====== 개인과제: 크래커 실험 F-test ======

^ **Table 1. Mean number of crackers eaten in each treatment condition** ^^^^
| | | Fullness ||
| | | Empty stomach | Full stomach |
|Weight|Normal|M=22 \\ SD=9.0029|M=15 \\ SD=8.1757|
|:::|Obese|M=17 \\ SD=8.3351|M=18 \\ SD=8.1628|

===1. Degrees of Freedom===

1. <latex>df_{total} = 80 - 1 = 79</latex>\\
2. <latex>df_{between} = 4 - 1 = 3</latex>\\
3. <latex>df_{within} = 80 - 3 = 76</latex>\\
4. <latex>df_A = 2 - 1 = 1</latex>\\
5. <latex>df_B = 2 - 1 = 1</latex>\\
6. <latex>df_{AxB} = 3 - 1 - 1 = 1</latex>\\

===2. Sum of Squares===

1. <latex>SS_{total} = 31936 - \frac{1440^2}{80} = 31936 - 25920 = 5916</latex>\\
2. <latex>SS_{between} = \frac{780^2}{40} + \frac{660^2}{40} + \frac{740^2}{40} + \frac{700^2}{40} - \frac{1440^2}{80} = 520</latex>\\
3. <latex>SS_{within} = SS_{total} - SS_{between} = 5916 - 520 = 5396</latex>\\
4. <latex>SS_A = \frac{740^2}{40} + \frac{700^2}{40} - \frac{1440^2}{80} = 20</latex>\\
5. <latex>SS_B = \frac{780^2}{40} + \frac{660^2}{40} - \frac{1440^2}{80} = 180</latex>\\
6. <latex>SS_{AxB} = SS_{between} - SS_A - SS_B = 520 - 20 - 180 = 320</latex>\\

===3. Mean Square===

1. <latex>MS_A = \frac{SS_A}{df_A} = \frac{20}{1} = 20</latex>\\
2. <latex>MS_A = \frac{SS_B}{df_B} = \frac{180}{1} = 180</latex>\\
3. <latex>MS_{AxB} = \frac{SS_{AxB}}{df_{AxB}} = \frac{320}{1} = 320</latex>\\
4. <latex>MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{5396}{76} = 71</latex>\\

===4. F-ratio===

1. <latex>F_A = \frac{MS_A}{MS_within} = \frac{20}{71} = 0.2817</latex>\\
2. <latex>F_B = \frac{MS_B}{MS_within} = \frac{180}{71} = 2.5352</latex>\\
3. <latex>F_{AxB} = \frac{MS_{AxB}}{MS_within} = \frac{320}{71} = 4.5070</latex>\\

===5. 결과===

^  **Result**  ^^^^^
| Source | SS | df | MS | F |
| Between Treatment |520|3| | |
| Factor A(weight) |20|1|20|0.2817|
| Factor B(fullness) |180|1|180|2.5352|
| A x B interaction |320|1|320|4.5070|
| Within treatment |5396|76|71| |
| Total |5916|79| | |
| weight x fullness factorial design |||||

1. <latex>F_A = 0.2817 < F_{CriticalValues}(1, 76)= 4.00 (p<0.5)</latex> **기각**\\
2. <latex>F_A = 2.5352 < F_{CriticalValues}(1, 76)= 4.00 (p<0.5)</latex> **기각**\\
3. <latex>F_{AxB} = 4.5070 > F_{CriticalValues}(1, 76)= 4.00 (p<0.5)</latex> **지지**\\

===6. 결론===

1번 가설인 weight가 크래커 갯수에 영향을 미친다는 가설은 기각되었다. 
또한 2번 가설인 fullness가 크래커 갯수에 영향을 미친다는 가설또한 기각되었다.
하지만 weight와 fullness 사이의 상호작용 효과가 크래커 갯수에 영향을 미친다는 3번 가설은 지지되었다.