Sampling

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Sampling

• Computations and Graphs

2
Review of Last Weeks example..

• Central Limit Theorem Even if the underlying
  population is not normal, the sampling
  distribution of a statistic calculated from
  random samples of the population will be normal.
  
• Normal Distribution Rule of Thumb 2/3 of cases
  or 1 sd from the mean 95 of the cases, or
  2 sd from the mean 99, or 3 sd of mean

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A population of 1500 cases

• Case number AGE
• 1 40.00
• 2 38.00
• 3 42.00
• 4 40.00
• 5 63.00
• 6 52.00
• 7 35.00
• 8 33.00
• 9 10.00
• 10 3.00
• 11 59.00
• 12 35.00
• 13 50.00
• 14 39.00
• 15 7.00
• 16 1.00
• 17 52.00
• 18 45.00

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Histogram or Density Display
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Basic Statistics for the Population

• AGE
• N of cases 1500
• Minimum 0.00
• Maximum 87.00
• Mean 25.68
• Standard Dev 17.49
• Median 24.5

6
But usually.

• The population statistics are unknown so we take
  a sample to estimate them
• Mean 22.87
• N of cases 100

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Take more samples of the population and we get
different means

• Mean 22.87
• N of cases 100
• Mean 30.66
• N of cases 100
• Mean 25.82
• N of cases 100
• Mean 25.81
• N of cases 100
• Mean 23.89

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A Dilemma?

• How do we decide which is the best estimate?
• Answer Use sampling theory to assign an
  interval and level of confidence to our point
  estimate.

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Sampling Distribution of Sample Means

• Variable AGE
• N of samples 100
• Minimum mean 22.46
• Maximum mean 30.66
• Mean of sample means 25.7
• Std Dev of sample means 1.68

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Sampling Distribution of Sample Means
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Sampling Distribution with Normal Curve
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The Standard Error Standard Deviation of the
Sampling Distribution

• The Standard Error tells us, given a confidence
  level, how far away the sample mean is from the
  population mean.
• SE Standard Deviation divided by the Square
  root of N..

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Sampling Distribution of Sample Means

• Variable AGE
• N of samples 100
• Minimum mean 22.46
• Maximum mean 30.66
• Mean of sample means 25.7
• Std Dev of sample means 1.68

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95 Confidence Interval

• Tells us that 95 of the time, the sample mean
  falls within the interval.
• Calculated by adding 1.96 standard errors to the
  sample mean for the upper bound, and subtracting
  1.96 standard errors for the lower bound.

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For one sample

• AGE
• N of cases 100
• Mean 27.33
• 95 CI Upper 30.96
• 95 CI Lower 23.70
• Std. Error 1.83
• Standard Dev 18.27

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Review..

• 1. The population distribution
• 2. The sampling distribution
• 3. The sampling distribution with the normal
  curve superimposed