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=0.05
Population statistics:σ1=75 and σ2=105
Sample Statistics: x̅1=2435, n1=35, x̅2=2432, n2=90

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that μ₁ ≥ μ₂, while the alternative hypothesis (H₁) states that μ₁ < μ₂. This is a one-tailed test since the claim is about μ₁ being less than μ₂.
2. Step 2: Determine the test statistic formula for comparing two population means when population standard deviations (σ₁ and σ₂) are known. The formula is: $\frac{(1_{}-2_{})}{\sqrt{(\frac{{{}_{}}^1}{2}{}_{}1+\frac{{{}_{}}^2}{2}{}_{}2)}}$
3. Step 3: Substitute the given values into the formula. Use x̅₁ = 2435, x̅₂ = 2432, σ₁ = 75, σ₂ = 105, n₁ = 35, and n₂ = 90. Calculate the numerator (x̅₁ - x̅₂) and the denominator (the square root of the sum of variances divided by sample sizes).
4. Step 4: Find the critical value for the test. Since α = 0.05 and this is a one-tailed test, use the z-distribution table to find the critical z-value corresponding to α = 0.05. This value will help determine the rejection region for the null hypothesis.
5. Step 5: Compare the calculated test statistic to the critical value. If the test statistic falls in the rejection region (i.e., it is less than the critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim μ₁ < μ₂.

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 0.05, meaning there is a 5% risk of concluding that μ1 is less than μ2 when it is not, guiding the decision-making process in hypothesis testing.

Two-Sample Z-Test

A two-sample Z-test is used to compare the means of two independent populations when the population variances are known. It calculates a Z statistic based on the difference between sample means, the population standard deviations, and the sample sizes. This test is appropriate here since the populations are normally distributed and the variances (σ1 and σ2) are provided, allowing for a valid comparison of the means.