Item 2
An article in Optical Engineering[“Operating Curve Extraction of a Correlator’s Filter” (2004, Vol.43, pp. 2775-2779)] reported on the use of an optical correlator to perform experimnent by varying brightness and contrast. The resulting modulation is charachterized by the useful range of gray levels.
## Useful.Range Contrast Brightness
## 1 96 56 54
## 2 50 80 61
## 3 50 70 65
## 4 112 50 100
## 5 96 65 100
## 6 80 80 100
## 7 155 25 50
## 8 144 35 57
## 9 255 26 54##
## Call:
## lm(formula = Data$Useful.Range ~ Data$Contrast + Data$Brightness)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32.334 -20.090 -8.451 8.413 69.047
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 238.5569 45.2285 5.274 0.00188 **
## Data$Contrast -2.7167 0.6887 -3.945 0.00759 **
## Data$Brightness 0.3339 0.6763 0.494 0.63904
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36.35 on 6 degrees of freedom
## Multiple R-squared: 0.7557, Adjusted R-squared: 0.6742
## F-statistic: 9.278 on 2 and 6 DF, p-value: 0.01459## Analysis of Variance Table
##
## Response: Data$Useful.Range
## Df Sum Sq Mean Sq F value Pr(>F)
## Data$Contrast 1 24196.3 24196.3 18.3128 0.005209 **
## Data$Brightness 1 322.1 322.1 0.2438 0.639043
## Residuals 6 7927.6 1321.3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
A.) Find the multiple linear regression model to these data
\[y = Xβ+ε\] \[β = (X'X)^{-1} (X'y)\]
Intercept \(β_0 = 238.5569\)
Contrast \(β_1 = -2.7167\)
Brightness \(β_2 = 0.3339\)
\[y = 238.5569 - 2.7167X_1 + 0.3339X_2\]
B.) Estimate variance
\(SS_E = SS_T - SS_R\)
\(SS_E = 7927.6\)
\[σ^2 = MS_E = \frac{SS_E}{(n-p)}\]
\(σ^2 = 1321.3\)
C.) compute the standard error of the regression coefficient
\[S_{β1} = \sqrt{\frac{S_{y12}^2}{Σ(X_1^2*(1-r_{12}^2))}}\]
\[S_{β2} = \sqrt{\frac{S_{y12}^2}{Σ(X_2^2*(1-r_{12}^2))}}\]
\(S_{β1} = 0.6887\)
\(S_{β2} = 0.6763\)
D.) Find the useful range when brightness = 80 and contrast = 75
\[y = 238.5569 - 2.7167X_1 + 0.3339X_2\]
\(y = 61.5164\)
E.) Test for significance of regression using α = 0.05. What is the P-value for this test
\(H_0: β_1 = β_2 = 0\)
\(H_1: β_1 or β_2 \neq 0\) \[f_0 = \frac{SS_R/k}{SS_E/(n-p)} = \frac{MS_R}{MS_E}\]
\(f_0 = 9.278135\)
\(f_0 > f_{0.05,2,6} = 5.1433\)
\(P(f_0) = 0.014587\)
We reject \(H_0\) which does not necesarily mean that both contrast and brightness will be a good indicator of the useful range.
F.) Construct a t-test on each regression coefficient. What conclusion can you draw about the variables in this model? use α = 0.05
\[H_0: \beta_j = \beta_{j0}\] \[H_1: \beta_j \neq \beta_{j0}\]
\[ T_0 = \frac{\hat{\beta_j}-\beta_{j0}}{\sqrt{\sigma^{2}C_{jj}}}\]
C =
##
## [1,] 1.54821553 -0.00676718 -0.01503639
## [2,] -0.00676718 0.00035901 -0.00017775
## [3,] -0.01503639 -0.00017775 0.00034616For \(\beta_1\) when \(\beta_2 = 0\), \(T_0 = -3.944461\)
\(|t_0| = 3.944461 > t_{0.025,6} = 2.447\)
\(P(t_0) = 0.00759 < \alpha = 0.05\)
We reject \(H_0\) for contrast # # For \(\beta_2\) \(\beta_1 = 0\), \(T_0 = 0.4937161\)
\(|t_0| = 0.4937161 < t_{0.025,6} = 2.447\)
\(P(t_0) = 0..639068 > \alpha = 0.05\)
We fail to reject \(H_0\) for brightness
Conclusion:
With this hypothesis we can say that the contrast greatly contributes to the useful range of the optical correlator over the brightness.
