Probability and the Sampling Distribution Quantitative Methods in HPELS HPELS 6210

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Introduction Recall: Next step  convert sample mean into a Z-score Any raw score can be converted to a Z-score Provides location relative to µ and  Assuming NORMAL distribution: Proportion relative to Z-score can be determined Z-score relative to proportion can be determined Previous examples have looked at single data points Reality  most research collects SAMPLES of multiple data points Next step  convert sample mean into a Z-score Why? Answer probability questions

Introduction Two potential problems with samples: Sampling error Difference between sample and parameter Variation between samples Difference between samples from same taken from same population How do these two problems relate?

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Distribution of Sample Means Distribution of sample means = sampling distribution is the distribution that would occur if: Infinite samples were taken from same population The µ of each sample were plotted on a FDG Properties: Normally distributed µM = the “mean of the means” M = the “SD of the means” Figure 7.1, p 202

Distribution of Sample Means Sampling error and Variation of Samples Assume you took an infinite number of samples from a population What would you expect to happen? Example 7.1, p 203

Assume a population consists of 4 scores (2, 4, 6, 8) Collect an infinite number of samples (n=2)

Total possible outcomes: 16 p(2) = 1/16 = 6.25% p(3) = 2/16 = 12.5% p(4) = 3/16 = 18.75% p(5) = 4/16 = 25% p(6) = 3/16 = 18.75% p(7) = 2/16 = 12.5% p(8) = 1/16 = 6.25%

Central Limit Theorem For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µM and a standard deviation of M, and will approach a normal distribution as the sample size (n) approaches infinity If it is NORMAL, it is PREDICTABLE!

Central Limit Theorem The CLT describes ANY sampling distribution in regards to: Shape Central Tendency Variability

Central Limit Theorem: Shape All sampling distributions tend to be normal Sampling distributions are normal when: The population is normal or, Sample size (n) is large (>30)

Central Limit Theorem: Central Tendency The average value of all possible sample means is EXACTLY EQUAL to the true population mean µM = µ If all possible samples cannot be collected? µM approaches µ as the number of samples approaches infinity

µ = 2+4+6+8 / 4 µ = 5 µM = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16

Central Limit Theorem: Variability The standard deviation of all sample means is denoted as M M = /√n Also known as the STANDARD ERROR of the MEAN (SEM)

Central Limit Theorem: Variability SEM Measures how well statistic estimates the parameter The amount of sampling error between M and µ that is reasonable to expect by chance The standard distance between the sample M and population µ

Central Limit Theorem: Variability SEM decreases when: Population  decreases Sample size increases Other properties: When n=1, M =  (Table 7.2, p 209) As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215) M = /√n

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Probability  Sampling Distribution Recall: A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

Probability  Sampling Distribution Example 7.2, p 209 Assume the following about SAT scores: µ = 500  = 100 n = 25 Population  normal What is the probability that the sample mean will be greater than 540? Process: Draw a sketch Calculate SEM Calculate Z-score Locate probability in normal table

Step 1: Draw a sketch Step 2: Calculate SEM SEM = M = /√n SEM = 100/√25 SEM = 20 Step 3: Calculate Z-score Z = 540 – 500 / 20 Z = 40 / 20 Z = 2.0 Step 4: Probability Column C p(Z = 2.0) = 0.0228

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Looking Ahead to Inferential Statistics Review: Single raw score  Z-score  probability Body or tail Sample mean  Z-score  probability What’s next? Comparison of means  experimental method

Textbook Assignment Problems: 14, 18, 24 In your words, explain the concept of a sampling distribution In your words, explain the concept of the Central Limit Theorum