Hypothesis Testing Using Rejection Regions In Exercises 7–12, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


Vaccinations In 2021, a reporter claims that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. In a random sample of 200 U.S. adults, 56% feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. At α=0.10, is there enough evidence to reject the reporter’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and the alternative hypothesis (Ha). The reporter claims that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. This translates to H0: p ≥ 0.55 (null hypothesis) and Ha: p < 0.55 (alternative hypothesis). The alternative hypothesis is based on testing whether the proportion is less than 55%.
2. Step 2: Determine the critical value(s) and rejection region(s). Since the significance level is α = 0.10 and the test is one-tailed (left-tailed), use a z-table to find the critical value corresponding to α = 0.10. The rejection region is z < critical value.
3. Step 3: Calculate the standardized test statistic z. Use the formula for the z-test for proportions: $z=\frac{(p-p}{}$₀)p₀(1-p₀)n, where p = 0.56 (sample proportion), p₀ = 0.55 (claimed proportion), and n = 200 (sample size). Substitute the values into the formula to compute z.
4. Step 4: Compare the calculated z-value to the critical value. If the calculated z-value falls within the rejection region (z < critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. Based on whether the null hypothesis is rejected or not, determine if there is enough evidence to reject the reporter’s claim that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall.

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.

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (α), which defines the probability of making a Type I error (rejecting H0 when it is true). In this case, with α=0.10, the rejection region will be based on the critical value(s) of the standardized test statistic, indicating where the sample evidence is strong enough to reject H0.

Standardized Test Statistic

The standardized test statistic, often denoted as z, measures how far the sample proportion is from the hypothesized population proportion under the null hypothesis, expressed in terms of standard deviations. It is calculated using the formula z = (p̂ - p0) / √(p0(1-p0)/n), where p̂ is the sample proportion, p0 is the hypothesized proportion, and n is the sample size. This statistic helps determine whether the observed sample proportion provides sufficient evidence to reject the null hypothesis.