Hypothesis Tests of a Proportion

See also: Inferential Statistics: Basic Cases, Case 2

See also: If you don’t know the numbered steps by heart yet, feel free to refer to Hypothesis Tests: Six Steps (Plus One).

Problem: Cancer Screening

The CDC’s Colorectal Cancer Screening Guidelines recommend a colonoscopy every ten years for adults aged 50 to 75. A public-health researcher believes that only a minority are following this recommendation. She interviews a simple random sample of 500 adults aged 50–70 in Metropolis (pop. 6.4 million) and finds that 219 of them have had a colonoscopy in the past ten years. At the 0.05 level of significance, is her belief correct?

Solution: Hypothesis Test about p

The population is adults aged 50–75 in Metropolis. You want to know whether less than half of them follow the colonoscopy guideline. Each person either does or does not, so you have binomial data, Case 2 in Inferential Statistics: Basic Cases.

Comment: There are lots of p’s in case 2 problems, so keep the notation straight:

• p = the proportion of success in the population. This is the parameter you’re testing, so it appears in your hypotheses. It has some definite value, but you don’t know what that value is.
• p_o = the number in your hypotheses, the proportion you are testing against. If you think 80% of the population have a certain characteristic (or more than 80%, or less than 80%, or different from 80%), then p_o is 0.80.
• p̂ = the proportion of success in your sample. This is x/n (number of “yes” in sample divided by sample size), and the 1-PropZTest computes it for you.
• p on the output screen of the calculator = the p-value. To reduce confusion, write it as “p-value” or “pval”.

Comment: Even though you already have the sample data in the problem, when you write the hypotheses, ignore the sample. In principle, you write the hypotheses, then plan the study and gather data. If you use any of the sample data in the hypotheses, something is wrong.

Comment: Often it’s helpful to add some words to each hypothesis. If nothing else, it makes your job easier when writing your conclusions in step 6. But don’t just rewrite the symbols in English: write down the deeper meaning or the implications.

Comment: The problem generally tells you which significance level to use.

Next is the requirements check. Even though it doesn’t have a number, it’s necessary.

Comment: Why use p_o in the requirements check, instead of the actual sample proportion p̂ as you did when computing a confidence interval? Because every hypothesis begins by assuming the null hypothesis to be true, and the null hypothesis is that the true proportion of successes in the population is p_o — in this case, 0.5.

Comment: Some authors express the third requirement as “at least five successes and at least five failures in the sample”. We’ll follow what we did in Chapter 8.

Comment: Usually, if requirements aren’t met you just have to give up. But for one-population binomial data, where the first two requirements are met but the third is not, you can use MATH200A part 3 to compute the p-value directly. There’s an example below.

Now it’s time to compute the test statistic (z) and the p-value:

What do you write down? The screen name, all inputs (including the alternative hypothesis), and the outputs. Omit the items on the output screen that duplicate the inputs. The convention is to round the test statistic to two decimal places and the p-value to four.

Please note: the screen name is 1-PropZTest, not “PropZTest”. We’ll have a 2-PropZTest later, so get in the habit of distinguishing them now.

Don’t get creative here. Use the decision rule exactly as it’s written in Step 5 of Hypothesis Tests: Six Steps (Plus One).

Comment: When the p-value turns out to be greater than the significance level, you write “p>α. Fail to reject H₀” and you do not mention H₁.

Your conclusion must include either the significance level or the p-value. p-values give more information, but most books seem to use the significance level.

Comment: When p is greater than α, you fail to reach a conclusion: “It’s impossible to say, at the 0.05 significance level, whether [insert H₁ here, in English] or not.” In this situation, you must use neutral language.

Small Samples

What if your sample is so small that np_o(1−p_o) < 10? You can no longer use 1-PropZTest with its normal approximation, but you can compute the binomial probability directly as long as the other two requirements are still met (SRS and 20n≤N). Only the calculation of the p-value changes.

Example: In 2001, 9.6% of Fictional County motorists said that fuel efficiency was the most important factor in their choice of a car. For her statistics project, Amber set out to prove that the percentage has increased. She interviewed 80 motorists in a systematic sample of those registering vehicles at the DMV, and 13 of them said that fuel efficiency was the most important factor in their choice of a car. Test her hypothesis, at the 0.05 significance level.

Solution:

What’s New

• 16 Apr 2013: Correct typos here, here, and here.
• 30 Mar 2013: Add conclusions with p-values to all examples. Change the examples of HT with binomial data, here and here. Remove the statement that was here about not doing a hypothesis test even though we could.
• 18 Mar 2013 Refer the test statistic back to the sampling error of the proportion.
• 8 Jul 2012: Add the section on small samples.
• 28 Aug 2011: new document