Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.

Verified step by step guidance

1. Step 1: State the null and alternative hypotheses. The null hypothesis (H₀) typically states that there is no difference between the population means (μ₁ = μ₂), while the alternative hypothesis (H₁) states that there is a difference (μ₁ ≠ μ₂, μ₁ > μ₂, or μ₁ < μ₂ depending on the context).
2. Step 2: Identify the sample statistics and population parameters. Gather the sample means (x̄₁ and x̄₂), sample sizes (n₁ and n₂), and the population standard deviations (σ₁ and σ₂) for both groups. Ensure the samples are independent and the population standard deviations are known.
3. Step 3: Calculate the test statistic (z). Use the formula: $z=\frac{(x_1-x_2)}{\sqrt{\frac{{\sigma }_1²}{n_1}+\frac{{\sigma }_2²}{n_2}}}$. This formula accounts for the difference in sample means and the variability of the two populations.
4. Step 4: Determine the critical value or p-value. Based on the significance level (α) and the type of test (one-tailed or two-tailed), find the critical z-value from the standard normal distribution table or calculate the p-value corresponding to the test statistic.
5. Step 5: Make a decision. Compare the test statistic to the critical value or use the p-value. If the test statistic exceeds the critical value or the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the results in the context of the problem.

Key Concepts

Two-Sample Z-Test

A two-sample z-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. This test is applicable when the population variances are known and the sample sizes are sufficiently large (typically n > 30). It compares the means by calculating a z-score, which indicates how many standard deviations the observed difference is from the expected difference under the null hypothesis.

Independent Samples

Independent samples refer to groups that are not related or paired in any way. In the context of a two-sample z-test, this means that the data collected from one sample does not influence or affect the data collected from the other sample. This independence is crucial for the validity of the test, as it ensures that the results are not biased by any relationship between the groups.

Population Means and Variances

Population means are the average values of a characteristic in a population, while population variances measure the spread of data points around the mean. In a two-sample z-test, knowing the population variances allows for the calculation of the standard error of the difference between the two means. This information is essential for determining the z-score and ultimately assessing whether the observed difference is statistically significant.