Chapter 22. Comparing Two Proportions. Objectives. Sampling distribution of the difference between two proportions 2-proportion z-interval Pooling 2-proportion z-test. Comparing Two Proportions.
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Proportions observed in independent random samples are independent. Thus, we can add their variances. So…
The standard deviation of the difference between two sample proportions is
Thus, the standard error is
Provided that the sampled values are independent, the samples are independent, and the samples sizes are large enough, the sampling distribution of
is modeled by a Normal model with
The approximately normal sampling distribution for ( 1 – 2):
When the conditions are met, we are ready to find the confidence interval for the difference of two proportions:
The confidence interval is
The critical value z* depends on the particular confidence level, C, that you specify.
The pooled proportion is
If the numbers of successes are not whole numbers, round them first. (This is the only time you should round values in the middle of a calculation.)
We then put this pooled value into the formula, substituting it for both sample proportions in the standard error formula:
The conditions for the two-proportion z-test are the same as for the two-proportion z-interval.
We are testing the hypothesis H0: p1 – p2 = 0, or, equivalently, H0: p1 = p2.
The alternative hypothesis is either; Ha: p1≠p2, Ha: p1<p2, or Ha: p1>p2.
Because we hypothesize that the proportions are equal, we pool them to find
We use the pooled value to estimate the standard error:
Now we find the test statistic:
When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value.
The P-value .5552 is too high, thus we fail to reject the null hypothesis. There is insufficient evidence to show a difference between the failure rates of rural and suburban students on the AP Statistics exam.
The p-value = .0035 is less than α= .01, thus we reject the null hypothesis. There is sufficient evidence at the 1% significance level to conclude that the proportion of working mothers who feel they have enough spare time for themselves is less than the corresponding proportion of working fathers.