For the given​ data, (a) find the test​ statistic, (b) find the standardized test​ statistic, (c) decide whether the standardized test statistic is in the rejection​ region, and​ (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent.
​Claim: mu 1less thanmu 2​, alphaequals0.01. Sample​ statistics: x overbar 1 equals 1245 comma n 1 equals 45 comma x overbar 2 equals 1205 comma and n 2 equals 85. Population​ parameters: sigma 1 equals 65 and sigma 2 equals 105.

To address the problem, we will follow these steps for hypothesis testing involving two independent samples:

1. **State the Hypotheses**:
   - Null hypothesis (\(H_0\)): \(\mu_1 \geq \mu_2\)
   - Alternative hypothesis (\(H_1\)): \(\mu_1 < \mu_2\)

2. **Find the Test Statistic**:
   We use a Z-test for the difference between two means since the population standard deviations (\(\sigma_1\) and \(\sigma_2\)) are known.

   The formula for the Z-test statistic is:
   \[
   Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
   \]

   Given:
   - \(\bar{x}_1 = 1245\)
   - \(\bar{x}_2 = 1205\)
   - \(n_1 = 45\)
   - \(n_2 = 85\)
   - \(\sigma_1 = 65\)
   - \(\sigma_2 = 105\)
   - \(\mu_1 - \mu_2 = 0\) (under \(H_0\))

3. **Find the Standardized Test Statistic (Z)**:
   - Plug the values into the formula to compute the Z statistic.

4. **Determine the Rejection Region**:
   - Given \(\alpha = 0.01\) and a left-tailed test, find the critical value from the standard normal distribution table corresponding to \(Z_{\alpha}\).

5. **Make a Decision**:
   - Compare the computed Z statistic with the critical value to decide whether to reject or fail to reject the null hypothesis.

Let's calculate these steps.

### (a) Test Statistic Calculation

\[
Z = \frac{(1245 - 1205) - 0}{\sqrt{\frac{65^2}{45} + \frac{105^2}{85}}}
\]

\[
= \frac{40}{\sqrt{\frac{4225}{45} + \frac{11025}{85}}}
\]

\[
= \frac{40}{\sqrt{93.89 + 129.71}}
\]

\[
= \frac{40}{\sqrt{223.60}}
\]

\[
= \frac{40}{14.95}
\]

\[
Z \approx 2.68
\]

### (b) Standardized Test Statistic

The standardized test statistic is \( Z \approx 2.68 \).

### (c) Rejection Region

For \(\alpha = 0.01\) in a left-tailed test, the critical value is \(Z_{\alpha} \approx -2.33\).

### (d) Decision

Since the computed \(Z \approx 2.68\) is not in the rejection region (it is greater than \(-2.33\)), we fail to reject the null hypothesis.

#### Conclusion:

There is not enough evidence at the 0.01 significance level to support the claim that \(\mu_1 < \mu_2\).

---

Would you like more details on any part of the solution or have any further questions?

Here are some related questions that might interest you:

1. How do you determine the critical value for a one-tailed test?
2. What is the difference between a one-tailed and a two-tailed test?
3. Why is the standard deviation used in the test statistic formula?
4. How does sample size affect the standard error and the test statistic?
5. What does a p-value represent in hypothesis testing?
6. How would the decision change if the test was two-tailed?
7. What assumptions must be met for the Z-test to be valid?
8. How do population variances influence the outcome of the hypothesis test?

**Tip:** Always check the assumptions of the statistical test you are using to ensure the validity of your results.

Learn how to conduct hypothesis testing using the Z-test statistic for two independent samples. Given sample statistics and population parameters, calculate the Z-test statistic, standardized test statistic, determine rejection regions, and make a decision based on the null hypothesis. Example problem with step-by-step solution.

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