Hypothesis Testing II

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Hypothesis Testing II

• The Two-Sample Case

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Introduction

• In this chapter, we will look at the difference
  between two separate populations
• As opposed to the difference between a sample and
  the population, which was Chapter 8
• example males and females or people with no
  children compared with people with at least one
  child
• You cannot test all males and all females, so
  need to draw a random sample from the population
• Will want to find that the difference between the
  samples is real (statistically significant)
  rather than due to random chance

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Summary of Chapter

• Difference between two groups means for large
  samples
• Difference between two groups means for small
  samples
• Difference between two groups proportions for
  large samples
• Will end the chapter with the limitations of
  hypothesis testing

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Hypothesis Testing with Sample Means

• Large Samples

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Assumptions

• We need to assume that each sample is random, and
  also that the two samples are independent of each
  other
• When random samples are drawn in such a way that
  the selection of a case for one sample has no
  effect on the selection of cases for another
  sample, the samples are independent
• To satisfy this requirement, you may randomly
  select cases from one list of the population,
  then subdivide that sample according to the trait
  of interest

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More Assumptions

• In the two-sample case, the null is still a
  statement of no difference, but now we are
  saying that the two populations are no
  different from each other
• The null stated symbolically

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Null Hypothesis

• We know that the means of our two samples are
  different, but we are stating in the null that
  they are theoretically the same in the two
  populations
• If the test statistic falls in the critical
  region, we as the researchers may conclude that
  the difference did not occur by random chance,
  and that there is a real difference between the
  two groups

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Test Statistic

• In this chapter, the test statistic will be the
  difference in sample means
• If sample size is large, meaning that the
  combined number of cases in the two samples is
  larger than 100, the sampling distribution of the
  differences in sample means will be normal in
  form and the standard normal curve can be used
  for critical regions
• Instead of plotting sample means or proportions
  in the sampling distribution, we will plot the
  difference between the means of each sample

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Formula for Z (Obtained)

• The Formula

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Revised Formula

• We do not know the means of the populations in
  this chapteronly know the means for the samples
• The expression for the difference in the
  population means is dropped from the equation
  because the expression equals zerowe assume in
  the null hypothesis that the values are the same

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New Formula for Z (Obtained)

• The Formula

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Pooled Estimate

• Use Formula 9.4 for the denominator if we do not
  know the population standard deviation (called
  the pooled estimate)

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Interpretation

• For the example in your book, you need to
  interpret the numbers
• Need a statistical interpretation
• Know that there is a difference between the means
  of the two groups
• Are doing the test of hypothesis to see if the
  difference is large enough to justify the
  conclusion that it did not occur by random chance
  alone but reflects a significant difference
  between men and women on this issue
• In your book, the Z (obtained) is -2.80, and Z
  (critical) is plus or minus 1.96
• So, can conclude that the difference did not
  occur by random chance
• The outcome falls in the critical region, so it
  is unlikely that the null is true

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Sociological Interpretation

• Begin by looking at which group has the lower
  mean
• In your book, we see that men have a lower
  average score on the Support for Gun Control
  Scale, so are less supportive of gun control than
  women
• We know that men and women are different in terms
  of their support for gun control
• Why would this be true?

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Hypothesis Testing with Sample Means

• Small Samples

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Distribution

• Cannot use the Z distribution for the sampling
  distribution of the difference between sample
  means
• Instead will use the t distribution to find the
  critical region for unlikely sample outcomes
• Will need to make two adjustments
• The degrees of freedom now will be (N1 N2) - 2

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Second Assumption

• With small samples, to justify the assumption of
  a normal sampling distribution and to form a
  pooled estimate of the standard deviation of the
  sampling distribution, we need to assume that the
  variances of the populations of interest are
  equal
• We may assume equal population variances if the
  sample sizes are approximately equal
• If one sample is large, and the other is small,
  we cannot use this test

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Formula for the Pooled Estimate

• Formula for the pooled estimate of the standard
  deviation of the sampling distribution is
  different for small samples than for large
  samples (see Formula 9.5)

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Formula 9.6 for t (obtained)

• It is the same as for Z (obtained)

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Interpretation of the Results

• The example in your book
• Statistical interpretation
• Will use a two-tailed test, since no direction
  has been predicted
• The test statistic falls in the critical region,
  so married people with no children and married
  people with at least one child are significantly
  different on the variable satisfaction with
  family life

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Sociological Interpretation

• Begin by comparing the means
• Higher scores indicate greater satisfaction
• Who is in each sample?
• The samples were divided into respondents with no
  children and respondents with at least one child
• Find that the respondents with no children scored
  higher on this attitude scale
• They are more satisfied with family life
• We know this difference is not due to chance, but
  is a real difference

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Hypothesis Testing With Sample Proportions (Large
Samples)

• The null hypothesis states that no significant
  difference exists between the populations from
  which the samples are drawn
• Will use the formulas for proportions when there
  is a percentage in the question

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Formula 9.8 for Z (obtained)
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The Limitations of Hypothesis Testing

• For All Tests of Hypothesis

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Probability of Rejecting the Null

• The probability of rejecting the null is a
  function of four independent factors
• The size of the observed differences
• The greater the difference, the more likely we
  reject the null
• The alpha level
• The higher the alpha level, the greater the
  probability of rejecting the null hypothesis

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Probability of Rejecting the Null

• The use of one- or two-tailed tests
• The use of the one-tailed test increases the
  probability of rejection of the null
• The size of the sample
• The value of all test statistics is directly
  proportional to sample size (not inversely
  proportional)
• The larger the sample, the higher the probability
  of rejecting the null hypothesis

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Two things to Remember about Sample Size

• Larger samples are better approximations of the
  populations they represent, so decisions based on
  larger samples about rejecting or failing to
  reject the null, can be regarded as more
  trustworthy
• It shows the most significant limitation of
  hypothesis testing

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Limitation of Hypothesis Testing

• Because a difference is statistically significant
  does not guarantee that it is important in any
  other sense
• Particularly with very large samples (Ns in
  excess of 1,000) where very small differences may
  be statistically significant
• Even with small samples, trivial differences may
  be statistically significant, since they
  represent differences in relation to the standard
  deviation of the population
• So, statistical significance is a necessary but
  not sufficient condition for theoretical
  importance
• Once a research result has been found to be
  significant, the researcher still faces the task
  of evaluating the results in terms of the theory
  that guides the inquiry

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Conclusion

• A difference between samples that is shown to be
  statistically significant may not be
  theoretically important, practically important,
  or sociologically important
• Logic will have to determine that
• And measures of association that show the
  strength of the association