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Inference for a Single Population Proportion for AP Statistics

By — McGraw-Hill Professional
Updated on Feb 4, 2011

Practice problems for these concepts can be found at:

We are usually more interested in estimating a population proportion with a confidence interval than we are in testing that a population proportion has a particular value. However, significance testing techniques for a particular population proportion exist and follow a pattern similar to those of the previous two sections. The main difference is that the only test statistic is z.

Notes on the preceding table:

  • The standard error for a hypothesis test of a single population proportion is different for a confidence interval for a population proportion. The standard error for a confidence interval,
  • is a function of , the sample proportion, whereas the standard error for a significance test,

    is a function of the hypothesized value of p.

  • Like the conditions for the use of a z-interval, we require that the np0 and n(1 – p0) be large enough to justify the normal approximation. As with determining the standard error, we use the hypothesized value of p rather than the sample value. "Large enough" means either np0 ≥ 5 and n(1 – p0) ≥ 5, or np0 ≥ 10 and n(1 – p0) ≥ 10 (it varies by text).

example: Consider a screening test for prostate cancer that its maker claims will detect the cancer in 85% of the men who actually have the disease. One hundred seventy-five men who have been previously diagnosed with prostate cancer are given the screening test, and 141 of the men are identified as having the disease. Does this finding provide evidence that the screening test detects the cancer at a rate different from the 85% rate claimed by its manufacturer?

solution:

example: Maria has a quarter that she suspects is out of balance. In fact, she thinks it turns up heads more often than it would if it were fair. Being a senior, she has lots of time on her hands, so she decides to flip the coin 300 times and count the number of heads. There are 165 heads in the 300 flips. Does this provide evidence at the 0.05 level that the coin is biased in favor of heads? At the 0.01 level?

solution:

Practice problems for these concepts can be found at:

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