Business Context
QuantEdge is hiring analysts and uses a short mental-arithmetic screen to assess fast quantitative reasoning. The recruiting team wants to know whether candidates perform better than random guessing on a timed set of multiple-choice questions.
Problem Statement
A candidate answered 20 independent mental-arithmetic questions. Each question had 4 answer choices, with exactly 1 correct answer. The candidate got 9 questions correct. Assume that under random guessing, the probability of a correct answer on each question is 0.25.
Given Data
| Metric | Value |
|---|
| Number of questions | 20 |
| Choices per question | 4 |
| Probability correct under guessing | 0.25 |
| Candidate correct answers | 9 |
| Significance level | 0.05 |
Model the number of correct answers as a binomial random variable.
Requirements
- Define the appropriate probability distribution and its parameters.
- Compute the expected number of correct answers and the variance under random guessing.
- Calculate the probability of getting exactly 9 correct by chance.
- Test whether the candidate performed better than guessing using a one-sided hypothesis test.
- Compute the p-value for observing 9 or more correct answers.
- State whether the result is statistically significant at the 5% level.
- Briefly interpret what this means for using this screen in hiring.
Assumptions
- Each question is independent.
- Each question has the same difficulty and the same 0.25 probability of a correct answer under guessing.
- The candidate did not partially solve questions; each response is either correct or incorrect.
