Question

the data to the right show the average retirement ages for a random sample of workers in country a and a random sample of workers in country b. the population standard deviations are given. complete parts a and b. let population 1 be the workers in country a and population 2 be the workers in country b. identify the null and alternative hypotheses. choose the correct answer below. calculate the appropriate test statistic. the test statistic is □.

Explanation:

Part a: Identifying Null and Alternative Hypotheses

We are comparing the average retirement ages of workers in Country A (population 1) and Country B (population 2). The null hypothesis \( H_0 \) typically assumes no difference or a specific relationship, and the alternative hypothesis \( H_1 \) is what we might be testing for. Here, we want to see if there's a difference (or a specific direction, but since the sample means are 64.2 (Country A) and 66.2 (Country B), we might be testing if \( \mu_1 - \mu_2 \) is less than 0 (since 64.2 < 66.2, so \( \mu_1 < \mu_2 \) implies \( \mu_1 - \mu_2 < 0 \)). Let's analyze the options:

• Option D: \( H_0: \mu_1 - \mu_2 \geq 0 \) (null assumes no difference or Country A's mean is at least Country B's) and \( H_1: \mu_1 - \mu_2 < 0 \) (alternative says Country A's mean is less than Country B's) which matches the context (since sample mean of A is less than B, we test if population mean of A is less than B). Other options: A tests if A's mean is greater, B is two - tailed, C is opposite direction, E is reversed, F is opposite direction. So correct is D.

We are dealing with a two - sample z - test (since population standard deviations \( \sigma_1 \) and \( \sigma_2 \) are known). The formula for the test statistic \( z \) in two - sample z - test is:

Step 1: Recall the formula for the test statistic

The formula for the test statistic \( z \) when comparing two population means with known population standard deviations is:

$$ z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)_0}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}} $$

where \( \bar{x}_1 = 64.2 \), \( \bar{x}_2 = 66.2 \), \( (\mu_1 - \mu_2)_0 = 0 \) (from the null hypothesis \( H_0: \mu_1 - \mu_2\geq0 \), we use 0 for the hypothesized difference), \( \sigma_1 = 4.6 \), \( \sigma_2 = 5.0 \), \( n_1 = 30 \), \( n_2 = 30 \)

Step 2: Substitute the values into the formula

First, calculate the numerator: \( (\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)_0=(64.2 - 66.2)-0=- 2 \)

Then, calculate the denominator:

$$ \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}=\sqrt{\frac{4.6^2}{30}+\frac{5.0^2}{30}}=\sqrt{\frac{21.16 + 25}{30}}=\sqrt{\frac{46.16}{30}}\approx\sqrt{1.5387}\approx1.2404 $$

Step 3: Calculate the z - statistic

$$ z=\frac{-2}{1.2404}\approx - 1.61 $$

Answer:

D. \( H_0: \mu_1 - \mu_2 \geq 0 \), \( H_1: \mu_1 - \mu_2 < 0 \)

Part b: Calculating the Test Statistic