3.9 Sampling Distributions for the Difference Between Two Population Proportions

Calculate the mean and standard deviation of the sampling distribution for the difference between two sample proportions.

• For two independent populations, with population proportions p1 and p2, when the sampled values are independent, the s ampling distribution for the difference in sample proportions, p p , has a mean, pp pp12 12 and standar 12 d deviation, pp pp n pp n12 11 1 22 2 11 .

Justify the appropriateness of conditions for the sampling distribution for the difference between two sample proportions.

• When sampling without replacement, two conditions must be met:
  • i. The randomization condition—the data should be collected using two independent random samples.
  • ii. The 10% condition—the size of each sample should be less than or equal to 10% of the respective population size: n11 10%N and n22 10%N , where N is the size of population 1 and 1 N2 is the size of population 2. The sample sizes are represented as n1 and n2.
• If the data come from an experiment, the data only need to meet the randomization condition. The treatments must be randomly assigned to the experimental units to meet the randomization condition.
• The sampling distribution for the difference between sample proportions, p 12 p , will have an approximately normal distribution provided both s ample sizes are large enough. To ensure that both samples are large enough, the data must meet the following conditions: n11p 10, n22 1 p 10, n11 11 p 0, n22p 10, and where n11p and n22p are the expected numb n2 1 p2 er of successes and n11 1 p and are the expected number of failures.

Interpret the mean, standard deviation, and probabilities for the sampling distribution for the difference between two sample proportions.

• The mean, standard deviation, and probabilities for the sampling distribution for the difference between two sample proportions should be interpreted within the context of two specific populations. Inference for Categorical Data: Proportions UNIT 3 102