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: Admissions Tests

From Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011), page 494:

There are two major college entrance exams that a majority of colleges accept for admission, the SAT and ACT. ACT looked at historical records and established 22 as the minimum score on the ACT math portion of the exam for a student to be considered prepared for college mathematics. ...

An official with the Illinois State Department of Education wonders whether less than half of the students in her state are prepared for College Algebra. She obtains a simple random sample of 500 records of students who have taken the ACT and finds that 219 are prepared for college mathematics, [meaning that they] scored at least 22 on the math portion of the ACT.

Does this represent significant evidence that less than half of the students in the state of Illinois are prepared for college mathematics upon graduation from a high school? Use the α = 0.05 level of significance.

Solution: Hypothesis Test about p

The population is graduating high-school seniors in Illinois. You want to know whether less than half of them are prepared for college math. Each person either is or is not prepared, so you have binomial data, Case 2 in Inferential Statistics: Basic Cases.

Comment: There are lots of p’s in case 2 problem, 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 your book does.

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 on pages 497–498 of Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011). However, we won’t use that procedure in this course. As far as we’re concerned, if the requirements for Case 2 aren’t met, just stop.

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: If the p-value had turned out to be greater than the significance level, you would write “p>α. Fail to reject H₀” and you would not mention H₁.

Your conclusion must include either the significance level or the p-value. p-values give more information, but your book generally includes significance levels, so I’ll follow that.

Comment: If p was greater than α, you would fail to reach a conclusion: “It’s impossible to say, at the 0.05 significance level, whether less than half of graduating seniors in Illinois are prepared for college mathematics or not.” Don’t fall into the trap of implying that it’s not less; you must use neutral language.