Treatment ni ỹi- s? Yit = u + Ti + Eit, t = 1, ..., ri, i = 1,..., v. A 6 5.0 10.0 u + Tị denotes the true mean response for 10.0 14.0 C 20.0 10.0 the ith treatment, D 6 7.0 14.0 Find a 95% confidence interval for T1 – T2;
Treatment ni ỹi- s? Yit = u + Ti + Eit, t = 1, ..., ri, i = 1,..., v. A 6 5.0 10.0 u + Tị denotes the true mean response for 10.0 14.0 C 20.0 10.0 the ith treatment, D 6 7.0 14.0 Find a 95% confidence interval for T1 – T2;
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Analysis of Variance (ANOVA) and Confidence Interval Calculation
#### Data Table
The table displays treatment data including sample sizes, means, and variances:
| Treatment | \( n_i \) | \( \bar{y}_i. \) | \( s_i^2 \) |
|-----------|-----------|------------------|-------------|
| A | 6 | 5.0 | 10.0 |
| B | 6 | 10.0 | 14.0 |
| C | 6 | 20.0 | 10.0 |
| D | 6 | 7.0 | 14.0 |
#### Model Equation
\[ Y_{it} = \mu + \tau_i + \epsilon_{it}, \]
where \( t = 1, \ldots, r_i \), \( i = 1, \ldots, v \), and \( \mu + \tau_i \) denotes the true mean response for the \( i^{th} \) treatment.
#### Task: Find a 95% Confidence Interval for \( \tau_1 - \tau_2 \)
#### Calculations
1. **Number of Treatments (\( k \))**:
\[
k = 4
\]
2. **Total Sample Number (\( N \))**:
\[
N = 24
\]
3. **Mean Square Error (MSE)**:
\[
MSE = \frac{SSE}{N-k} = \frac{240}{24-4} = \frac{240}{20} = 12
\]
4. **Confidence Interval Formula**:
\[
\bar{y}_1. - \bar{y}_2. \pm t_{N-k, \alpha/2} \sqrt{\frac{2 \cdot MSE}{n}}
\]
5. **Substitute Values**:
\[
= 5 - 10 \pm t_{20, 0.025} \sqrt{\frac{2 \cdot 12}{6}}
\]
6. **Calculate Values**:
\[
= -5 \pm t_{20, 0.025} \cdot \sqrt{4}
\]
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F048330fb-37a4-4649-bc7e-2ea6f54a2a1a%2Faf4fdce3-adc3-42ae-a355-b606ff79556c%2Frwhjfsk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analysis of Variance (ANOVA) and Confidence Interval Calculation
#### Data Table
The table displays treatment data including sample sizes, means, and variances:
| Treatment | \( n_i \) | \( \bar{y}_i. \) | \( s_i^2 \) |
|-----------|-----------|------------------|-------------|
| A | 6 | 5.0 | 10.0 |
| B | 6 | 10.0 | 14.0 |
| C | 6 | 20.0 | 10.0 |
| D | 6 | 7.0 | 14.0 |
#### Model Equation
\[ Y_{it} = \mu + \tau_i + \epsilon_{it}, \]
where \( t = 1, \ldots, r_i \), \( i = 1, \ldots, v \), and \( \mu + \tau_i \) denotes the true mean response for the \( i^{th} \) treatment.
#### Task: Find a 95% Confidence Interval for \( \tau_1 - \tau_2 \)
#### Calculations
1. **Number of Treatments (\( k \))**:
\[
k = 4
\]
2. **Total Sample Number (\( N \))**:
\[
N = 24
\]
3. **Mean Square Error (MSE)**:
\[
MSE = \frac{SSE}{N-k} = \frac{240}{24-4} = \frac{240}{20} = 12
\]
4. **Confidence Interval Formula**:
\[
\bar{y}_1. - \bar{y}_2. \pm t_{N-k, \alpha/2} \sqrt{\frac{2 \cdot MSE}{n}}
\]
5. **Substitute Values**:
\[
= 5 - 10 \pm t_{20, 0.025} \sqrt{\frac{2 \cdot 12}{6}}
\]
6. **Calculate Values**:
\[
= -5 \pm t_{20, 0.025} \cdot \sqrt{4}
\]
\[
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