15 1 1 6 means scale same 1 57.73.5 15 1.5 Histogram of means from uniform.5 15 1.5 Histogram of means from normal 1 1 s s.5.5 4 6 8 1 1 14 16 18 same classes 4 6 8 1 1 14 16 18 same classes 47

7. The Sampling Distribution of Sample Means With a population mean μ and standard deviation σ. Random samples of size n with replacement, for large n, the distribution of the sample means quickly becomes normally distributed with, n Generally n 3 is sufficiently large, but can be as small as15 or as big as 5 depending upon the shape of distribution! 48

Lecture Chapter 7.3 49

3: Descriptive Analysis and Bivariate Data 3.1 Bivariate Data: Scatter Diagram Our data. Height vs. Weight Recall TuTh 1 n=48 responses Find yourself. If not here then removed () or you did not respond. 5

TuTh 1 7.3 Application of the Sampling Distribution of Sample Means Assume that this is a population of data. Does this population look normally distributed? 68.4 4. Put normal with same μ and σ. Height 34

7.3 Application of the Sampling Distribution of Sample Means If I m interested in the mean of a sample of size n=15, then by SDSM and, in addition, n by the CLT is (hopefully) normally distributed. I wrote a computer program to take a sample of n=1,3,5,15 from the population of N heights with replacement 1 6 times. 35

7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and 4.. 1 Histogram of the 1 million means n=1 TuTh 1 N=48 values 68.4 4. Put normal with same and. 36

7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and.4. 3 Histogram of the 1 million means n=3 TuTh 1 N=48 values 68.4 4. Put normal with same and. 37

TuTh 1 7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and 1.9. 5 Histogram of the 1 million means n=5 N=48 values 68.4 4. Put normal with same and. 38

7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and 1.1. 15 Histogram of the 1 million means n=15 TuTh 1 N=48 values 68.4 4. Put normal with same and. By CLT becomes normal 39

7.3 Application of the Sampling Distribution of Sample Means Now that we believe that the mean from a sample of n=15 is normally distributed with mean and standard deviation n, we can find probabilities. a b b a P( a b) P( b ) P( a) 4

7.3 Application of the Sampling Distribution of Sample Means To find these probabilities, we first convert to z scores a b b a P( a b) P( b ) P( c z d) same area P( d z) same area P( a) P( z c) same area z a b, c, d and use the table in book. 58

7.3 Application of the Sampling Distribution of Sample Means Eample: What is probability that sample mean from a random sample of n=15 heights is greater than 7 when 68.4 and 4.? 59

Questions? Homework: Chapter 7 # 6, 1, 3, 9, 33, 35 64

Discussion: Chapters 65

Discussion on Course We re moving into a new phase of the course Part III on Inferential Statistics. Parts I and II were all foundational material for Part III. Before we discussed 66

Discussion on Course Part I: Descriptive Statistics Chapter 1: Statistics Background material. Definitions. Chapter : Descriptive Analysis and Presentation of single variable data Graphs, Central Tendency, Dispersion, Position Chapter 3: Descriptive Analysis and Presentation of bivariate data Scatter plot, Correlation, Regression 67

Discussion on Course Part II: Probability Chapter 4: Probability Conditional, Rules, Mutually Eclusive, Independent Chapter 5: Probability Distributions (Discrete) Random variables, Probability Distributions, Mean & Variance, Binomial Distribution with Mean & Variance Chapter 6: Probability Distributions (Continuous) Normal Distribution, Standard Normal, Applications, Notation Chapter Sampling Distributions, SDSM, CLT 68

Discussion on Course Part III: Inferential Statistics Chapter 8: Introduction to Statistical Inferences Confidence Intervals, Hypothesis testing Chapter 9: Inferences Involving One Population Mean μ (σ unknown), proportion p, variance σ Chapter 1: Inferences Involving Two Populations Difference in means μ 1 -μ, proportions p 1 -p, variances Part IV: More Inferential Statistics Chapter 11: Applications of Chi-Square Chi-square statistics.. We will discuss later. / 1 69

Discussion on Course Net Lecture will be on Chapter 8. Chapter 8: Introduction to Statistical Inferences Confidence Intervals and Hypothesis testing 7

Go over Eam 3. 71