The null hypothesis on true/false tests is that the student...

The null hypothesis on true/false tests is that the student is guessing, and the proportion of right answers is $0.50$. A student taking a five-question true/false quiz gets 4 right out of 5 . She says that this shows that she knows the material, because the one-tailed p-value from the one-proportion $z$ -test is $0.090$, and she is using a significance level of $0.10 .$ What is wrong with her approach?

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Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter by comparing observed data to what is expected under a specified hypothesis, known as the null hypothesis. In this context, the null hypothesis represents the scenario of guessing, which specifies a fixed probability of success that must be rigorously tested against an alternative hypothesis.

Appropriateness of Test Statistics

Test statistics such as the z-test are typically used when data satisfy certain conditions—usually related to sample size and distributional approximations (e.g., the central limit theorem). Applying a z-test to a very small sample, such as a five-question quiz, can violate these assumptions because the normal approximation may not hold, leading to unreliable p-values.

Discrete Data and Exact Distributions

When dealing with a small number of discrete outcomes (like true/false questions), the binomial distribution accurately models the probability of a given number of successes under the null hypothesis. In such cases, an exact binomial test should be used instead of a normal approximation because it properly accounts for the discrete nature of the data, avoiding potential inaccuracies in the p-value calculation.

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