Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1=μ2, α=0.01, Assume (σ1)^2=(σ2)^2

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). Since the claim is μ₁ = μ₂, the null hypothesis is H₀: μ₁ = μ₂, and the alternative hypothesis is Hₐ: μ₁ ≠ μ₂ (two-tailed test).
2. Step 2: Determine the test statistic to use. Since the population variances are assumed to be equal (σ₁² = σ₂²), use the pooled t-test formula for the test statistic: $t=\frac{\left(x_1-x_2\right)}{\sqrt{\frac{\left(n_1-1\right)s_1²+\left(n_2-1\right)s_2²}{n_1+n_2-2}}}$, where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
3. Step 3: Calculate the degrees of freedom (df) for the test. Since the population variances are equal, use the formula: $\mathrm{df}=n_1+n_2-2$.
4. Step 4: Determine the critical t-value(s) from the t-distribution table for a two-tailed test at the significance level α = 0.01 and the calculated degrees of freedom. These critical values will define the rejection region.
5. Step 5: Compare the calculated test statistic (t) to the critical t-value(s). If the test statistic falls in the rejection region, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. 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 data to determine whether to reject H0. In this case, the null hypothesis states that the means of two populations are equal (μ1=μ2), while the alternative suggests they are not.

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 0.01, indicating a 1% risk of concluding that there is a difference between the population means when there is none.

Assumption of Equal Variances

The assumption of equal variances, often referred to as homoscedasticity, is crucial when comparing two population means. It implies that the variances of the two populations are the same (σ1^2 = σ2^2), which allows for the use of specific statistical tests, such as the pooled t-test. This assumption simplifies the analysis and increases the reliability of the test results.