
Treatment
ATreatment
BTreatment
C

31

45

34

29

42

37

29

44

36

26

45

37

30

49

41

Sample mean

29

45

37

Sample variance

3.50

6.50

6.50



Use Fisher's LSD procedure to develop a 95% confidence interval estimate of the difference between the means of treatments A and B. (Use 

*x*A − *x*B.

 Round your answers to two decimal places.)

  to  

To calculate the 95% confidence interval estimate of the difference between the means of treatments A and B using Fisher's Least Significant Difference (LSD) procedure, we can follow these steps:

### 1. **Calculate the pooled variance:**

The pooled variance (\( s_p^2 \)) is calculated by taking the average of the sample variances of the groups. The formula for pooled variance is:

\[
s_p^2 = \frac{(n_A - 1) \cdot s_A^2 + (n_B - 1) \cdot s_B^2 + (n_C - 1) \cdot s_C^2}{n_A + n_B + n_C - 3}
\]

Where:
- \( s_A^2 \), \( s_B^2 \), and \( s_C^2 \) are the sample variances for treatments A, B, and C.
- \( n_A = n_B = n_C = 5 \) (since we have 5 samples in each treatment group).

Given variances:
- \( s_A^2 = 3.50 \)
- \( s_B^2 = 6.50 \)
- \( s_C^2 = 6.50 \)

### 2. **Compute the t-critical value:**
For a 95% confidence interval and degrees of freedom of \( df = n_A + n_B + n_C - 3 = 5 + 5 + 5 - 3 = 12 \), we need to find the t-critical value, \( t_{\alpha/2} \), from a t-distribution table.

### 3. **Calculate the confidence interval:**

The formula for the confidence interval of the difference between two means is:

\[
CI = (\bar{x}_A - \bar{x}_B) \pm t_{\alpha/2} \cdot \sqrt{s_p^2 \left(\frac{1}{n_A} + \frac{1}{n_B}\right)}
\]

Where:
- \( \bar{x}_A = 29 \) (sample mean of treatment A)
- \( \bar{x}_B = 45 \) (sample mean of treatment B)

Let me calculate the values.The 95% confidence interval for the difference between the means of treatments A and B using Fisher's LSD procedure is approximately:

\[
-19.23 \text{ to } -12.77
\]

This means the difference in means between Treatment A and Treatment B is estimated to lie between -19.23 and -12.77 with 95% confidence.

Would you like further details or clarification?

Here are 5 related questions you might find useful:
1. How is the pooled variance calculated in more detail?
2. What assumptions does Fisher's LSD procedure make?
3. How would the confidence interval change if the sample sizes were different?
4. What is the significance of a negative confidence interval in this context?
5. How does Fisher's LSD compare to other post-hoc tests, like Tukey's HSD?

**Tip:** The Fisher's LSD method is valid only when the overall ANOVA test is significant.

Use Fisher's LSD procedure to develop a 95% confidence interval estimate of the difference between the means of treatments A and B. (Use *x*A − *x*B. Round your answers to two decimal places.)

Learn how to use Fisher's Least Significant Difference (LSD) procedure to calculate a 95% confidence interval for the difference between the means of Treatments A and B. This problem includes key statistical concepts such as pooled variance, confidence intervals, and hypothesis testing. Suitable for undergraduate students.

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