Confidence Intervals for Two Proportions. Section 6.1. Section 6.1 CI for Two Proportions. We are interested in confidence intervals for the difference p 1 – p 2 between the unknown vlaues of two population proportions.
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For two independent samples of sizes n1 and n2 with sample proportion of successes 1 and 2 respectively, an approximate level C confidence interval for p1 – p2is
C is the area under the standard normal curve between −z* and z*.
Use this method when
Football coaches often employ the “icing the kicker” strategy. To ice the kicker the opposing coach calls for a timeout just before the kicker attempts a field goal, hoping that the delay interrupts the kicker’s concentration and causes him to miss the kick.
Standard error of the difference p1−p2:
So the 95% CI is 0.024 ± 0.0672 = (0.0432, 0.0912)
We are 95% confident that the interval 4.32% to 9.12% captures the true difference in the ABILITY of kickers to make a field goal when iced and their ABILITY to make a field goal when not iced. Because 0 is in the interval, we do not have convincing evidence that there is a significant difference in the ABILITY of kickers to make field goals when iced and when not iced.
A common mistake is to calculate a one-sample confidence interval for p1, a one-sample confidence interval for p2, and to then conclude that
p1and p2 are equal if the confidence intervals overlap.
This is WRONG because the variability in the sampling distribution for
from two independent samples is more complex and must take into account variability coming from both samples. Hence the more complex formula for the standard error.
INCORRECT Two single-sample 95% confidence intervals: The confidence interval for the rightie BA and the confidence interval for the leftie BA overlap, suggesting no significant difference between Ryan Howard’s ABILITY to hit right-handed pitchers and his ABILITY to hit left-handed pitchers.
Rightie interval: (0.274, 0.366)
Leftie interval: (0.170, 0.280)
The age at which a woman gives birth to her first child may be an important factor in the risk of later developing breast cancer. An international study conducted by WHO selected women with at least one birth and recorded if they had breast cancer or not and whether they had their first child before their 30th birthday or after.
The parameter to be estimated is p1 – p2.
p1 = cancer rate when age at 1st birth >30
p2 = cancer rate when age at 1st birth <=30
We estimate that the cancer rate when age at first birth > 30 is between .05 and .082 higher than when age <= 30.