Is the difference between the mean annual salaries of statisticians in Region 1 and Region 2 more than $ 4000 question mark To​ decide, you select a random sample of statisticians from each region. The results of each survey are shown to the right. At alpha equals 0.10 comma what should you​ conclude?Determine the critical​ value(s).
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​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.)

To determine whether the difference between the mean annual salaries of statisticians in Region 1 and Region 2 is more than $4000, we conduct a hypothesis test. The hypotheses for this test are:

- **Null hypothesis ($H_0$)**: The difference in mean salaries is less than or equal to $4000, $(\mu_1 - \mu_2) \leq 4000$
- **Alternative hypothesis ($H_1$)**: The difference in mean salaries is greater than $4000, $(\mu_1 - \mu_2) > 4000$

This is a one-tailed test, where we are interested in whether the mean salary difference is greater than a specific amount.

### Given Information
To proceed with the test, we need the following information from the sample dataTo determine whether the difference between the mean annual salaries of statisticians in Region 1 and Region 2 is more than $4000, we conduct a hypothesis test. The hypotheses for this test are:

- **Null hypothesis ($H_0$)**: The difference in mean salaries is less than or equal to $4000, $(\mu_1 - \mu_2) \leq 4000$
- **Alternative hypothesis ($H_1$)**: The difference in mean salaries is greater than $4000, $(\mu_1 - \mu_2) > 4000$

This is a one-tailed test, where we are interested in whether the mean salary difference is greater than a specific amount.

### Given Information
To proceed with the test, we need the following information from the sample 

1. Sample means ($\bar{x}_1$ and $\bar{x}_2$)
2. Sample sizes ($n_1$ and $n_2$)
3. Population standard deviations ($\sigma_1$ and $\sigma_2$) or sample standard deviations if population values are not available.

Since the exact values are not provided, I'll outline the general steps for conducting this test:

1. **Calculate the Test Statistic**:
   We use the Z-test statistic formula for the difference between two means:

   \[
   Z = \frac{(\bar{x}_1 - \bar{x}_2) - 4000}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
   \]

   If population standard deviations are unknown, use sample standard deviations and the t-distribution.

2. **Determine the Critical Value(s)**:
   For a one-tailed test at $\alpha = 0.10$, find the critical value from the standard normal distribution table.

   - **Critical value (Z$_{\alpha}$)**: This corresponds to the z-score where the cumulative probability to the right is 0.10. The critical value for $\alpha = 0.10$ is $Z_{\alpha} \approx 1.28$.

3. **Compare the Test Statistic to the Critical Value**:
   - If the computed $Z$ value exceeds 1.28, reject the null hypothesis.
   - If the computed $Z$ value is less than or equal to 1.28, fail to reject the null hypothesis.

### Conclusion

Based on the computed Z statistic and the critical value:
- **Reject $H_0$** if $Z > 1.28$
- **Fail to reject $H_0$** if $Z \leq 1.28$

If you have the specific sample data, we can compute the exact test statistic and make a precise conclusion.

**Critical value(s) to report**: $ Z_{\alpha} = 1.28 $

Would you like more details on this process, or do you have any specific sample data to use for a more precise calculation?

Here are some related questions you might consider:

1. What are the implications of rejecting the null hypothesis in this context?
2. How does the level of significance ($\alpha$) affect the conclusion of a hypothesis test?
3. Why do we use the standard normal distribution (Z-distribution) in this test?
4. How would the conclusion change if we were conducting a two-tailed test?
5. What is the importance of sample size in determining the reliability of hypothesis testing?
6. What are some limitations of using a Z-test in real-world data analysis?
7. How would the use of a t-test change the analysis if population standard deviations were unknown?
8. What factors could potentially bias the results of this salary comparison?

**Tip:** When conducting hypothesis tests, always report the test statistic, critical values, and p-values (if calculated), as these provide a complete picture of the decision-making process.

Explore hypothesis testing to determine if the difference between the mean annual salaries of statisticians in Region 1 and Region 2 is more than $4000. Learn about the Z-test, critical values, and how to make conclusions based on statistical significance. Suitable for advanced high school and college-level students.

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