You Explain It! ESPSuppose an acquaintance claims to have the ability to determine the birth month of randomly selected individuals. To test such a claim, you randomly select 80 individuals and ask the acquaintance to state the birth month of the individual. If the individual has the ability to determine birth month, then the proportion of correct birth months should exceed 1/12 ≈ 0.083, the rate one would expect from simply guessing.
b. Suppose the individual was able to guess nine correct birth months. The P-value for such results is 0.1726. Explain what this P-value means and write a conclusion for the test.

Verified step by step guidance

1. Understand the null hypothesis (\(H_0\)) and alternative hypothesis (\(H_a\)): Here, \(H_0\) states that the individual is just guessing, so the true proportion of correct guesses is \(p = \frac{1}{12} \approx 0.083\). The alternative hypothesis \(H_a\) is that the individual has some ability, so \(p > 0.083\).
2. Interpret the P-value: The P-value of 0.1726 represents the probability of observing 9 or more correct guesses out of 80 purely by chance if the null hypothesis is true (i.e., if the person is just guessing).
3. Explain what a P-value means in hypothesis testing: A P-value measures how compatible the observed data is with the null hypothesis. A small P-value (usually less than a significance level like 0.05) suggests the data is unlikely under \(H_0\), leading us to reject \(H_0\). A large P-value suggests the data is consistent with \(H_0\).
4. Based on the P-value of 0.1726, which is greater than common significance levels (e.g., 0.05), we do not have strong evidence to reject the null hypothesis. This means the observed number of correct guesses could reasonably occur by random chance.
5. Write a conclusion: We conclude that there is insufficient evidence to support the claim that the individual can determine birth months better than random guessing. The data does not show a statistically significant ability to guess birth months.

Key Concepts

P-value

The P-value measures the probability of obtaining results at least as extreme as the observed data, assuming the null hypothesis is true. A small P-value indicates strong evidence against the null hypothesis, while a large P-value suggests the data is consistent with the null. It helps decide whether to reject or fail to reject the null hypothesis.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) represents the default assumption, often stating no effect or no difference—in this case, that the individual guesses birth months correctly at the chance rate of 1/12. The alternative hypothesis (Ha) claims the individual’s accuracy is greater than chance. Hypothesis testing evaluates evidence to support or refute these claims.

Statistical Significance and Decision Making

Statistical significance determines if observed results are unlikely under the null hypothesis, typically using a significance level (e.g., 0.05). If the P-value is less than this level, we reject H0; otherwise, we fail to reject it. Here, a P-value of 0.1726 is greater than 0.05, so we do not have enough evidence to conclude the individual can guess birth months better than chance.