Question

Mist (airborne droplets or aerosols)is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and work-piece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. The article "Variables Affecting Mist Generation from Metal Removal Fluids" (Lubrication Engr., 2002: 10-17) gave the accompanying data on x- fluid flow velocity for a 5% soluble oil (cm/sec) and y = the extent of mist droplets having diameters smaller than some value: x: 89 177 189 354 362 442 965 y: .40 .60 .48 .66 .61 .69 .99

Answer 1

## c)

> x=c(89,177,189,354,362,442,965)
> y=c(0.40,0.60,0.48,0.66,0.61,0.69,0.99)
>
> ### linear model
> model=lm(y~x)
> summary(model)

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5 6 7
-0.05940 0.08595 -0.04151 0.03602 -0.01895 0.01136 -0.01346

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 4.041e-01 3.459e-02 11.684 8.07e-05 ***
x 6.211e-04 7.579e-05 8.195 0.00044 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05405 on 5 degrees of freedom
Multiple R-squared: 0.9307, Adjusted R-squared: 0.9168
F-statistic: 67.15 on 1 and 5 DF, p-value: 0.0004403

s_e is the Residual standard error from the model and its estimated value is 0.05405. s_e is the standard deviation of the model.

## d) 95% confidence interval

> confint(model, confidence=0.95)
2.5 % 97.5 %
(Intercept) 0.3152097913 0.4930377793
x 0.0004262474 0.0008159042

Comment: The estimated confidence interval of slope of x does not include zero. Hence, x has the significant effect on y at 0.05 level of significance.

## e)

> predict(model, newdata=data.frame(x=250), interval="confidence", level=0.95)
fit lwr upr
1 0.5593927 0.5020485 0.616737

## f)

> predict.lm(model, newdata=data.frame(x=250), interval="prediction", level=0.95)
fit lwr upr
1 0.5593927 0.4090954 0.7096901