3.2 Sampling Distributions for Sample Proportions

Calculate the mean and standard deviation of a sampling distribution for a sample proportion.

• For a population with population proportion p, when the sampled values are independent, the sampling distribution of a sample proportion p has a mean μp = p and a standard deviation p pp n  1 .

Justify the appropriateness of conditions for the sampling distribution of a sample proportion.

• Sampling without replacement requires that two conditions be met:
  • i. The randomization condition—the data should be collected using a random sample.
  • ii. The 10% condition—the population size must be at least 10 times larger than the sample size nN≤10 , where N is the size of the population and n is the sample size.
• The sampling distribution of the sample proportion p is approximately normal provided the sample size is large enough. To ensure the sample size is lar ge enough, the following condition must be met: np 10 and n 1 p 10, where np is the expected number of successes and n 1 p is the expected number of failures.

Interpret the mean, standard deviation, and probabilities for a sampling distribution of a sample proportion.

• The mean, standard deviation, and probabilities for a sampling distribution of a sample proportion should be interpreted in the context of a specific population.