In this section we will turn our attention to comparing two population proportions. Once again, there are times when we aren't necessarily focused on the exact proportion, but rather how proportions from two populations compare, that is, if they are equal, or if one is larger than the other.
When we were comparing population means, we constructed a confidence interval for the difference between the two population means. Similarly, when comparing two population proportions, we use a confidence interval for the difference between the population proportions. The best point estimate for the difference is pˆ1−pˆ2
. In this section we will restrict our discussion to comparing two population proportions when the following conditions are met. Notice that the conditions are similar to those discussed for estimating a single population proportion.
All possible samples of a given size have an equal probability of being chosen; that is, simple random samples are used. The samples are independent. The conditions for a binomial distribution are met for both samples. The sample sizes are large enough to ensure that n1pˆ1≥5
, n1(1−pˆ1)≥5, n2pˆ2≥5, and n2(1−pˆ2)≥5
When these conditions are met, we can apply the Central Limit Theorem to the sampling distribution of the differences between the sample proportions for two independent samples. This means that we will use the standard normal distribution to calculate the margin of error of a confidence interval for the difference between two population proportions. You can assume that the necessary criteria are met for all examples and exercises in this lesson.
p=xN=# of successespopulation size
pˆ=xn=# of successessample size
Properties of a Binomial Distribution
The experiment consists of a fixed number, n, of identical trials. Each trial is independent of the others. For each trial, there are only two possible outcomes. For counting purposes, one outcome is labeled a success, and the other a failure. For every trial, the probability of getting a success is called p. The probability of getting a failure is then 1−p
The binomial random variable, X, counts the number of successes in n trials.