The following table presents measurements of mean noise levels in dBA (y), roadw

Question

The following table presents measurements of mean noise levels in dBA (y), roadway width in m (x_1), and mean speed in km/h (x_2) for 10 locations in Bangkok, Thailand, as reported in the article "Modeling of Urban Area Traffic Noise" (P. Pamanikabud and C Tharasawatipipat, Journal of Transportation Engineering, 1999:152-159). Construct a good linear model to predict mean noise levels using roadway width, mean speed, or both, as predictors. Provide the standard deviations of the coefficient estimates and the P-values for testing that they are different from 0. Explain how you chose your model.

Explanation / Answer

The model was fitted using both the predictors. The result of the fit is provided below:

lm(formula = y ~ x1 + x2, data = data1)

Residuals:
Min 1Q Median 3Q Max
-1.3474 -0.1618 0.2112 0.3806 0.7411

Coefficients:

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7014 on 7 degrees of freedom
Multiple R-squared: 0.5608,   Adjusted R-squared: 0.4353
F-statistic: 4.469 on 2 and 7 DF, p-value: 0.05615

The p value of each of the estimates is obtained in th result. Here we test the null hypothesis that the coefficient of the estimates are equal to zero i.e. it have no effect. If the p value is less than 0.05 then we reject the null hypothesis and for that variables the estimates are significantly different from zero and may be included in the regression model. Here we can see that only the intercept and the variable x2 is significant. The effect of variable x1 is not significant as the p value is > 0.05. The statistically significant variables are automatically showed in the result using the *(asteriks) symbol.

The standard deviation and the p values of the significant estiamtes are given below:

Therefore the fitted model is:

y = 73.94355 + 0.17935*x2 or

mean noise levels in dBA =  73.94355 + 0.17935*Mean noise in km/hr

Here we have not included the variable x1 i.e. the roadway width in m as it is not statistically significant. We may fit a model with both the predictors taken together as we may eliminate the not significant variables by looking at the p value.

I have used R 3.3.2 to fit the regression model. The code for the same is included here:

data1 = read.csv("Book1.csv") # reading data into R

model1 = lm(y~x1+x2, data=data1)
summary(model1)

Estimate Std. Error t value Pr(>|t|) (Intercept) 73.94355 2.14230 34.516 4.44e-09 *** x1 -0.10513 0.09572 -1.098 0.308 x2 0.17935 0.06309 2.843 0.025 *

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