A manufacturing company wants to compare the average production output of two of its
plants. Plant X has a sample of 40 days with an average daily production of 150 units and a standard
deviation of 15 units. Plant Y has a sample of 35 days with an average daily production of 145
units and a standard deviation of 12 units. Test at the 1% significance level whether there is a
significant difference in the average daily production output between the two plants.

To test whether there is a significant difference in the average daily production output between Plant X and Plant Y at the 1% significance level, we can use a **two-sample t-test for independent samples**. This test is appropriate when comparing the means of two independent groups.

### Steps:

1. **Set up the hypotheses**:
   - Null hypothesis (\(H_0\)): There is no significant difference in the average daily production output between the two plants, i.e., \( \mu_X = \mu_Y \).
   - Alternative hypothesis (\(H_A\)): There is a significant difference in the average daily production output, i.e., \( \mu_X \neq \mu_Y \) (this is a two-tailed test).

2. **Calculate the test statistic**:
   The formula for the two-sample t-test when the population variances are not assumed to be equal (Welch's t-test) is:

   \[
   t = \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{s_X^2}{n_X} + \frac{s_Y^2}{n_Y}}}
   \]

   where:
   - \( \bar{X} = 150 \) (mean of Plant X),
   - \( \bar{Y} = 145 \) (mean of Plant Y),
   - \( s_X = 15 \) (standard deviation of Plant X),
   - \( s_Y = 12 \) (standard deviation of Plant Y),
   - \( n_X = 40 \) (sample size of Plant X),
   - \( n_Y = 35 \) (sample size of Plant Y).

3. **Degrees of freedom (df)**:
   Since we are using Welch's t-test, the degrees of freedom can be approximated using the following formula:

   \[
   df = \frac{\left(\frac{s_X^2}{n_X} + \frac{s_Y^2}{n_Y}\right)^2}{\frac{\left(\frac{s_X^2}{n_X}\right)^2}{n_X - 1} + \frac{\left(\frac{s_Y^2}{n_Y}\right)^2}{n_Y - 1}}
   \]

4. **Determine the critical t-value**:
   For a two-tailed test at the 1% significance level, we need to find the critical t-value from the t-distribution with the calculated degrees of freedom.

5. **Make a decision**:
   Compare the calculated t-value with the critical t-value. If the absolute value of the calculated t is greater than the critical t, we reject the null hypothesis, indicating a significant difference.

Let me calculate these values.### Results:

- **Test statistic (t-value)**: 1.602
- **Degrees of freedom**: 72.45
- **Critical t-value**: 2.645 (for a two-tailed test at the 1% significance level)
- **p-value**: 0.113

### Conclusion:
Since the calculated t-value (1.602) is less than the critical t-value (2.645) and the p-value (0.113) is greater than the significance level of 0.01, we **fail to reject the null hypothesis**. This means there is **no statistically significant difference** in the average daily production output between Plant X and Plant Y at the 1% significance level.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the test change if we used a 5% significance level?
2. What is the impact of increasing the sample sizes on the t-test result?
3. How would the results differ if the standard deviations were significantly different?
4. Could a one-tailed test be more appropriate for this comparison?
5. How does Welch's t-test differ from Student's t-test in this scenario?

**Tip**: Always check if assumptions (like equal variances or normal distribution) are met before choosing a t-test type.

A manufacturing company wants to compare the average production output of two of its plants. Plant X has a sample of 40 days with an average daily production of 150 units and a standard deviation of 15 units. Plant Y has a sample of 35 days with an average daily production of 145 units and a standard deviation of 12 units. Test at the 1% significance level whether there is a significant difference in the average daily production output between the two plants.

This example compares the average production output of two manufacturing plants using a two-sample t-test at a 1% significance level. Learn about hypothesis testing, the t-test formula, and calculating degrees of freedom using Welch's approximation. Suitable for undergraduate statistics students.

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