Inference about Two Population Proportions Section 10'3

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Inference aboutTwo Population Proportions
Section 10.3

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

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Objectives 10.3

• Conduct hypothesis tests on the difference
  between two population proportions
• Construct confidence intervals for the difference
  between two population proportions
• Determine sample size for the difference between
  two population proportions

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Example

• In clinical trials of Nasonex, 26 of patients
  receiving the drug reported a headache as a side
  effect, and 22 of patients receiving a placebo
  reported a headache as a side effect.
• Is the proportion of patients receiving the drug
  and complaining of headaches significantly higher
  than the proportion for those receiving a placebo?

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Sampling Distribution for Difference of 2
Proportions

• A simple random sample of size n1 is taken from a
  population where x1 individuals have a specified
  characteristic. And likewise, n2, x2. The
  sampling distribution of is
  approximately normal with mean
  and standard deviation

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Sampling Distribution for Difference of 2
Proportions

• If we standardize (subtract the mean
  and divide by the standard deviation), we get
• which has an approximately standard normal
  distribution.

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Test Statistic Difference of 2 Proportions

• The null hypothesis is , so
  let
• , then

We need an estimate for p
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Test Statistic Difference of 2 Proportions

• p is unknown, so we need a point estimate.
• The best point estimate of p is called the pooled
  estimate of p and is found by adding all of those
  with the characteristic and dividing by the sum
  of the two populations

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Test Statistic Difference of 2 Proportions

• Substituting the pooled estimate of p into the
  formula for Z gives our test statistic

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RequirementsHypothesis Test for Difference
between 2 Proportions

• If a claim is made regarding two population
  proportions, p1 and p2

• Independent simple random samples
• Sample size no more than 5 of population

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Example 1, p 462-3

• In clinical trials of Nasonex, 3774 patients were
  randomly divided into two groups. In Group 1
  (experimental group-drug), 547 of 2103 patients
  reported headaches as a side effect. In Group 2
  (control group-placebo), 368 of 1671 patients
  reported headaches.
• The claim is that the proportion of Nasonex users
  experiencing headaches is greater than the
  proportion from the control group. Use a 0.05
  level of significance.

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Example 1, p 462-3

• Check requirements

• Independent simple random samples

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Example 1, p 462-3

• Check requirements

• Independent simple random samples

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Example 1, p 462-3

• Check requirements

• Independent simple random samples
• Sample size is less than 5 of 10 million
  population

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Example 1, p 462-3
The claim is that the experimental group
proportion is greater than the control group
proportion (so their difference is positive) at a
0.05 level of significance. Null
Hypothesis H0 p1 p2 or H0 p1 - p2 0
versus Alternative Hypothesis Reject H0
if H1 p1 gt p2 Right-Tailed Z gt z0.05
1.645 or H1 p1 - p2 gt 0
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Example 1, p 462-3
We will use the calculator STAT?TESTS?6
2-PropZTest Enter the values of x1 547, n1
2103, x2 368, n2 1671
Do we reject H0?
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Practical Significance

• A result in statistics can be statistically
  significant but not practically significant.
• The proportion of Group 1 reporting headaches,
  26, is statistically significantly greater than
  the proportion of Group 2, 22. It is unlikely to
  have occurred by chance.
• But the 4 difference is not practically
  significant to someone wanting allergy relief.

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Practical Significance

• Small differences in statistics can be
  statistically significant but may not be large
  enough to have any practical significance.

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Confidence Intervals

• The same 3 requirements must be met.
• A (1a)100 confidence interval about p1 - p2 is
  given by
• We are NOT pooling the sample proportions because
  we make no assumption that they are equal when we
  construct confidence intervals.

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Example 14b, p. 468

• In two different surveys, people were asked if
  life existed elsewhere in the universe. In the
  first survey in 1996, 385 out of 535 answered
  yes. In the second survey in 1999, 326 out of
  535 answered yes.
• Construct a 90 confidence interval for the
  difference between the two population proportions
  p1996 p1999

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Example 14b, p. 468

• Verify requirements
• Simple random sample, less than 5 of total
  population

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Example 14b, p. 468

• We will use the calculator
• STAT?TESTS?B 2-PropZInt
• Enter the values of x1 385, n1 535, x2 326,
  n2 535
• The confidence interval is (0.06, 0.16)

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Determining Sample Size

• To determine the sample size n for a (1a)100
  confidence interval about p1 - p2 with margin of
  error, E

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Example 18, p. 469

• An educator wants to determine the difference
  between the proportion of males and females that
  have completed at least four years of college.
  What should the sample size be if the educator
  wants the estimate to be within two percentage
  points with 90 confidence, assuming that

• Prior estimates of 27.5 male and 23.1 female
  are used
• No prior estimates are used

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Example 18, p. 469

• Prior estimates of 27.5 male and 23.1 female
  are used
• Round up to 2551
• No prior estimates are used
• Round up 3383

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Questions

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