Homework 6 (B

Question listHomework 6 (B.2)

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Question 1, 8.2.3-T

Part 1 of 11

In order to compare the means of two populations, independent random samples of 385 observations are selected from nach papulation, with the rasults found in the table to the right. Complete parts a through e below.

Question 1, 8.2.3-T

Part 1 of 11

In order to compare the means of two populations, independent random samples of 385 observations are selected from nach papulation, with the rasults found in the table to the right. Complete parts a through e below.

It seems like you're working on a problem that involves comparing the means of two populations based on independent random samples. Although I don’t have access to the table with sample data (as it seems to be from your homework), I'll guide you through the general approach to solve this type of problem. 

Here’s the common structure for solving these problems:

### Steps to Compare Means of Two Independent Populations

1. **Identify Given Information**:
    - Sample sizes for both populations (in this case, both sample sizes are 385).
    - Sample means (\( \bar{X_1}, \bar{X_2} \)).
    - Standard deviations (\( s_1, s_2 \)) or variances.

2. **Determine the Null and Alternative Hypotheses**:
    - \( H_0 \): \( \mu_1 = \mu_2 \) (The population means are equal).
    - \( H_A \): \( \mu_1 \neq \mu_2 \) (The population means are not equal), or it could be \( \mu_1 > \mu_2 \) or \( \mu_1 < \mu_2 \), depending on the problem.

3. **Select the Test Statistic**:
   Use the formula for a two-sample t-test or z-test for independent samples:
   \[
   t = \frac{(\bar{X_1} - \bar{X_2}) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
   \]
   Where:
   - \( \bar{X_1}, \bar{X_2} \) = sample means of the two populations.
   - \( s_1^2, s_2^2 \) = sample variances.
   - \( n_1, n_2 \) = sample sizes (both are 385 in your case).
   - \( \mu_1 - \mu_2 \) = the hypothesized difference in population means (usually 0).

4. **Calculate the Test Statistic**:
    Plug in the values from the table and solve for \( t \).

5. **Determine the p-value**:
    Based on the calculated test statistic, use a t-distribution (or z-distribution if variances are known and sample sizes are large) to find the p-value.

6. **Decision Rule**:
    - Compare the p-value to the significance level (\( \alpha \)) to determine whether to reject or fail to reject the null hypothesis.

7. **Interpret the Results**:
    - If the p-value is less than \( \alpha \), reject \( H_0 \), suggesting that there is a statistically significant difference between the means of the two populations.
    - If the p-value is greater than \( \alpha \), fail to reject \( H_0 \), suggesting that there is not enough evidence to conclude a difference in means.

### Additional Information You Would Need:
- Sample means (\( \bar{X_1}, \bar{X_2} \)).
- Sample standard deviations or variances (\( s_1, s_2 \)).
- Significance level (\( \alpha \)).

If you provide the sample data (means, standard deviations), I can help you compute the exact test statistic and interpret the result.

Would you like to provide the additional details, or should we go through a more detailed hypothetical example?

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### 5 Related Questions:
1. How would the method change if the sample sizes were unequal?
2. What if the variances of the two populations are not equal—what adjustment is needed?
3. How do you interpret the results if the p-value is exactly equal to the significance level?
4. How do you check if the assumption of normality holds for the samples?
5. Can you use the same method for comparing more than two populations?

### Tip:
Always visualize the data before running the test. A simple box plot or histogram can give insight into the distribution and spread of the data.

In order to compare the means of two populations, independent random samples of 385 observations are selected from each population, with the results found in the table. Complete parts a through e below.

Learn how to compare the means of two populations using a two-sample t-test. This problem involves independent random samples of 385 observations, hypothesis testing, and p-value interpretation. Includes step-by-step guidance suitable for undergraduate students studying statistics.

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