To solve this problem, we are finding the standardized test statistic (z) for testing the claim that $\mu_1 = \mu_2$, assuming that the two samples are independent. The population standard deviations ($\sigma_1$ and $\sigma_2$) are given.

Step 1: Formula for the test statistic

The formula for the standardized test statistic (z) for two independent samples is:

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

Where:

• $\bar{x}_1$ and $\bar{x}_2$: sample means
• $n_1$ and $n_2$: sample sizes
• $\sigma_1$ and $\sigma_2$: population standard deviations
• $\mu_1 - \mu_2 = 0$ under the null hypothesis

Step 2: Substitution of given values

From the problem:

• $\bar{x}_1 = 4$, $\bar{x}_2 = 5$
• $n_1 = 40$, $n_2 = 35$
• $\sigma_1 = 2.5$, $\sigma_2 = 2.8$

Substitute these values into the formula:

1. Difference of sample means: $\bar{x}_1 - \bar{x}_2 = 4 - 5 = -1$

2. Standard error (SE): $SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ First, calculate each term under the square root:

   • $\frac{\sigma_1^2}{n_1} = \frac{2.5^2}{40} = \frac{6.25}{40} = 0.15625$
   • $\frac{\sigma_2^2}{n_2} = \frac{2.8^2}{35} = \frac{7.84}{35} = 0.224$

   Now sum these: $SE = \sqrt{0.15625 + 0.224} = \sqrt{0.38025} \approx 0.6167$

3. Compute z: $z = \frac{(\bar{x}_1 - \bar{x}_2)}{SE} = \frac{-1}{0.6167} \approx -1.62$

Step 3: Match the closest option

The closest option to $-1.62$ is:

$-1.6$

Final Answer: -1.6

Let me know if you'd like a breakdown of any specific step!

Related Questions:

1. What is the formula for comparing means of two samples with unknown variances?
2. How do you calculate the critical z-value for a given significance level?
3. What is the difference between z-tests and t-tests for comparing two means?
4. How does increasing sample size affect the standard error of the test statistic?
5. What assumptions are made when performing a two-sample z-test?

Tip:

Always check whether population standard deviations are known to determine whether to use the z-test or t-test for two independent samples.