Question

Let’s continue to use the Toluca.txt dataset. We now assume a normal error model: Yi = β0 + β1xi + i (i = 1, . . . , n), i ∼ N (0, σ2 ), where xi = lotSize and Yi = workHrs. a. Obtain an estimate for σ 2 . Let’s call this estimator ˆσ 2 . You can either calculate this using the formula directly or obtain this from the R output using lm. b. Construct a 90% confidence interval for β0. Remember to use the t-distribution here. Interpret your confidence interval for β0. c. Test H0 : β1 = 0 vs. H1 : β1 > 0 at 0.05 significance level. State the test statistic, its null distribution, the decision rule, and your conclusion.

lotSize workHrs

80 399

30 121

50 221

90 376

70 361

60 224

120 546

80 352

100 353

50 157

40 160

70 252

90 389

20 113

110 435

100 420

30 212

50 268

90 377

110 421

30 273

90 468

40 244

80 342

70 323

Answer 1

R output:

Call:
lm(formula = workHrs ~ lotSize)

Residuals:
Min 1Q Median 3Q Max
-83.876 -34.088 -5.982 38.826 103.528

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 62.366 26.177 2.382 0.0259 *
lotSize 3.570 0.347 10.290 4.45e-10 ***
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 48.82 on 23 degrees of freedom
Multiple R-squared: 0.8215, Adjusted R-squared: 0.8138
F-statistic: 105.9 on 1 and 23 DF, p-value: 4.449e-10

Analysis of Variance Table

Response: workHrs
Df Sum Sq Mean Sq F value Pr(>F)
lotSize 1 252378 252378 105.88 4.449e-10 ***
Residuals 23 54825 2384
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1