In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1<μ2; α=0.03
Population statistics:σ1=136 and σ2=215
Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference or μ₁ ≥ μ₂. The alternative hypothesis (Hₐ) states that μ₁ < μ₂. This is a left-tailed test.
2. Step 2: Calculate the test statistic using the formula for the z-test for two population means: z = (x̄₁ - x̄₂) / √((σ₁² / n₁) + (σ₂² / n₂)). Substitute the given values: x̄₁ = 5004, x̄₂ = 4895, σ₁ = 136, σ₂ = 215, n₁ = 144, and n₂ = 156.
3. Step 3: Determine the critical value for the z-test at the given significance level α = 0.03. Use a z-table or statistical software to find the z-value corresponding to a left-tailed test with α = 0.03.
4. Step 4: Compare the calculated test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: State the conclusion in the context of the problem. Based on the comparison in Step 4, determine whether there is sufficient evidence to support the claim that μ₁ < μ₂ at the 0.03 significance level.

Key Concepts

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) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the claim is that the mean of population one (μ1) is less than the mean of population two (μ2), which sets the stage for testing this hypothesis.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether the observed data is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, α is set at 0.03, indicating a 3% risk of concluding that μ1 is less than μ2 when it is not, which guides the decision-making process in hypothesis testing.

Standard Error and Z-Test

The standard error measures the variability of the sample mean estimates and is crucial for conducting a Z-test, which compares the means of two populations. It is calculated using the population standard deviations (σ1 and σ2) and the sample sizes (n1 and n2). In this case, the Z-test will help determine if the difference between the sample means (x̅1 and x̅2) is statistically significant, given the specified significance level.