Probability and the Sampling Distribution

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Probability and the Sampling Distribution

• Quantitative Methods in HPELS
• HPELS 6210

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Agenda

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

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Introduction

• Recall
• 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

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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?

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Agenda

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

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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

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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

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Assume a population consists of 4 scores (2, 4,
6, 8)
Collect an infinite number of samples (n2)
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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
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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!

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Central Limit Theorem

• The CLT describes ANY sampling distribution in
  regards to
• Shape
• Central Tendency
• Variability

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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 (gt30)

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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

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µ 2468 / 4 µ 5
µM 2334445555666778 / 16 µM 80
/ 16 5
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Central Limit Theorem Variability

• The standard deviation of all sample means is
  denoted as ?M
• ?M ?/vn
• Also known as the STANDARD ERROR of the MEAN
  (SEM)

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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

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Central Limit Theorem Variability

• SEM decreases when
• Population ? decreases
• Sample size increases
• Other properties
• When n1, ?M ? (Table 7.2, p 209)
• As SEM decreases the sampling distribution
  tightens (Figure 7.7, p 215)

?M ?/vn
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Agenda

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

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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

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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

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Step 1 Draw a sketch
Step 2 Calculate SEM SEM ?M ?/vn SEM
100/v25 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
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Agenda

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

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Looking Ahead to Inferential Statistics

• Review
• Single raw score ? Z-score ? probability
• Body or tail
• Sample mean ? Z-score ? probability
• Body or tail
• Whats next?
• Comparison of means ? experimental method

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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