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*(*n p), 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

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