Hypothesis Testing: The Pumpkin Patch Problem

Using discrete probability to evaluate a claim of random selection.

A friendly Labrador Retriever service dog wearing a red vest, looking attentively.

Scenario A: Strong Evidence

A farm has 20 pumpkins. 4 are prize-winners (>=50lbs) and 16 are regular. The farmer claims that when they send pumpkins to the market, they grab 3 at random. However, an inspector observes that all 3 pumpkins at the market are prize-winners. Test the claim that the farmer is actually selecting randomly versus cherry-picking the best ones.

Here are the weights (in lbs) of all the pumpkins for this scenario:

14, 18, 21, 25, 26, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 42, 55, 58, 61, 63

The observation: All 3 pumpkins selected for the display were "prize-winners" (weighing 50 lbs or more).

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Simulation: What Does Random Look Like?

We can simulate the random selection process thousands of times to see what a typical sample of pumpkins looks like. This helps us judge if our observed sample was truly unusual.