Question

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).

Short answer

In plain English, the null hypothesis is that Muriel can't tell whether the tea or milk was poured first, while the alternative hypothesis is that she can. The null hypothesis in terms of \( p \) is \( p = 0.5 \), expressing that Muriel's correct guesses happen by pure chance, and the alternative hypothesis is \( p > 0.5 \) denoting her ability to guess correctly more often than by chance.

Step-by-step solution

Identify and Explain the Hypotheses in Plain English

In this case, the null hypothesis is Fisher's skeptical assumption that Muriel can't distinguish whether tea or milk was put in the cup first. Hence, any correct guess would be based solely on chance. In contrast, the alternative hypothesis is Muriel's claim that she is able to correctly identify whether milk or tea was poured first in the cup.

Formulate the Hypotheses in terms of Probability

The hypotheses can now be expressed in statistical terms. The null hypothesis, denoted as H_0, is the assertion that Muriel's guesses are by pure chance. Therefore, the probability of her guessing correctly, \( p \), is 0.5. On the other hand, the alternative hypothesis (H_a) espouses Muriel's claim that she gets it right more frequently than by mere luck, indicating \( p > 0.5 \).

Key concepts

null hypothesis

Imagine you're at a tea party, just like in the 1920's scenario with Muriel Bristol-Roach, and you hear an extraordinary claim. Someone says they can tell whether milk was poured before the tea, just by tasting. Now, if you were the statistician R.A. Fisher, you'd likely be skeptical and assume that the person is just guessing. This assumption of no special ability is what we call the null hypothesis. It's essentially the default statement suggesting that there is no effect or no difference, and it's what we seek to test against evidence.

In statistical tests, proving or disproving the null hypothesis is a crucial step. It embodies skepticism and requires evidence to be shown false, which would support the alternate claim. If we can't find sufficient evidence, then we fail to reject the null hypothesis. In this tea-tasting scenario, the null hypothesis would be that Muriel's correct guesses are due to chance alone, and this can be surprisingly powerful in leading to insights about the reality of a claim.

alternative hypothesis

Now, let's talk about the challenger to the null hypothesis—the alternative hypothesis. In our tea party story, this would be Muriel's bold claim that she can indeed discern the order in which milk and tea are poured into a cup. When you form an alternative hypothesis, you're suggesting that there is an effect or difference that contradicts the null hypothesis.

The alternative hypothesis can take different forms depending on the scenario. For Muriel, it's about performance better than chance—her success rate is greater than what random guesses would produce. In statistical testing, the alternative hypothesis needs to be specific and is usually formulated based on prior knowledge, theory, or the direct opposite of what the null hypothesis states. By setting up both hypotheses, researchers and skeptics can use data to see if there's enough evidence to lean towards the alternative claim or stay with the null.

probability

Probability is a way of expressing uncertainty or certainty. It's a numerical value between 0 and 1, where 0 means an event is impossible and 1 signifies absolute certainty. In hypothesis testing, probability helps us make judgments based on statistical evidence.

For instance, if Muriel's ability were due solely to chance, the probability (denoted by p) of her correctly identifying the order of milk and tea being poured would be 0.5 — akin to flipping a fair coin. The power of probability in hypothesis testing lies in comparing what we observe (the data) with what we expect under the null hypothesis. If the observed data are highly improbable under the null hypothesis (with a very small probability), this might indicate that the alternative hypothesis is more plausible.

statistical significance

One of the most exciting parts of hypothesis testing is figuring out if our results are just random chance or if they're statistically significant. Statistical significance is like a stamp of confidence on our findings—it tells us that what we observed is probably not due to just random variability and is likely a genuine effect.

In Muriel's case, if she consistently identifies the correct order of milk and tea being poured far more often than chance would allow, this would be considered statistically significant. This significance is often determined by a p-value, which is the probability of observing a statistic as extreme as the one we got, or more, if the null hypothesis were true. If this p-value is below a predetermined threshold (like 0.05), we start thinking, 'Hey, something interesting might be happening here!' and we may reject the null hypothesis, moving a step closer to believing the alternative hypothesis.