Hypothesis Test for Means of Paired Observations

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Hypothesis Test for Means of Paired Observations

• In the hypothesis tests discussed thus far, the
  underlying assumption is that samples are
  independent
• In several cases, this is not true
• Example
• We examine a firms dividend policy before and
  after a change in the tax code regarding taxation
  of dividends
• We examine average returns of two mutual funds
  over the same period

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Hypothesis Test for Means of Paired Observations

• In these two examples, we are referring to the
  case of matched pairs of observations
• In the first example, we have pairs of before
  and after observations
• In the second example, there may be common
  factors, capturing market conditions in each
  period, that affect the performance of the two
  funds

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Hypothesis Test for Means of Paired Observations

• Denote by A and B the before and after
  observations or the observations on two groups of
  entities
• Suppose that the difference between the ith pair
  of observations, i 1, 2, , n, is given by di
  xAi xBi
• The mean difference is denoted by ?d

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Hypothesis Test for Means of Paired Observations
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Hypothesis Test for Means of Paired Observations

• The hypotheses can be formulated as follows
• H0 ?d ?d0 vs. H1 ?d ? ?d0
• H0 ?d ? ?d0 vs. H1 ?d gt ?d0
• H0 ?d ? ?d0 vs. H1 ?d lt ?d0
• Following the same steps as with previous
  hypothesis tests, the t-statistic with (n 1)
  degrees of freedom is

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Hypothesis Test for Means of Paired Observations

• The sample mean difference and standard error of
  mean difference are

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Hypothesis Test for Means of Paired Observations

• Suppose we want to test the hypothesis that the
  mean quarterly returns on two portfolios are
  equal
• Given that we observe their returns during the
  same period, the returns of the two portfolios
  will not be independent because they will be
  influenced by common factors (market conditions)
• The testable hypotheses (at 5 significance
  level) are
• H0 ?d 0 vs. H1 ?d ? 0

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Example of Hypothesis Test forMean Differences
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Example of Hypothesis Test forMean Differences

• The sample mean difference is 3.69 and the
  standard deviation is 6.90
• The standard error of the mean difference is
  
• The t-statistic is (-3.69 0)/2.44 -1.51 with
  7 degrees of freedom
• The cutoff of the t with 7 degrees of freedom at
  the 5 significance level is 2.365 (rejection
  rule is that either t gt 2.365 or t lt -2.365)
• Thus, we fail to reject the null hypothesis

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Hypothesis Test for the Variance of aSingle
Population

• To test hypotheses about the variance of a normal
  population, we use the chi-squared statistic
• Note If the underlying distribution is not
  normal, tests based on the chi-squared statistic
  will produce incorrect results

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Hypothesis Test for the Variance of aSingle
Population
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Hypothesis Test for Equality of two Variances

• To test hypotheses about the variances of two
  populations, we use an F-statistic
• Suppose we obtain two samples with observations
  n1 and n2 and sample variances s12 and s22 and
  the samples are random, independent and obtained
  from normal distributions
• The appropriate test statistic for tests between
  variances of two populations is

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Hypothesis Test for Equality of two Variances

• The F-statistic follows the F-distribution with
  (n1 1) degrees of freedom in the numerator and
  (n2 1) degrees of freedom in the denominator

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Hypothesis Test for Equality of two Variances