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
Sample statistics:
x̅1=33.7, s1=3.5 , n1=12 and x̅2=35.5 , s2=2.2 , n2=17

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) is that the population means are equal, μ₁ = μ₂. The alternative hypothesis (Hₐ) is that the population means are not equal, μ₁ ≠ μ₂. This is a two-tailed test.
2. Step 2: Determine the test statistic formula. Since the population variances are assumed to be equal (σ₁² = σ₂²), we use the pooled t-test formula. The test statistic is given by: t = (x̄₁ - x̄₂) / √(sp²(1/n₁ + 1/n₂)), where sp² is the pooled variance.
3. Step 3: Calculate the pooled variance (sp²). The formula for sp² is: sp² = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2). Substitute the given values for n₁, n₂, s₁, and s₂ to compute sp².
4. Step 4: Compute the test statistic (t). Substitute the values of x̄₁, x̄₂, sp², n₁, and n₂ into the t-test formula to calculate the t-value.
5. Step 5: Compare the calculated t-value to the critical t-value. Use the t-distribution table with degrees of freedom df = n₁ + n₂ - 2 and significance level α = 0.01 (two-tailed) to find the critical t-value. If the absolute value of the calculated t is greater than the critical t-value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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 states that the means of the two populations are equal (μ1=μ2).

Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. 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. This level of significance helps determine the threshold for making statistical inferences.

Pooled Variance

Pooled variance is used when comparing two sample means under the assumption that the populations have equal variances. It combines the variances of both samples to provide a more accurate estimate of the common variance. In this question, the assumption (σ1)^2=(σ2)^2 allows for the calculation of a pooled variance, which is essential for conducting the t-test for the difference between the two means.