21: Chapter Comparing Two Proportions: Key Ideas Sampling distribution of the difference in sample ^ ^ proportions p1 - p2 successes from two populations. Then the sampling ^ ^ distribution of p1 - p2 : o Is approximately normal, o Has mean p1 - p2 (the difference in the population proportions of successes) o Has standard deviation
^ ^ Let p1 and p2 be the sample proportions of
p1 (1 - p1 ) p2 (1 - p2 ) + n1 n2
o Assumptions: The data are from an SRSs from the corresponding populations. The populations are at least 10 times as large as the corresponding samples. The sample sizes are large. N.B.: Specific guidelines for significance tests and confidence intervals are given below.
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Testing the equality of proportions from two samples: z test for H 0 : p1 = p2 1. State the problem in context 2. Formulate a. Define the parameters of interest b. State H 0 and Ha . 3. Solve a. Check that the conditions are satisfied (see below). b. Compute two-sample z statistic
z =
^ ^ p1 - p2 ^ ^ p (1 - p ) 1 n1 + 1 n2
^ where p is the pooled sample proportion defined as
^ p = count of successes in both samples combined count of observations in both samples combined .
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4. Conclude in the context of the stated problem. using the Rossman-Chance applet for critical normal values.
Be sure to specific the alternative hypothesis correctly. Note: This test can carried be out by hand calculation
Conditions for use of the z significance test for comparing two proportions: o The data are from two independent SRSs. o The population is at least 10 times as large as the sample. o The counts of success and failures are each 5 or more in both samples. Plus four z confidence interval for p1 - p2
1. State the parameter estimation problem in context
2. Formulate
a. Define the parameter of interest b. State the level of confidence to be used
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3. Solve a. Check that the conditions are satisfied (see guidelines below). b. Calculate the level C confidence interval for p1 - p2 using the plus-four interval:
( p1 - p2 ) z
where
*
p1 (1 - p1 ) p2 (1 - p2 ) + n1 n2
n1 = n1 + 2 and n2 = n2 + 2 p1 and p represent the plus four estimate
2
of p and p respectively:
1 2
p1 =
...