Exercise 4.29 on page 271 introduces a historical scenario...

Exercise 4.29 on page 271 introduces a historical scenario in which a British woman, Muriel Bristol-Roach, claimed to be able to tell whether milk had been poured into a cup before or after the tea. An experiment was conducted in which Muriel was presented with 8 cups of tea, and for each cup she correctly guessed whether the milk was added first or the tea added first. Let's assume that Muriel did not know beforehand how many of the 8 cups had tea first and how many had milk first. Let $p$ represent the true proportion of times Muriel can guess correctly. Our hypotheses are $H_{0}: p=0.5$ (random guessing) and $H_{a}: p>0.5$. (a) Give the value and notation for the sample statistic. (b) Use StatKey or other technology to generate a randomization distribution, and find the p-value.

Exercise 4.29 on page 271 introduces a historical scenario in which a British woman, Muriel Bristol-Roach, claimed to be able to tell whether milk had been poured into a cup before or after the tea. An experiment was conducted in which Muriel was presented with 8 cups of tea, and for each cup she correctly guessed whether the milk was added first or the tea added first. Let's assume that Muriel did not know beforehand how many of the 8 cups had tea first and how many had milk first. Let $p$ represent the true proportion of times Muriel can guess correctly. Our hypotheses are $H_{0}: p=0.5$ (random guessing) and $H_{a}: p>0.5$.
(a) Give the value and notation for the sample statistic.
(b) Use StatKey or other technology to generate a randomization distribution, and find the p-value.

Key Concepts

p-value

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed result, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a small p-value indicates that the observed data are unlikely under the null hypothesis, thus providing support for the alternative hypothesis.

Randomization Distribution

A randomization distribution is generated by repeatedly simulating the process under the assumption that the null hypothesis is true. This distribution represents the expected variability in the sample statistic from random chance alone and is used to determine how unusual the observed statistic is, helping in assessing evidence against the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical procedure in which competing hypotheses about a population parameter are evaluated based on sample data. In this context, the null hypothesis (H?) represents a statement of no effect or no difference (for example, that a person is merely guessing), while the alternative hypothesis (H?) reflects the claim that the person has some ability beyond random chance.

Sample Proportion

The sample proportion is a statistic that represents the fraction of successes (e.g., correct guesses) in a sample. It is typically denoted as p?. In hypothesis testing, p? is used as an estimate of the true proportion p to assess whether the observed outcome deviates significantly from what would be expected under the null hypothesis.

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