Question 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 an experiment by varying brightness and contrast. The resulting modulation is characterized by the useful range of gray levels. The data follow:

##   usefulrange brightness contrast
## 1          96         54       56
## 2          50         61       80
## 3          50         65       70
## 4         112        100       50
## 5          96        100       65
## 6          80        100       80
## 7         155         50       25
## 8         144         57       35
## 9         255         54       26

plot(correlator.data)

multiple.regression <- lm(usefulrange ~ brightness + contrast, data=correlator.data)
summary(multiple.regression)

## 
## Call:
## lm(formula = usefulrange ~ brightness + contrast, data = correlator.data)
## 
## 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 **
## brightness    0.3339     0.6763   0.494  0.63904   
## contrast     -2.7167     0.6887  -3.945  0.00759 **
## ---
## 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

A. Fit a multiple linear regression model to these data.

Formula for multiple linear regression model:

\[ y = \beta_0 + \beta_1 X_1 + \beta_2 X_2\] where

y = Useful range (ng)
B0 = y-intercept
B1 = slope 1 B2 = slope 2 X1 = Brightness (%)
X2 = Contrast (%)

Inputting the values solved by our code chunk,

\[ y = 238.56 + 0.3339 X_1 + (-2.7167) X_2 \\ \\ = 238.56 + 0.3339 X_1 -2.7167 X_2\]

B. Estimate 𝜎2.

(summary(multiple.regression)$sigma)**2

## [1] 1321.273

C. Compute the standard errors of the regression coefficients.

As seen from the results by our code chunk,

\[ SE(\beta_0)= 45.2285 \\ SE(\beta_1)= 0.6763 \\ SE(\beta_2)= 0.6887\]

D. Predict the useful range when brightness= 80 and contrast= 75.

We will use our multiple regression model and will be inputting our values knowing that X1 = 80 and X2 = 75.

\[ y = 238.56 + 0.3339 X_1 -2.7167 X_2 \\ = 238.56 + 0.3339(80) - 2.7167(75) \\ y = 61.5195 \]

E. Test for significance of regression using 𝛼=0.05. What is the P-value for this test?

As seen in our code chunk, the p-value for this test is 0.01459. As we should know, the p-value for a variable is less than our significance level, we have enough evidence to reject our null hypothesis. Since we are given the significance level 𝛼=0.05,

\[ 0.01459 < 0.05 \]

Therefore,

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