FINANCIAL ESSAY
Joss_Smitx01
SixSigma4.pptANALYZE PHASE
Polytechnic University of Puerto Rico
Department of Industrial Engineering
MMP 6130
Six Sigma
Hypothesis Test of the Variance of a Population
- Chi-Square Distribution
- Used when we want to know if the standard deviation or variance of a process is greater, equal, or lower than a specific value.
Example 1
Variance Specifications
Ingredient A = .36 lbs
Ingredient B= .10 lbs
Ingredient C=3.5 lbs
Alpha = 5%
| Blending Process | ||
| Ingredient A | Ingredient B | Ingredient C |
| 19.2 lbs | 1.2 lbs | 140.7 lbs |
| 19.5 lbs | 1.4 lbs | 139.6 lbs |
| 19.0 lbs | 1.6 lbs | 137.3 lbs |
| 18.9 lbs | 1.4 lbs | 142.5 lbs |
| 19.3 lbs | 1.5 lbs | 138.7 lbs |
| 19.2 lbs | 1.4 lbs | 139.6 lbs |
Hypothesis Test of the Variance of Two Population
- F-Distribution
- Used to compare the standard deviation or variance of two processes.
- If n is the same,
- If is n not the same,
Example 2
| Blending Process | ||
| Process A | Process B | Process C |
| 100 lbs/min | 108 lbs/min | 109 lbs/min |
| 120 lbs/min | 107 lbs/min | 108 lbs/min |
| 105 lbs/min | 104 lbs/min | 110lbs/min |
| 107 lbs/min | 110 lbs/min | 107 lbs/min |
| 114 lbs/min | 110 lbs/min | 116 lbs/min |
| 108lbs/min | 106 lbs/min | 113 lbs/min |
Linear Regression
- Used to find the equation that is closer and most precise to be used as a tool for projection.
- Objective:
- Determine the linear function that assimilates to the behavior of the data.
Sum of Minimum Squares
- Linear Function
- Independent Variable x
- Dependent Variable y
- Variation of the combination of both variables
- Slope of the Function
- Intercept in y
Coefficient of Correlation
- Used to determine the precision of the function with relation to the actual values.
Coefficient of Correlation
- If r, is near +1 (0.5 and 1) therefore there is a strong linear relationship between the dependent and independent variable with a positive slope.
- If r, is near -1 (-0.5 and -1) therefore there exists a strong linear relationship between the dependent and independent variable with a negative slope.
- If r, is near 0 (-0.49 and .49) therefore there is no linear relationship. This means that the value of Y does not depend of X.
Anova
| Variation Form | Sum of the Squares | Liberty Degrees | Mean Square | Value of Fexp |
| Regression | 1 | |||
| Error | n-2 |
Example 3
| Blending Process | |
| Temperature | Output |
| 80° | 108 lbs/min |
| 79° | 107 lbs/min |
| 75° | 104 lbs/min |
| 82° | 110 lbs/min |
| 84° | 110 lbs/min |
| 80° | 106 lbs/min |
Multiple Regression
- When we have a process where the dependent variable follows a linear behavior that the independent variable changes it is possible to determine the linear function it represents.
Multiple Regression
N Σ X Σ Z Σ Y
Σ X Σ X^2 Σ XZ Σ XY
Σ Z Σ XZ Σ Z^2 Σ ZY
Multiple Regression
Usando la tecnica de Kramer de matrices, sacamos cuatro determinandes:
- I A I Determinante de la matriz original
- I a I Determinante de la matriz impactada por el Intercepto.
- I x I Determinante de la matriz impactada por pendiente en X.
- I z I Determinante de la matriz impactada por pendiente en Z.
A = Intercepto en Y = IaI / IAI
Bx= Pendiente en X = IxI / IAI entonces, Y = a + x(bx) + z(bz)
Bz= Pendiente en Z = IzI / IAI
Multiple Regression
ANOVA: Syy = Σ Y2 - (Σ Y)2
n
SSR = ΣBj Gj - (Σ Y)2
n
SSE = Syy- SSR
Se2 = SSE / (n-k-1)
Ho: Regresión no significativa
H1: Regresión significativa
F exp: SSR / k
SSE / (n-k-1)
Example 4
Weight(lbs) Volume(cft) Output
10 5 12
9 4 15
11 5 14
13 5 16
12 6 11
Two Factor Design
- Used to determine if their exists or not any relationship between more than one independent or dependent variable.
- This analysis could measure if the dependent variable could have been affected with two independent variables.
Two Factor Design
Where:
- a=# of treatments
- b=# of blocks
- n=# repetitions
- Ti= Total treatments
- Tij= Total of treatments
per treatment/block
- T=total of all the results
Two Factor Design
- To convert the sum of minimum squares into a variance
Two Factor Design
Example 5
| Solution Quantity | Product Yield Process Temperature | ||
| 80°cl | 85°cl | 90°cl | |
| 1.2 gr | 95 93 | 91 93 | 92 90 |
| 1.6 gr | 94 95 | 90 91 | 90 91 |
| 2.0 gr | 96 92 | 92 93 | 89 92 |
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