SAPLING DISTRIBUTIONS
Average U.S. Height in Inches (ales; 20-29yr) Individual Values 61 62 63 64 64 65 65 66 66 66 61 64 67 69 68 69 68 69 67 68 69 67 68 69 67 68 69 67 68 69 x 71 71 71 71 71 71 71 72 72 72 72 72 72 73 73 73 73 74 74 74 75 75 76 76 X 3 X 77 78 79 73 76 79
Dilemma: I want to collect sample data from a population and I want to identify the extent to which the sample outcome is consistent with the underlying population. I have a few major concerns: 1) Will a sample outcome resemble the original population? 2) To what extent will different sample outcomes vary from each other on average? 3) How can I use information about the population to make predictions of future sample outcomes?
Population (μ x ) 1 2 3 4 5 6 7
Average U.S. Height in Inches (ales; 20-29yr) Individual Values 1 69.33 2 71.17 3.33 4 71.17 61 62 63 5 69.83 6 76.83 7 69.67 64 64 65 65 66 66 66 61 64 67 69 68 69 68 69 67 68 69 67 68 69 67 68 69 67 68 69 x 71 71 71 71 71 71 71 72 72 72 72 72 72 73 73 73 73 74 74 74 75 75 76 76 X 3 X 77 78 79 73 76 79
Point estimation The process of using a sample statistic to estimate the corresponding population parameter x Sampling error The difference between the value of a sample statistic and its corresponding population parameter x
Average U.S. Height in Inches (ales; 20-29yr) Sample eans 69.73 69.12 69.80.83 X 69.26 69.98.78 68.80 69.53.18.61 71.33 68.78 69.06.26.59 71.12 68.42 68.96 69.24 69.94.54 71.41 71.74 67.68 68.16 68.61 69.03 69.87.66 71.20 71.81 72.12 67.13 67.85 68.39 68.77 69.19.13.75 71.03 71.69 72.35 72.89
Sampling distribution of the mean A probability distribution of sample means that would occur if all possible samples of a fixed sample size (with replacement) were drawn from a population
Sampling distribution of the mean A probability distribution of sample means that would occur if all possible samples of a fixed sample size with replacement were drawn from a population Population (μ x ) 1 N = 5 3 4 N = 50 2 N = 20 N = 30
μ ean of the sampling distribution of the mean The mean of all sample means for a given sampling distribution σ Standard error of the mean (S.E. or S.E..) The mean deviation of a given sample mean () from the population mean (μ X ) Standard deviation of the sampling distribution of the mean
A sampling distribution of the mean is formed by creating all possible random samples from a population and determining all of the sample mean values. If access to the entire population (i.e. all possible samples) is not available, this is an impossible process. What can we do?
Central Limit Theorem A statistical theory that describes the numerical characteristics and shape of a sampling distribution as a function of the chosen sample size For a given population with a mean ( x ) and variance ( 2 x ), the resulting sampling distribution of the mean will possess the following: 1 μ μ 2 X N 2 X X N 3 With a sufficiently large sample size (N 30) the sampling distribution of the mean will take the shape of a normal distribution regardless of the shape of the population from which the sample was drawn
How does variability differ between individual and sample outcomes? Individual Values X 3 X 61 64 67 Sample eans N = 30 3 0.55 30 73 76 79 μ 0.55 68.35 68.90 69.45.55 71.10 71.65
Individual Values X 3 X 61 64 67 73 76 79 Sample eans N = 30 3 0.55 30 μ 0.55 68.35 68.90 69.45 Sample eans N = 100 3 100 0.30.55 71.10 71.65 μ 0.30 69.10 69.40 69..30.60.90
Central Limit Theorem A statistical theory that describes the characteristics and distribution shape of a sampling distribution as a function of the chosen sample size For a given population with a mean ( x ) and standard deviation ( x ), the resulting sampling distribution of the mean will possess the following: 1 μ μ 2 X 2 X X N N 3 With a sufficiently large sample size (N 30) the sampling distribution of the mean will resemble the shape of a normal distribution regardless of the shape of the population from which the sample was drawn
Interactive Demonstration of Central Limit Theorem http://onlinestatbook.com/stat_sim/sampling_dist/index.html
Changes in Population ean Determine the expected mean and standard error if creating samples (N = 25) from a population that has a mean of x = 65 and a standard deviation of x = 7. 65 7 25 1.40 Determine the expected mean and standard error if creating samples (N = 25) from a population that has a mean of x = 45 and a standard deviation of x = 7. 45 7 25 1.40
Changes in Population Variability Determine the expected mean and standard error if creating samples (N = 25) from a population that has a mean of x = 48 and a standard deviation of x = 4. 48 4 25 0.80 Determine the expected mean and standard error if creating samples (N = 25) from a population that has a mean of x = 48 and a standard deviation of x = 13. 48 13 25 2.60
Changes in Sample Size Determine the expected mean and standard error if creating samples (N = 25) from a population that has a mean of x = 78 and a standard deviation of x = 15. 78 15 25 3.00 Determine the expected mean and standard error if creating samples (N = 75) from a population that has a mean of x = 78 and a standard deviation of x = 15. 78 15 1.73 75
Sampling Distributions & Expected Outcomes Raw Scores z X X X X Sample eans z μ σ
Assume a normally distributed population of values with x = 60 and x = 9. What is the probability of an individual obtaining a score of 62 or higher? 1. Determine z score 2. Locate area of curve 3. Identify proportion 62 60 z 0.22 X 9 p X 62 p =.4129 z = 0.00 0.22
Assume a normally distributed population of values with x = 60 and x = 9. What is the probability of obtaining a sample mean of 62 or higher based on a sample size of 36 individuals? 1. Determine z score 2. Locate area of curve 3. Identify proportion p 62 62 60 z 1.33 9 / 36 p =.0918 z = 0.00 1.33
Assume a normally distributed population of values with x = 84 and x = 14. What is the probability of obtaining a sample mean of 77 or higher based on a sample size of 45 individuals? 1. Determine z score 2. Locate area of curve 3. Identify proportion p 77 77 84 z 3.35 14 / 45 p =.9996 p =.4996 p =.5000-3.35 z = 0.00
Assume a normally distributed population of values with x = 754 and x = 241. What is the probability of obtaining a sample mean equal or higher to 847 or equal or lower than 661 based on a sample size of 31 individuals? p 1. Determine z score 2. Locate area of curve 3. Identify proportion 661, 847 661 754 z 2.15 241/ 31 847 754 z 2.15 241/ 31 p =.0316 p =.0158 p =.0158-2.15 z = 0.00 2.15
Dilemma: I want to collect sample data from a population and I want to identify the extent to which the sample outcome is consistent with the underlying population. I have a few major concerns: 1) Will a sample outcome resemble the original population? Sample means will vary as a function of sampling error. 2) To what extent will different sample outcomes vary from each other on average? The standard error of the mean provides expected sample variability. 3) How can I use information about the population to make predictions of future sample outcomes? By using z-scores, the shape of the sampling distribution of means (via CLT) provides expectations for future mean outcomes.
Terminology Sampling error Sampling distribution of the mean ean of the sampling distribution of the mean and standard error Concepts and Skills Understand the nature of the sampling process and sampling error Be aware of the purpose of sampling distributions Be able to determine the characteristics of sampling distributions via central limit theorem Be able to evaluate the probability of obtaining specific sample outcomes from a population using z scores