Testing Claims About Proportions
In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.


Belief in Ghosts In a Harris Interactive poll of 2250 adults, 42% of the respondents said that they believe in ghosts. Use a 0.01 significance level to test the claim that more than of adults believe in ghosts.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents the claim that the proportion of adults who believe in ghosts is equal to or less than 42% (p ≤ 0.42). The alternative hypothesis represents the claim that the proportion of adults who believe in ghosts is greater than 42% (p > 0.42).
2. Step 2: Identify the sample proportion (p̂), sample size (n), and hypothesized population proportion (p₀). Here, p̂ = 0.42 (42%), n = 2250, and p₀ = 0.42. These values will be used to calculate the test statistic.
3. Step 3: Calculate the test statistic using the formula: z = (p̂ - p₀) / √((p₀(1 - p₀)) / n). Substitute the values of p̂, p₀, and n into the formula to compute the z-score.
4. Step 4: Determine the P-value corresponding to the calculated z-score. Since this is a one-tailed test (right-tailed), find the area to the right of the z-score using a standard normal distribution table or statistical software.
5. Step 5: Compare the P-value to the significance level (α = 0.01). If the P-value is less than 0.01, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Based on this decision, state the conclusion about the null hypothesis and whether the claim that more than 42% of adults believe in ghosts is supported.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if the P-value is less than the significance level (e.g., 0.01), the null hypothesis is rejected.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this case, a significance level of 0.01 means that there is a 1% risk of concluding that a claim is true when it is actually false, guiding the decision-making process in hypothesis testing.