Sampling Distributions of a Sample Proportion for AP Statistics

Practice problems for these concepts can be found at:

• Binomial Distributions, Geometric Distributions, and Sampling Distributions Multiple Choice Practice Problems for AP Statistics
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Free Response Practice Problems for AP Statistics
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Cumulative Review Problems
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Rapid Review for AP Statistics

If X is the count of successes in a sample of n trials of a binomial random variable, then the proportion of success is given by = X/n. is what we use for the sample proportion (a statistic). The true population proportion would then be given by p.

We learned in Section 8.1 that, if X is a binomial random variable, the mean and standard deviation of the sampling distribution of X are given by

We know that if we divide each term in a dataset by the same value n, then the mean and standard deviation of the transformed dataset will be the mean and standard deviation of the original dataset divided by n. Doing the algebra, we find that the mean and standard deviation of the sampling distribution of are given by:

Like the binomial, the sampling distribution of will be approximately normally distributed if n and p are large enough. The test is exactly the same as it was for the binomial: If X has B(np), and = X /n, then has approximately

provided that np ≥ 10 and n(1 – p) ≥ 10 (or np ≥ 5 and n(1 – p) ≥ 5).

example: Harold fails to study for his statistics final. The final has 100 multiple choice questions, each with 5 choices. Harold has no choice but to guess randomly at all 100 questions. What is the probability that Harold will get at least 30% on the test?

solution: Since 100(0.2) and 100(0.8) are both greater than 10, we can use the normal approximation to the sampling distribution of . Since .

Therefore,

. The TI-83/84 solution is given by normalcdf(0.3,100,0.2,0.040)=0.0062.

Harold should have studied.

Practice problems for these concepts can be found at:

• Binomial Distributions, Geometric Distributions, and Sampling Distributions Multiple Choice Practice Problems for AP Statistics
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Free Response Practice Problems for AP Statistics
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Cumulative Review Problems
• Binomial Distributions, Geometric Distributions, and Sampling Distributions Rapid Review for AP Statistics