A researcher compared three treatment conditions with a separate sample of $ n = 8 $ in each treatment. The results are shown in the following summary. Fill in all missing values in the table, which are noted as #1, #2, and so on.| Source | SS | df | MS | ||----------|-----|----|-----|---|| Between | #1 | #3 | #6 | F = 4.0 || Within | 63 | #4 | #7 | || Total | #2 | #5 | | |### Blanks- Blank #1- Blank #2- Blank #3- Blank #4- Blank #5- Blank #6- Blank #7**Explanation:**- **SS (Sum of Squares):** Represents the variation or dispersion between and within the treatment groups.- **df (Degrees of Freedom):** Corresponds to the number of independent values that can vary in the calculation of a statistic.- **MS (Mean Square):** Calculated by dividing the SS by the corresponding df.- **F (F-ratio):** A statistic used to compare variances and assess the significance of results.### Instructions:Fill in the missing values using the relationships between SS, df, MS, and F-ratio calculations.

Answered: A researcher compared three treatment…

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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Question

A researcher compared three treatment conditions with a separate sample of $ n = 8 $ in each treatment. The results are shown in the following summary. Fill in all missing values in the table, which are noted as #1, #2, and so on.

| Source | SS | df | MS | |
|----------|-----|----|-----|---|
| Between | #1 | #3 | #6 | F = 4.0 |
| Within | 63 | #4 | #7 | |
| Total | #2 | #5 | | |

### Blanks
- Blank #1
- Blank #2
- Blank #3
- Blank #4
- Blank #5
- Blank #6
- Blank #7

**Explanation:**

- **SS (Sum of Squares):** Represents the variation or dispersion between and within the treatment groups.
- **df (Degrees of Freedom):** Corresponds to the number of independent values that can vary in the calculation of a statistic.
- **MS (Mean Square):** Calculated by dividing the SS by the corresponding df.
- **F (F-ratio):** A statistic used to compare variances and assess the significance of results.

### Instructions:
Fill in the missing values using the relationships between SS, df, MS, and F-ratio calculations.

Expert Solution

Given

The formulas are given as:

$T o t a l S S = S S b e t w e e n + S S w i t h i n T o t a l d f = d f_{b e t w e e n} + d f_{w i t h i n} M S b e t w e e n = \frac{S S b e t w e e n}{d f_{b e t w e e n}} M S w i t h i n = \frac{S S w i t h i n}{d f_{w i t h i n}} F = \frac{M S B e t w e e n}{M S w i t h i n}$

Also,

$T h e r e a r e t h r e e t r e a t m e n t s, h e n c e, k = 3 A n d t o t a l s a m p l e s i z e i s : n = 8 f o r e a c h o f t h e t h r e e t r e a t m e n t s i e . N = 24$

Therefore,

$d f_{b e t w e e n} : k - 1 = 2 d f_{w i t h i n} : N - k = 24 - 3 = 21 T o t a l d f = d f_{b e t w e e n} + d f_{w i t h i n} = 2 + 21 = 23$

Step 2

$w e h a v e, M S w i t h i n = \frac{S S w i t h i n}{d f_{w i t h i n}} = \frac{63}{21} = 3$

Therefore,

$\begin{array}{rcl} F & = & \frac{M S B e t w e e n}{M S w i t h i n} \\ \Rightarrow & 4 = \frac{M S B e t w e e n}{3} \\ \Rightarrow & M S B e t w e e n = 12 \end{array}$

And

$\begin{array}{rcl} M S b e t w e e n & = & \frac{S S b e t w e e n}{d f_{b e t w e e n}} \\ 12 & = & \frac{S S b e t w e e n}{2} \\ S S b e t w e e n & = & 24 \end{array}$