In a recent poll, 30 percent of adults said they wanted to "cut down or be free of gluten," according to The NDP Group, the market-research company that conducted the poll. A researcher wonders if a smaller proportion of students at her university would respond in the same fashion.
Suppose the researcher conducts a survey of a random sample of 25 students at her university and five of them say they want to at least reduce gluten. A statistician carries out a significance test of the null hypothesis that the proportion wanting to reduce gluten at the university is the same as for all adults versus the alternative hypothesis that a smaller proportion p of students would say they want to reduce or be free of gluten.
What is the value of the standardized test statistic for this significance test?

Question

Quizplus · Accepted Answer

The null hypothesis is that the proportion of students wanting to reduce gluten is the same as for all adults, so p = 0.3. The alternative hypothesis is that a smaller proportion of students want to reduce gluten, so p < 0.3.

The sample proportion of students who want to reduce gluten is 5/25 = 0.2.

The test statistic for a one-tailed test is:

z = (sample proportion - hypothesized proportion) / standard error

z = (0.2 - 0.3) / sqrt(0.3 * 0.7 / 25) = -1.091

Therefore, the standardized test statistic for this significance test is -1.091.

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