Question

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 = 0.01 population parameters: σ₁ = 3.5, σ₂ = 1.7 sample statistics: x̄₁ = 15, n₁ = 29, x̄₂ = 17, n₂ = 30 hₐ: μ₁ ≠ μ₂ determine the standardized test statistic. z = -2.78 (round to two decimal places as needed.) determine the p - value. p - value = 0.005 (round to three decimal places as needed.) what is the proper decision? a. reject h₀. there is not enough evidence at the 1% level of significance to reject the claim. b. fail to reject h₀. there is not enough evidence at the 1% level of significance to reject the claim. c. reject h₀. there is enough evidence at the 1% level of significance to reject the claim. d. fail to reject h₀. there is enough evidence at the 1% level of significance to reject the claim.

Explanation:

Step1: Recall Decision Rule

For a hypothesis test, we compare the P - value with the significance level \(\alpha\). The decision rule is: If \(P - value<\alpha\), we reject the null hypothesis \(H_0\); if \(P - value\geq\alpha\), we fail to reject \(H_0\).
Here, \(\alpha = 0.01\) and the \(P - value=0.005\).

Step2: Compare P - value and \(\alpha\)

Since \(0.005<0.01\) (i.e., \(P - value<\alpha\)), we reject the null hypothesis \(H_0\). The claim is \(H_0:\mu_1=\mu_2\) (since the original claim is \(\mu_1 = \mu_2\) and the null hypothesis for a two - tailed test when the claim is of equality is \(H_0:\mu_1=\mu_2\) and the alternative is \(H_a:\mu_1
eq\mu_2\)). Rejecting \(H_0\) means there is enough evidence at the 1% level of significance to reject the claim.

Answer:

C. Reject \(H_0\). There is enough evidence at the 1% level of significance to reject the claim.