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The Sampling Distribution. Introduction to Hypothesis Testing and Interval Estimation. Outline. Distinctions Sampling Distribution The Central Limit Theorem Confidence Intervals. Random Sampling. Key things to keep in mind. Population- what we want to talk about

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the sampling distribution

The Sampling Distribution

Introduction to Hypothesis Testing and Interval Estimation

  • Distinctions
  • Sampling Distribution
  • The Central Limit Theorem
  • Confidence Intervals
key things to keep in mind
Key things to keep in mind
  • Population- what we want to talk about
  • Sample- what we have with our data
  • Sampling distribution- the means by which we will go from our sample to the population
sampling distribution
Sampling Distribution
  • Sampling distributions concern any statistic we can come up with. Examples:
    • Measures of Central Tendency
    • Measures of Variability
    • Measures of Relationship
    • Ratios
  • Sample != sampling distribution
  • Recall also that sampling distributions can be theoretical (used in most studies) or empirical (seeing wider use via bootstrapping).
  • It is the properties of the sampling distribution
central limit theorem clt
Central Limit Theorem (CLT)
  • Suppose X is
    • random
    • mean 
    • standard deviation
    • not necessarily normal
terms concerning sampling distribution of the mean
Terms Concerning Sampling Distribution of the Mean
  • Standard Error of the mean:
    • Is just the standard deviation of the sampling distribution.
      • i.e. it is a particular standard deviation
  • Sampling error
    • The sample cannot be fully representative of the population
    • As such, there is variability due to chance
    • We could have a thousand sample means and none of them equal exactly the population mean. However…
clt continued
CLT (continued)
  • Properties of the sampling distribution of the mean
    • random
    • has a mean of 
    • has a standard error
    • Distributed approximately normal for large samples
    • Normal for all samples if the variable X is normal
the central limit theorem
The Central Limit Theorem
  • For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution as the sample size (N) gets larger.
    • This of course begs the question of what is ‘large enough’
  • Furthermore, the sampling distribution of the mean will have a mean equal to µ (the population mean), and a standard deviation equal to
central limit theorem
Central Limit Theorem
  • With the mean, we can use sample data and the normal curve to reach conclusions about the population of interest
  • We of course desire large, random samples in order to do
    • Non-random selection can result in under-selection or over-selection of subsections of the population.
      • e.g. carry out a telephone opinion poll
in summary sample means
In summary: sample means
  • are random
  • are normally distributed for large sample sizes
  • distribution has mean 
  • distribution has standard error (standard deviation)
confidence intervals
Confidence intervals
  • Draw a sample, gives us a mean
  • is our best guess at µ
  • For most samples will be close to µ
  • is a ‘point’ estimate
  • However, we can also give a range or interval estimate that takes into account the uncertainty involved in that estimate
    • Using the normal distribution
confidence interval equation
Confidence interval equation


= sample mean

Z = z value from normal curve

= standard error of the mean

95 confidence interval
95% confidence interval
  • Let’s say we want a 95% confidence interval.
  • Obtain1 the ‘critical’ z-score for p =.025
    • 2.5% above +z, and 2.5% below -z
  • p = .025 then z = 1.96
  • When the population standard deviation is not known, we use the t critical value instead
confidence interval example
Confidence interval example
  • Randomly selected a group of 10 of you folks with a mean score of 89 (s = 6) on the midterm.
  • What guess can we make as to the true mean of the class?
89 + 2.26*
  • 89 + 2.26(1.90)
  • (89 - 4.294) < < (89 + 4.294)
  • 84.71 < < 93.294
  • This seems pretty wide; it essentially covers a full letter grade. Why do you think that is?
important what a confidence interval means
Important: what a confidence interval means
  • A 95% confidence interval means that:

95% of the confidence intervals calculated on repeated sampling of the same population will contain µ

  • Note that the population value does not vary i.e. it’s not a 95% chance that it falls in that specific interval1
  • In other words, the CI attempts to capture the true population mean, but we would have a different interval estimate for each sample drawn
  • In R


question to think about
Question to think about
  • How does one know if the confidence interval calculated actually contains the true population mean?