In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1>μ2; α=0.05


Population statistics: σ1= 0.30 and σ2= 0.23


Sample statistics: x̅1 = 1.28, n1 = 96, and x̅2= 1.34, n2= 85

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ₁ ≤ μ₂, and the alternative hypothesis is H₁: μ₁ > μ₂. This is a one-tailed test since the claim is about μ₁ being greater than μ₂.
2. Step 2: Calculate the test statistic using the formula for the z-test for the difference between two population means when population standard deviations are known: z = ((x̄₁ - x̄₂) - (μ₁ - μ₂)) / √((σ₁² / n₁) + (σ₂² / n₂)). Here, μ₁ - μ₂ = 0 under the null hypothesis.
3. Step 3: Substitute the given values into the formula. Use x̄₁ = 1.28, x̄₂ = 1.34, σ₁ = 0.30, σ₂ = 0.23, n₁ = 96, and n₂ = 85. Compute the standard error (SE) first: SE = √((σ₁² / n₁) + (σ₂² / n₂)). Then calculate the z-test statistic.
4. Step 4: Determine the critical value for the z-test at the significance level α = 0.05 for a one-tailed test. Use a z-table or standard normal distribution to find the critical z-value corresponding to α = 0.05.
5. Step 5: Compare the calculated z-test statistic to the critical z-value. If the test statistic is greater than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

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 null hypothesis would state that the means are equal (μ1 ≤ μ2), while the alternative claims that μ1 is greater than μ2.

Level of Significance (α)

The level of significance, 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 0.05, indicating a 5% risk of concluding that μ1 is greater than μ2 when it is not.

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). The Z-test statistic is then computed to assess the difference between the sample means relative to the standard error, allowing for the evaluation of the hypothesis.