Researchers studied the impact of temperature (x_1) and concentration (x_2) on t

Question

Researchers studied the impact of temperature (x_1) and concentration (x_2) on the percentage of impurities Y for a new chemical process. The data is below: there are n = 14 observations. The researchers would like to consider the statistical model Y_i + beta_0 + beta_1 x_1i + beta_2 x_2i + elementof_i for i = 1, 2, ..., 14, under the standard assumptions for the errors elementof_i. (a) Calculate the multiple linear regression equation that relates percent impurities (Y) to temperature (x_1) and concentration (x_2). (b) Test to see whether the impurity percentage is linearly related to temperature (after adjusting for the effect of concentration). You can do this by performing a relevant hypothesis test or by writing a relevant confidence interval (c) Test to see whether the impurity percentage is linearly related to concentration (after adjusting for the effect of temperature). You can do this by performing a relevant hypothesis test or by writing a relevant confidence interval. (d) Calculate R^2 and interpret its value clearly. (Don't just give me the R output from the summary of your linear fit. Use the equation we discussed in class to get this value.) (e) Perform residual diagnostics for the model fit, specifically, construct a normal qq-plot for the residuals and plot the residuals versus the fitted values. Interpret each plot. You can use the following code to input the data in R.

Explanation / Answer

impurities <- c(14.9,16.9,17.4,16.9,16.9,16.7,17.1,16.9,16.7,16.9,16.7,17.1,17.6,16.9)

temp <- c(85.8,83.8,84.5,86.3,85.2,83.8,86.1,85.9,85.7,86.3,83.5,85.8,85.9,84.2)

concentration <- c(42.3,43.4,42.7,462,43.2,43.7,43.3,43.4,43.3,42.6,44.0,42.8,43.1,43.5)

model <- lm(concentration~temp+impurities)
> summary(model)

Call:
lm(formula = concentration ~ temp + impurities)

Residuals:
Min 1Q Median 3Q Max
-68.53 -53.81 -33.08 14.03 350.87

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2964.112 2850.547 -1.040 0.321
temp 34.057 31.507 1.081 0.303
impurities 8.053 52.426 0.154 0.881

Residual standard error: 115.6 on 11 degrees of freedom
Multiple R-squared: 0.09712, Adjusted R-squared: -0.06704
F-statistic: 0.5916 on 2 and 11 DF, p-value: 0.5701

a) Y = -2964.112+34.057*x1 +8.053*x2

b) p-value for x1 is 0.331

which is more than 0.05 , hence we reject the null and conclude that there is not significant linear relation

between y and temp

c) p-value for x2 is 0.881 > 0.05

hence we reject the null and conclude that there is not significant linear relation between y and concentration

d) R^2 = 0.09712 which means this model explains 9.7 % of variation in y .

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