Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.4.9

Chapter 8, Problem 8.4.9

Young Adults In a survey of 3500 males ages 20 to 24 whose highest level of education is some college, but no bachelor’s degree, 80.2% were employed. In a survey of 2000 males ages 20 to 24 whose highest level of education is a bachelor’s degree or higher, 86.4% were employed. At α=0.01, can you support the claim that there is a difference in the proportion of those employed between the two groups? (Adapted from National Center for Education Statistics)

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Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference in the proportion of employed individuals between the two groups: p₁ = p₂. The alternative hypothesis (Hₐ) states that there is a difference in the proportions: p₁ ≠ p₂.

Step 2: Identify the sample proportions and sample sizes. For the first group (some college, no bachelor’s degree), the sample size is n₁ = 3500 and the sample proportion is p̂₁ = 0.802. For the second group (bachelor’s degree or higher), the sample size is n₂ = 2000 and the sample proportion is p̂₂ = 0.864.

Step 3: Calculate the pooled proportion (p̂) under the null hypothesis. Use the formula: p̂ = (x₁ + x₂) / (n₁ + n₂), where x₁ = n₁ * p̂₁ and x₂ = n₂ * p̂₂. This pooled proportion represents the overall proportion of employed individuals across both groups.

Step 4: Compute the test statistic (z). Use the formula: z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]. This formula accounts for the difference in sample proportions and the variability due to sampling.

Step 5: Compare the test statistic to the critical value or p-value at α = 0.01. For a two-tailed test, find the critical z-values corresponding to α/2 = 0.005. If the absolute value of the test statistic exceeds the critical value, or if the p-value is less than 0.01, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. In this context, the null hypothesis would state that there is no difference in employment proportions between the two educational groups.

Proportion Comparison

Comparing proportions involves analyzing the differences between two or more groups regarding a categorical outcome. In this case, we are comparing the employment rates of two groups of young males based on their education levels. This comparison can be conducted using statistical tests such as the z-test for proportions, which assesses whether the observed difference is statistically significant.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining 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 scenario, α is set at 0.01, indicating a 1% risk of concluding that there is a difference in employment proportions when there is none, thus requiring strong evidence to support the claim.

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