Statistics 112 Simple Linear Regression Fuel Consumption Example March 1, 2004 E. Bura

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1 Statistics 112 Simple Linear Regression Fuel Consumption Example March 1, 2004 E. Bura Fuel Consumption Case: reducing natural gas transmission fines. In 1993, the natural gas industry was deregulated. In consequence, the natural gas companies became responsible for acquiring the natural gas needed to heat the homes and businesses they serve. Natural gas companies place orders for natural gas to be transmitted by pipeline transmission systems to their cities. For placing an order, the natural gas companies need to make a prediction of the city s natural gas need for that period. In order to encourage natuaral gas companies to make accurate predictions and to help control costs, pipeline transmission systems charge in addition to their usual fees, transmission fines if the order is below need or above need. There is of course some leeway; i.e., there is a minimum amount of errors that go unfined. Suppose a management consulting firm is responsible to make predictions for need of gas for a natural gas company serving a small city. The problem is to predict weekly fuel consumption (y) on the basis of average hourly temperature (x). For this we observed y and x for eight weeks: Week x y x 2 xy n i=1 x i = n i=1 y i =81.7 n i=1 x2 i = n i=1 x iy i = The plot of y versus x suggests that the simple linear regression model may provide a good fit to the data. Hence, we hypothesize that y i = β 0 + β 1 x i + ɛ i

2 with 1. E(ɛ i )=0 2. Var(ɛ i )=σ 2 3. ɛ i N(0,σ 2 ) 4. The errors are independent of one another. That is, the data are a random sample. The least squares estimates of the parameters of the model, β 0 and β 1, are where SS xy = = n i=1 n i=1 ˆβ 1 = SS xy ˆβ 0 =ȳ ˆβ 1 x x i y i ( n i=1 x i)( n i=1 y i) = n x 2 i ( n i=1 x i) 2 n = Also, ȳ = and x = These yield, ˆβ 1 = SS xy = =.1279 ˆβ 0 =ȳ ˆβ 1 x = = The fitted line is given by 1 The Meaning of ˆσ = s ŷ i = x i ˆσ = s = SSE MSE = n 2 Since y N(β 0 + β 1 x, σ 2 ) we expect most of the observed responses (roughly 95%) to fall within 2s from the fitted line.

3 The Pearson Correlation Coefficient is a measure of the strengh of the linear relationship between x and y. It is defined to be r = r satisfies 1 r 1. For the fuel consumption data, r = SS xy SSxx SS yy = SS xy SSxx SS yy = Testing the significance of the Slope Question: Is there a statistically significant linear relationship between y and x? To answer this question we have to test H 0 :β 1 =0 versus H 1 : β 1 0 When the errors, and hence the responses, are normally distributed we have Therefore, ˆβ 1 N(β 1, σ 2 ) ˆβ 1 β 1 σ SSxx N(0, 1) Since σ is unknown and is estimated by s, weobtain ˆβ 1 β 1 s SSxx t n 2 Consequently, to test whether β 1 =0weuse t = ˆβ 1 0 s SSxx t n 2 We reject the null at level α, if t >t α/2 (n 2)

4 In general, to test H 0 :β 1 = β versus H 1 :β 1 β β 1 >β β 1 <β use the test statistic Reject the null at level α if ˆβ 1 β s SSxx t n 2 t >t α/2 (n 2) t>t α (n 2) t< t α (n 2) with respect to the analogous alternative. Also, a 100(1-α)% confidence interval for β 1 is given by ˆβ 1 ± t α/2 (n 2) s SSxx Observe that if β 1 = 0 then the population correlation coefficient, ρ, is also equal to zero. Therefore, the t-test for the slope of the model can be also used to test whether ρ = Back to the fuel consumption example SSE = n i=1 (y i ŷ i ) 2 = , and MSE = SSE n 2 = = So, s = MSE = To test whether β 1 =0versusβ 1 0, we compute the test statistic t = ˆβ 1 s = = 7.33 SSxx Since t = 7.33, the p-value of the test is smaller twice the area to the right of That is, p-value < =.001. This is a highly significant result so we reject the null in favor of the alternative. The linear model is useful for modelling the mean of fuel consumption, y.

5 3 The coefficient of determination The coefficient of determination, R 2, is a measure of the contribution of x or the model in predicting y. R 2 = explained variability total variability in y about its mean ȳ SS yy SSE SS yy =1 SSE SS yy In other words, R 2 represents the proportion of variability in y explained by the fitted model. In the case of the simple linear regression, that is when the hypothesized model is of the form y = β 0 + β 1 x + ɛ, R 2 = r 2,wherer is the correlation coefficient of y and x. 3.1 Back to the fuel consumption example R 2 =1 SSE = =.8995 SS yy Hence, 89.95% of the variability in the y values about their mean ȳ is explained by the fitted simple linear regression model. 4 Estimation and Prediction The regression model has two main uses: Estimating the mean y value for a specific value of x Predicting a new individual y value for a given x is The standard error of ŷ as an estimator of the mean y-value when x = x p 1 σŷ = σ n + (x p x) 2 The standard error of ŷ as a predictor of the individual y-value when x = x p is

6 σ y ŷ = σ 1+ 1 n + (x p x) 2 Remark: The population standard deviation σ is unknown. Its estimate s = MSE is used instead. A 100(1 α)% CI for the mean y-value at x = x p is given by 1 ŷ ± t α/2 (n 2)s n + (x p x) 2 A 100(1 α)% CI for the predicted y-value at x = x p is given by ŷ ± t α/2 (n 2)s 1+ 1 n + (x p x) Back to the Fuel Consumption Example Compute a 95% CI for the mean value of weekly fuel consumption when the average hourly temperature is x p = 40 degrees F: ŷ = = ( )2 ŷ ± t.025 (6)s ± ±.59 = (10.13, 11.31) A 95% prediction interval for the actual value of weekly fuel consumption when the average hourly temperature is x p = 40 degrees F: ŷ ± t.025 (6)s 1+ 1 ( ) ± ± 1.71 = (9.01, 12.43) Important Remark: Observe that the prediction interval is wider than the confidence interval for the mean y-value. This is always the case: prediction intervals are wider than confidence intervals for the same confidence level.