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=0.10
Population statistics:σ1=40 and σ2=15
Sample Statistics: x̅1=500, n1=100, x̅2=495, n2=75

Verified step by step guidance

1. Identify the null and alternative hypotheses based on the claim μ1 > μ2. The null hypothesis is \(H_0: \mu_1 \leq \mu_2\) and the alternative hypothesis is \(H_a: \mu_1 > \mu_2\).
2. Determine the significance level, which is given as \(\alpha = 0.10\). This will be used to find the critical value for the test.
3. Calculate the test statistic for the difference between two population means when population standard deviations are known. Use the formula: \[ Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \] Since under the null hypothesis \(\mu_1 - \mu_2 = 0\), simplify accordingly.
4. Find the critical value from the standard normal distribution corresponding to the right-tailed test at \(\alpha = 0.10\). This critical value will be used to compare with the calculated test statistic.
5. Compare the calculated test statistic to the critical value: if the test statistic is greater than the critical value, reject the null hypothesis; otherwise, do not reject the null hypothesis. This will determine if there is sufficient evidence to support the claim \(\mu_1 > \mu_2\).

Key Concepts

Hypothesis Testing for Two Population Means

This involves comparing two population means to determine if there is enough evidence to support a specific claim. The null hypothesis typically states no difference (μ1 = μ2), while the alternative reflects the claim (μ1 > μ2). The test uses sample data to decide whether to reject the null hypothesis at a given significance level.

Significance Level (α)

The significance level, α, is the threshold probability for rejecting the null hypothesis when it is true (Type I error). Here, α = 0.10 means there is a 10% risk of incorrectly rejecting the null. It sets the critical value or rejection region for the test statistic.

Z-Test for Difference Between Means with Known Population Standard Deviations

When population standard deviations (σ1 and σ2) are known and samples are independent and normal, a Z-test is used. The test statistic measures how far the sample mean difference is from zero, scaled by the standard error. This helps determine if the observed difference is statistically significant.