Samples and populations

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Samples and populations

• Estimating with uncertainty

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Review - order of operations
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Review - order of operations

• Parentheses
• Exponents and roots
• Multiply and divide
• Add and subtract

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Review - order of operations
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Review - order of operations
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Review - types of variables

• Categorical variables
• For example, country of birth
• Numerical variables
• For example, student height

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Review - types of variables

• Categorical variables
• Numerical variables

Discrete
Continuous
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Review - types of variables
Nominal

• Categorical variables
• Numerical variables

Ordinal
Discrete
Continuous
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Review - types of variables

• Categorical variables
• Nominal - no natural order
• Ordinal - can be placed in an order

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Review - types of variables

• Categorical variables
• Nominal - no natural order
• Example - country of birth
• Ordinal - can be placed in an order

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Review - types of variables

• Categorical variables
• Nominal - no natural order
• Example - country of birth
• Ordinal - can be placed in an order
• Example - educational experience
• Some high school, high school diploma, some
  college, college degree, masters degree, PhD

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Sampling from a population

• We often sample from a population
• Consider random samples
• Each individual has an equal and identical
  probability of being selected

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Body mass of 400 humans
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Random sample of 10 people
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Population mean ? 70.8 kg
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Population mean ? 70.8 kg
Sample mean x 76.7 kg
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Another sample
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Population mean ? 70.8 kg
Sample mean x 69.2 kg
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What if we do this many times?

• Example gene length

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n 20,290
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n 20,290 ? 2622.0 ? 2037.9
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Sample histogram
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n 100 Y 2675.4 s 1539.2
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Y 2767.2 s 2044.7
Y 2675.4 s 1539.2
Y 2588.8 s 1620.5
Y 2702.4 s 1727.1
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Sampling distribution of the mean
Y 2767.2 s 2044.7
Y 2675.4 s 1539.2
Y 2588.8 s 1620.5
Y 2702.4 s 1727.1
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Sampling distribution of the mean
1000 samples
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Sampling distribution of the mean
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Sampling distribution of the mean
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? 2622.0
Sampling distribution of the mean
Mean of means 2626.4
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Y 2767.2 s 2044.7
Y 2675.4 s 1539.2
Y 2588.8 s 1620.5
Y 2702.4 s 1727.1
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Sampling distribution of the standard deviation
s 2044.7
s 1539.2
s 1620.5
s 1727.1
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Sampling distribution of the standard deviation
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Sampling distribution of the standard deviation
100 samples Population ? 2036.9 Mean sample s
1962.6
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Sampling distribution of the standard deviation
1000 samples Population ? 2036.9 Mean sample s
1929.7
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Sampling distribution of the mean, n10
Sampling distribution of the mean, n100
Sampling distribution of the mean, n 1000
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Sampling distribution of the mean, n10
Sampling distribution of the mean, n100
Sampling distribution of the mean, n 1000
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Larger sample size
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Group activity 2

• Form groups of size 2-5
• Get out a blank sheet of paper
• Write everyones full name on the paper

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How many toes do aliens have?
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Instructions

• You have measurements from a population of 400
  aliens
• Use your random number table to select a sample
  of ten measurements
• Calculate your sample mean and, if you have a
  calculator or a large brain, your sample standard
  deviation
• On your paper, answer the following
• What was your sample mean and standard deviation?
• How did you randomly choose your sample?

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Distribution of the sample mean

• No matter what the frequency distribution of the
  population
• The sample mean has an approximately bell-shaped
  (normal) distribution
• Especially for large n (large samples)

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How precise is any one estimated sample mean?
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The standard error of an estimate is the
standard deviation of its sampling distribution.
The standard error predicts the sampling error of
the estimate.
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Standard error of the mean
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Estimate of the standard error of the mean
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Confidence interval

• Confidence interval
• a range of values surrounding the sample estimate
  that is likely to contain the population
  parameter
• 95 confidence interval
• plausible range for a parameter based on the data

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The 2SE rule-of-thumb
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Confidence interval
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Pseudoreplication
The error that occurs when samples are not
independent, but they are treated as though they
are.
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Example The transylvania effect
A study of 130,000 calls for police assistance
in 1980 found that they were more likely than
chance to occur during a full moon.
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Example The transylvania effect
A study of 130,000 calls for police assistance
in 1980 found that they were more likely than
chance to occur during a full moon.
Problem There may have been 130,000 calls in
the data set, but there were only 13 full moons
in 1980. These data are not independent.