Comparing

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Chapter 22

• Comparing
• Two Proportions

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Yet Another Standard Deviation (YASD)

• Standard deviation of the sampling distribution
• The variance of the sum or difference of two
  independent random variables is the sum of their
  variances

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Standard Deviation of the Difference Between Two
Proportions

• Proportions observed in independent random
  samples are independent
• Sample proportions

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Assumptions Conditions

• Independence Assumption
• Within each group, the data should be based on
  results for independent individuals
• Randomization
• The data in each group should be drawn
  independently and at random from a homogeneous
  population or generated by a randomized
  comparative study

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Assumptions Conditions

• 10 condition
• When the data are sampled without replacement,
  the sample should not exceed 10 of the
  population.
• Independent Samples condition
• The two groups that are being compared must be
  independent of one another.
• Success/failure
• Both samples are big enough that at least 10
  successes and and least 10 failures have been
  observed.

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The Sampling Distribution

• The sampling distribution model for a difference
  between two independent proportions
• Provided that the sampled values are independent,
  the samples are independent, and the sample sizes
  are large enough

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A Two-proportion z-interval

• When the conditions are met, find the confidence
  interval
• The critical value depends on the confidence
  level, C, that you specify.

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Intelligence An Example

• Identify the parameter and choose a confidence
  level

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Intelligence An Example

• Check the conditions.
• Randomization Gallup drew a random sample of
  U.S. adults
• 10 the sample size for each groups was
  certainly less than the U.S. population of men
  and women
• Independent Samples the sample of women and the
  sample of men are independent of each other

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Intelligence An Example

• Check the conditions.
• Success/Failure
• Both samples exceed the minimum size.

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Intelligence An Example

• State the sampling distribution model
• Under these conditions, the sampling distribution
  of the difference between the sample proportions
  is approximately Normal with a mean of pM pF,
  the true difference between the population
  proportions.
• Find a two-proportion z-interval.

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Intelligence An Example

• Calculator

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Intelligence An Example

• Interpret the results
• We are 95 confident that the proportion of
  American men that the attribute of intelligent
  applies more to men than to women is between 9
  and 19 more than American women who think that,

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Pooling

• Pooling combines the counts to get an overall
  proportion
• When we have counts for each group
• When we have only proportions
• Round to whole numbers

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Two-proportion z-test

• The conditions for the two-proportion z-test are
  the same as for the two-proportion z-interval
• Test the hypothesis
• Because we hypothesize that the proportions are
  equal, pool to find

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Two-proportion z-test

• Standard Error
• Find the test statistic

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Snoring Rates An Example

• Hypothesis
• HOThere is no difference in snoring rates
  between those who are 18 29 years old and those
  who are 30 years old.
• HA The rates are different.

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Snoring Rates An Example

• Check the conditions
• Randomization the patients were randomly
  selected and stratified by sex and region
• 10 the number of adults surveyed is certainly
  less than 10 of the population.
• Independent samples the two groups are
  independent of each other
• Success/failure
• Younger group 48 snored, 136 didnt
• Older group 318 snored, 493 didnt

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Snoring Rates An Example

• State the null model and choose your method
• Because the conditions are met, we can model the
  sampling distribution of the difference in
  proportions with a Normal model
• Perform a two-proportion z-test

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Snoring Rates An Example

• Calculator

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Snoring Rates An Example

• State your conclusion
• The P-value of 0.0008 says that if there really
  were no difference in snoring rates between the
  two age groups, then the difference observed
  would only happen 8 out of 10,000 times.
• This is rare enough for us to reject the null
  hypothesis of no difference and conclude that
  there is a difference between older and younger
  adults. It appears that older adults are more
  likely to snore.

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What Can Go Wrong???

• Dont use two-sample proportion methods when the
  samples arent independent
• Make sure there is no relationship between the
  two groups
• When the assumption of independence is violated,
  this method gives wrong answers
• Dont apply inference methods when you dont have
  random samples
• Dont interpret a significant difference in
  proportions causally.