Hypothesis tests for difference between means

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Hypothesis tests for difference between means
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Hypothesis tests for difference between means

• Hypothesis testing
• Central limit theorem
• Sampling distribution of difference between means
• Null and alternative hypotheses
• Type I and type II errors
• Tests for difference between means
• two-sample t-test (equal variances)
• one sample t-test
• paired two-sample t-test
• two-sample z-test
• two-sample t-test (unequal variances)
• One-tailed and two-tailed tests

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Hypothesis Testing

• Where we formally test the objective of our
  scientific investigation
• As we would speak
• Male and female shrimps are different sizes
• In statistics speak
• H0 There is no difference in size between male
  and female shrimps
• H1 There is a difference in size between male
  and female shrimps
• More formally
• H0 ?1 ?2
• H1 ?1 ? ?2 or ?1 - ?2 ? 0

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Test for Differences Between Means
x
Question are these two means calculated from
samples of the same population?
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Test for Differences Between Means
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Central Limit Theorem

• If the sample size is sufficiently large
• the sampling distribution of the means
  approximates the normal probability distribution
• the underlying population does not have to be
  normally distributed
• If the sample size is small
• the sampling distribution of the means
  approximates the t-distribution, but only if the
  sample is normally distributed

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Central Limit Theorem

• We can extend the Central Limit Theorem to the
  sampling distribution of other statistics as
  well
• including the sampling distribution of the
  difference between two means.

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Test for Differences Between Means
?1 - ?2 is a statistic which has a sampling
distribution
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The sampling distribution of ?1 - ?2
??1-?2
?1- ?2
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The standard error of the difference between
means
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The sampling distribution of ?1 - ?2

• if samples are from same population, any
  deviations of ?1 - ?2 from zero are due to
  sampling
• the larger the value of ?1 - ?2 the less likely
  they are to be drawn from the same population
• the sampling distribution of ?1 - ?2 follows the
  t distribution

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Example two-sample test
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Example two-sample test
n1 20 ?1 16.67 s1 4.2
n2 20 ?2 13.75 s2 5.1
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Example two-sample test
H0 ?1 ?2 H1 ?1 - ?2 ? 0
n1 20 ?1 16.67 s1 4.2
n2 20 ?2 13.75 s2 5.1
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Example two-sample test
H0 ?1 ?2 H1 ?1 - ?2 ? 0
n1 20 ?1 16.67 s1 4.2
n2 20 ?2 13.75 s2 5.1
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Example two-sample test
H0 ?1 ?2 H1 ?1 - ?2 ? 0
n1 20 ?1 16.67 s1 4.2
n2 20 ?2 13.75 s2 5.1
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Example two-sample test
get critical value for t from the table degrees
of freedom n1 n2 2 20 20 2
38 significance (?) 0.05 tcrit 2.02
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Example two-sample test
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Hypothesis Testing for Sample Means

• State Null Hypothesis (H0)
• State Alternative Hypothesis (H1)
• Decide on Level of Significance
• Choose Test Distribution
• Define Rejection Regions
• State Decision Rule
• Calculations
• Make Statistical Decision

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Type I and Type II Errors
H0 ?1 ?2 H1 ?1 ? ?2
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Probability of Type I and Type II Errors

• Probability of a Type I Error is the significance
  level
• Probability of a Type I Error is unknown, but
• increases as significance level decreases

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One-sample t-test

• Test a sample mean against some known or
  suggested population mean
• H0 ? reported value
• H1 ? ? reported value

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Example one-sample test

• Water authority mean nitrate value 17.34 mg/l
• Sample ? 21.89 s 3.04 n 20
• Significance 0.05

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Example one-sample test
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Example one-sample test

• tcrit ?
• df n 1 20 1 19
• ? 0.05
• tcrit 2.093
• tcalc gt tcrit so we reject H0

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t-test for paired samples

• sometimes two samples are not independent
• then we test for difference between paired
  measurements

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t-test for paired samples
is the difference between each pair of
measurements is the mean of all the individual
differences is the best estimate of the standard
deviation of d.
degrees of freedom is n-1
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One-tailed and two-tailed tests

• Two-tailed test
• H0 ?1 ? 2 (or ? 1 - ? 2 0)
• H1 ? 1 ? ? 2 (or ? 1 - ? 2 ? 0)

One-tailed test H0 ? 1 ? 2 (or ? 1 - ? 2
0) H1 ? 1 gt ? 2 (or ? 1 - ? 2 gt 0)
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Z-test for Means

• For large samples (gt30)
• But might as well use t-test, as produce
  approximately the same results
• See notes for more details

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Key assumptions to t-test

• for small samples, the parent data from which the
  samples are drawn are normally distributed
• for large samples, the parent data can have any
  distribution
• the two samples come from distributions that may
  differ in their mean value, but not in the
  standard deviation (or variance) and
• the observations are random, and the samples are
  independent of each other (not the paired test).