Hypothesis Testing Using a P-Value In Exercises 13–16, (a) identify the claim and state H0 and Ha, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim.


Stray Cats An animal advocate claims that 25% of U.S. households have taken in a stray cat. In a random sample of 500 U.S. households, 105 say they have taken in a stray cat. At α=0.05, is there enough evidence to reject the advocate’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that 25% of U.S. households have taken in a stray cat. This translates to the null hypothesis (H₀): p = 0.25, where p is the proportion of households that have taken in a stray cat. The alternative hypothesis (Hₐ) depends on whether the problem specifies a one-tailed or two-tailed test. Since the problem does not specify, assume a two-tailed test: Hₐ: p ≠ 0.25.
2. Step 2: Calculate the test statistic. Use the formula for the z-test for proportions: z = (p̂ - p₀) / √((p₀(1 - p₀)) / n), where p̂ is the sample proportion (105/500), p₀ is the hypothesized proportion (0.25), and n is the sample size (500). Plug in the values to compute the z-statistic.
3. Step 3: Use technology or statistical software to find the P-value. The P-value is the probability of observing a test statistic as extreme as the one calculated in Step 2, assuming the null hypothesis is true. For a two-tailed test, double the area in the tail of the standard normal distribution corresponding to the absolute value of the z-statistic.
4. Step 4: Compare the P-value to the significance level α = 0.05. If the P-value is less than α, reject the null hypothesis (H₀). If the P-value is greater than or equal to α, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If you rejected H₀, conclude that there is enough evidence to reject the advocate's claim that 25% of U.S. households have taken in a stray cat. If you failed to reject H₀, conclude that there is not enough evidence to reject the advocate's claim.

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 (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.

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 H0 is true. A smaller P-value indicates stronger evidence against H0, and if the P-value is less than the significance level (α), we reject H0.

Significance Level (α)

The significance level, denoted as α, is a threshold set by the researcher before conducting a hypothesis test. It defines the probability of making a Type I error, which occurs when H0 is incorrectly rejected when it is true. Commonly set at 0.05, this level helps determine whether the P-value indicates sufficient evidence to reject the null hypothesis in the context of the study.