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.


Working Students An education researcher claims that 65% of full-time college students work year-round. In a random sample of 105 college students, 66 say they work year-round. At α=0.10, is there enough evidence to reject the researcher’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and alternative hypothesis (Ha). The claim is that 65% of full-time college students work year-round. This translates to H0: p = 0.65 (null hypothesis) and Ha: p ≠ 0.65 (alternative hypothesis, since we are testing for a difference).
2. Step 2: Determine the significance level (α) and find the critical value(s). The significance level is given as α = 0.10. Since the test is two-tailed (Ha: p ≠ 0.65), divide α by 2 to find the critical values for both tails. Use a z-table or statistical software to find the z-scores corresponding to α/2 = 0.05 in each tail.
3. Step 3: Calculate the standardized test statistic z. Use the formula: $z=\frac{(p-p}{}$₀)p₀(1-p₀)n, where p = 66/105 (sample proportion), p₀ = 0.65 (claimed proportion), and n = 105 (sample size). Substitute the values into the formula to compute z.
4. Step 4: Compare the calculated z-value to the critical values to decide whether to reject or fail to reject the null hypothesis. If the z-value falls within the rejection region (outside the range defined by the critical values), reject H0. Otherwise, fail to reject H0.
5. Step 5: Interpret the decision in the context of the original claim. If H0 is rejected, conclude that there is enough evidence to reject the researcher’s claim that 65% of full-time college students work year-round. If H0 is not rejected, conclude that there is not enough evidence to reject the 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.

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 results would be considered statistically significant.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an element is from the mean. In hypothesis testing, it is calculated using the sample proportion and the hypothesized population proportion. This statistic helps determine how far the sample data deviates from the null hypothesis, allowing researchers to assess whether the observed results are statistically significant.