Compare Candy-Color Guessing Experiments with Different Sample Sizes

A person claims they can identify the color of a candy purely by taste. There are five possible colors. Compare the evidence from correctly identifying 2 of 3 candies with correctly identifying 40 of 100 candies. ### Part 1 — Two Correct Out of Three Formulate a null and alternative hypothesis and calculate an appropriate one-sided probability under random guessing. Explain whether the result is convincing at a stated significance level. #### What This Part Should Cover The null success probability, binomial assumptions, an upper-tail calculation rather than an exact-count probability alone, and a qualified conclusion. ### Part 2 — Forty Correct Out of One Hundred Apply the same framework to the larger experiment and explain why a lower observed accuracy can provide stronger statistical evidence. #### What This Part Should Cover The role of sample size, the null sampling distribution, and the distinction between statistical evidence and proof of the claimed sensory mechanism. ### Constraints Assume independent trials, a predeclared sample size, no access to visual or other identifying cues, and uniform random guessing among five labels under the null, giving success probability 0.2. If those assumptions do not hold, discuss how the test changes. The significance level is not supplied; state one for illustration. ### Clarifying Questions - Are candy colors balanced or otherwise controlled, and is guessing behavior truly uniform under the null? - Were trials independent, and was the experiment stopped after looking at the results? - Is the claim simply better-than-chance identification or specifically identification by taste? ```hint Compare each result with its own null distribution The percentages 2/3 and 40/100 use different sample sizes, so their distance from chance must be assessed with different sampling uncertainty. ``` ### What a Strong Answer Covers - A correctly specified one-sided binomial test and assumptions. - Exact treatment of the small experiment and a valid tail expression for the large one. - Careful interpretation, study design, and stronger evidence without overclaiming causality. ### Follow-up Questions - How would repeatedly testing until a favorable result appears affect the conclusion? - What if one color appears much more often and the person always guesses that color?