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=0.1
Population statistics:σ1=3.4 and σ2=1.5
Sample Statistics: x̅1=16, n1=29, x̅2=14, n2=28

Verified step by step guidance

1. Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Since the claim is \(\mu_1 = \mu_2\), the hypotheses are: \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 \neq \mu_2\) (two-tailed test).
2. Determine the significance level \(\alpha = 0.1\) and find the corresponding critical z-values for a two-tailed test. These critical values define the rejection regions for the test statistic.
3. Calculate the standard error of the difference between the two sample means using the formula: \(SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\) where \(\sigma_1 = 3.4\), \(n_1 = 29\), \(\sigma_2 = 1.5\), and \(n_2 = 28\).
4. Compute the test statistic (z-score) using the formula: \(z = \frac{\bar{x}_1 - \bar{x}_2 - (\mu_1 - \mu_2)}{SE}\) Since under \(H_0\), \(\mu_1 - \mu_2 = 0\), this simplifies to: \(z = \frac{16 - 14}{SE}\).
5. Compare the calculated z-value with the critical z-values. If the test statistic falls into the rejection region, reject \(H_0\); otherwise, do not reject \(H_0\). This will determine whether there is sufficient evidence to reject the claim that \(\mu_1 = \mu_2\) at the \(\alpha = 0.1\) significance level.

Key Concepts

Hypothesis Testing for Two Population Means

This involves testing a claim about the difference between two population means using sample data. The null hypothesis typically states that the means are equal (μ1 = μ2), while the alternative suggests a difference. The test evaluates whether observed sample differences are statistically significant at a chosen significance level (α).

Level of Significance (α)

The level of significance, α, is the threshold probability for rejecting the null hypothesis when it is true (Type I error). In this problem, α = 0.1 means there is a 10% risk of incorrectly rejecting the claim that the population means are equal. It determines the critical value for the test statistic.

Known Population Standard Deviations and Z-Test

When population standard deviations (σ1 and σ2) are known, the Z-test is used to compare two means. The test statistic is calculated using sample means, population standard deviations, and sample sizes. This approach assumes normality and independence of samples to determine if the difference is significant.