Final answer:
The null hypothesis for the conjecture that the mean time to prepare dinner is 20 minutes is H0: μ = 20. The alternative hypothesis could vary as H1: μ ≠ 20, H1: μ > 20, or H1: μ < 20 depending on the specific claim being tested against the null hypothesis.
Step-by-step explanation:
The conjecture mentioned in the paragraph suggests that the mean time to prepare dinner is expected to be 20 minutes. To test this claim, we would set up null and alternative hypotheses. In statistical notation, the null hypothesis (H0) would state that the mean time, μ, to prepare dinner is 20 minutes.
Therefore, H0: μ = 20. Conversely, the alternative hypothesis (H1) would state that the mean time is different from 20 minutes, so H1: μ ≠ 20. If we were specifically testing whether it takes longer than conjectured, H1 would be μ > 20. Similarly, if we suspected it takes less time, H1 would be μ < 20.
To solve the mathematical problem completely, we must conduct a hypothesis test using the sample data. In cases like this, where the population standard deviation is known and the sample size is large (n ≥ 30), we would use a z-test. If the population standard deviation is unknown and the sample is small (n < 30), a t-test would be appropriate, assuming the sample comes from a normally distributed population. The choice between one-tailed or two-tailed tests depends on the alternative hypothesis: one-tailed for μ > 20 or μ < 20 and two-tailed for μ ≠ 20.