Question

like any seeds, watermelon seeds lose their viability over time. if they've been left sitting for years, it's possible that only a small proportion of the seeds may sprout when planted. if they have been stored in a cool, dark, dry location, they have a better chance of sprouting. drake has a large quantity of old watermelon seeds he's been saving, and now wants to know how many of them may still be able to sprout. he randomly selects 50 seeds from the set, places them between damp paper towels, and puts them in a plastic baggie in a warm spot on the counter. he finds that 27 out of the 50 seeds do sprout. does this provide good evidence that over half of all of the seeds will sprout?

Answer 1

Drake's experiment does not provide strong evidence that more than 50% of all the old watermelon seeds will sprout.

How to determine evidence basis?

To determine if Drake's experiment provides good evidence that over half of all the seeds will sprout, perform a hypothesis test. Specifically, use a binomial test to assess the probability of observing at least 27 successes (sprouted seeds) in 50 trials (seeds tested), given a null hypothesis that the true proportion of seeds that can sprout is 50% or less.

The null hypothesis (H₀) and alternative hypothesis (H₁) can be stated as follows:

H₀: p ≤ 0.5 (The proportion of seeds that can sprout is 50% or less.)

H₁: p > 0.5 (The proportion of seeds that can sprout is more than 50%.)

In this test, calculate the probability (p-value) of observing 27 or more sprouted seeds out of 50 if the true sprouting rate is 50% or less.

Given:

Number of trials (seeds tested), n = 50.

Number of successes (sprouted seeds), k = 27.

Null hypothesis proportion, p₀ = 0.5.

The p-value calculation involves summing the probabilities of getting 27 or more sprouts out of 50 seeds, given that the true sprouting rate is 50%. Mathematically, this is represented as:

p-value = P(X ≥ 27) when p = 0.5.

This is calculated using the cumulative distribution function of the binomial distribution, considering all outcomes from 27 successes to 50 successes.

The p-value for the hypothesis test is approximately 0.336.

A p-value greater than 0.05 indicates that we do not have sufficient evidence to reject the null hypothesis at the 95% confidence level. Therefore, based on this test, we cannot conclude that more than half of all the seeds will sprout.