Assignment 2 - Linear Regression - Chap 3: 2, 9, 10, and 12
Marla Seth, ibc346
7/9/2021
Problem 2
While both are non-parametric models that use the nearest K points to estimate a value, The KNN classifier is used in classification to classify a categorical or qualitative response variable. The KNN regression uses regression to predict a quantitative response variable.
Problem 9a
library (ISLR)## Warning: package 'ISLR' was built under R version 4.0.5names(Auto)## [1] "mpg" "cylinders" "displacement" "horsepower" "weight"
## [6] "acceleration" "year" "origin" "name"pairs(Auto)
cor(Auto [,1:8])## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7776175 -0.8051269 -0.7784268 -0.8322442
## cylinders -0.7776175 1.0000000 0.9508233 0.8429834 0.8975273
## displacement -0.8051269 0.9508233 1.0000000 0.8972570 0.9329944
## horsepower -0.7784268 0.8429834 0.8972570 1.0000000 0.8645377
## weight -0.8322442 0.8975273 0.9329944 0.8645377 1.0000000
## acceleration 0.4233285 -0.5046834 -0.5438005 -0.6891955 -0.4168392
## year 0.5805410 -0.3456474 -0.3698552 -0.4163615 -0.3091199
## origin 0.5652088 -0.5689316 -0.6145351 -0.4551715 -0.5850054
## acceleration year origin
## mpg 0.4233285 0.5805410 0.5652088
## cylinders -0.5046834 -0.3456474 -0.5689316
## displacement -0.5438005 -0.3698552 -0.6145351
## horsepower -0.6891955 -0.4163615 -0.4551715
## weight -0.4168392 -0.3091199 -0.5850054
## acceleration 1.0000000 0.2903161 0.2127458
## year 0.2903161 1.0000000 0.1815277
## origin 0.2127458 0.1815277 1.0000000
Problem 9c
lm_auto1 <- lm(mpg~.-name, data = Auto)
summary(lm_auto1)##
## Call:
## lm(formula = mpg ~ . - name, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5903 -2.1565 -0.1169 1.8690 13.0604
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.218435 4.644294 -3.707 0.00024 ***
## cylinders -0.493376 0.323282 -1.526 0.12780
## displacement 0.019896 0.007515 2.647 0.00844 **
## horsepower -0.016951 0.013787 -1.230 0.21963
## weight -0.006474 0.000652 -9.929 < 2e-16 ***
## acceleration 0.080576 0.098845 0.815 0.41548
## year 0.750773 0.050973 14.729 < 2e-16 ***
## origin 1.426141 0.278136 5.127 4.67e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.328 on 384 degrees of freedom
## Multiple R-squared: 0.8215, Adjusted R-squared: 0.8182
## F-statistic: 252.4 on 7 and 384 DF, p-value: < 2.2e-16The p-value for the model is less than 2.2e-16, so there is a significant relationship between at least one of the predictors and the response variable. Looking at the individual t-test results of the model, the significant variables appear to be displacement (p-vaue = 0.00844), weight (p-value < 2e-16), year (p-value < 2e-16), and origin (p-value = 4.67e-07). The coefficient for year is 0.750773. This means that each year, on average, new cars get approximately 0.75 more miles to the gallon as compared to the previous year’s cars when all other variables are held constant. Another way to describe it is that the predicted mpg will be 0.75 higher for each unit year increase when all other variables are held constant.
Problem 9d
par(mfrow = c(2,2))
plot(lm_auto1)
 The residuals plot shows that there is an issues with heteroscedasticity. There is a slight curve to the line which suggests the relationship might not be linear. It shows observations 327 and 394 as outliers in addition to some other unreadable points that will be identified below. The Q-Q plot suggests that there could be an issue with normality at one tail. The leverage plot identifies observation 14 with unusually high leverage.
 Outliers:
rstudent(lm_auto1)[which(rstudent(lm_auto1)>=3)]## 245 323 326 327
## 3.390068 4.029537 3.494823 3.690246
Problem 9e
lm_auto2 <- lm(mpg~year*origin, data = Auto)
summary (lm_auto2)##
## Call:
## lm(formula = mpg ~ year * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.3141 -3.7120 -0.6513 3.3621 15.5859
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -83.3809 12.0000 -6.948 1.57e-11 ***
## year 1.3089 0.1576 8.305 1.68e-15 ***
## origin 17.3752 6.8325 2.543 0.0114 *
## year:origin -0.1663 0.0889 -1.871 0.0621 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.199 on 388 degrees of freedom
## Multiple R-squared: 0.5596, Adjusted R-squared: 0.5562
## F-statistic: 164.4 on 3 and 388 DF, p-value: < 2.2e-16lm_auto3 <- lm(mpg~year*weight, data = Auto)
summary (lm_auto3)##
## Call:
## lm(formula = mpg ~ year * weight, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.0397 -1.9956 -0.0983 1.6525 12.9896
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.105e+02 1.295e+01 -8.531 3.30e-16 ***
## year 2.040e+00 1.718e-01 11.876 < 2e-16 ***
## weight 2.755e-02 4.413e-03 6.242 1.14e-09 ***
## year:weight -4.579e-04 5.907e-05 -7.752 8.02e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.193 on 388 degrees of freedom
## Multiple R-squared: 0.8339, Adjusted R-squared: 0.8326
## F-statistic: 649.3 on 3 and 388 DF, p-value: < 2.2e-16lm_auto4 <- lm(mpg~.-name+year:weight, data = Auto)
summary (lm_auto4)##
## Call:
## lm(formula = mpg ~ . - name + year:weight, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.9995 -1.8495 -0.1559 1.6061 11.7042
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.186e+02 1.338e+01 -8.864 < 2e-16 ***
## cylinders -1.218e-01 3.032e-01 -0.402 0.6881
## displacement 1.293e-02 7.019e-03 1.842 0.0663 .
## horsepower -2.877e-02 1.286e-02 -2.236 0.0259 *
## weight 3.044e-02 4.652e-03 6.543 1.94e-10 ***
## acceleration 1.447e-01 9.196e-02 1.574 0.1164
## year 2.084e+00 1.732e-01 12.033 < 2e-16 ***
## origin 1.174e+00 2.597e-01 4.519 8.30e-06 ***
## weight:year -4.879e-04 6.097e-05 -8.002 1.47e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.084 on 383 degrees of freedom
## Multiple R-squared: 0.847, Adjusted R-squared: 0.8439
## F-statistic: 265.1 on 8 and 383 DF, p-value: < 2.2e-16lm_auto5 <- lm(mpg~.-name+year:horsepower, data = Auto)
summary (lm_auto5)##
## Call:
## lm(formula = mpg ~ . - name + year:horsepower, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.6314 -1.8896 -0.0724 1.5781 11.6140
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.723e+01 1.008e+01 -9.648 < 2e-16 ***
## cylinders 8.146e-02 3.027e-01 0.269 0.788
## displacement 5.183e-03 7.072e-03 0.733 0.464
## horsepower 8.070e-01 9.496e-02 8.498 4.33e-16 ***
## weight -5.288e-03 6.112e-04 -8.651 < 2e-16 ***
## acceleration -2.894e-02 9.121e-02 -0.317 0.751
## year 1.830e+00 1.318e-01 13.887 < 2e-16 ***
## origin 1.178e+00 2.558e-01 4.604 5.66e-06 ***
## horsepower:year -1.140e-02 1.302e-03 -8.754 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.042 on 383 degrees of freedom
## Multiple R-squared: 0.8512, Adjusted R-squared: 0.8481
## F-statistic: 274 on 8 and 383 DF, p-value: < 2.2e-16lm_auto6 <- lm(mpg~.-name+acceleration:horsepower, data = Auto)
summary (lm_auto6)##
## Call:
## lm(formula = mpg ~ . - name + acceleration:horsepower, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.0329 -1.8177 -0.1183 1.7247 12.4870
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -32.499820 4.923380 -6.601 1.36e-10 ***
## cylinders 0.083489 0.316913 0.263 0.792350
## displacement -0.007649 0.008161 -0.937 0.349244
## horsepower 0.127188 0.024746 5.140 4.40e-07 ***
## weight -0.003976 0.000716 -5.552 5.27e-08 ***
## acceleration 0.983282 0.161513 6.088 2.78e-09 ***
## year 0.755919 0.048179 15.690 < 2e-16 ***
## origin 1.035733 0.268962 3.851 0.000138 ***
## horsepower:acceleration -0.012139 0.001772 -6.851 2.93e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.145 on 383 degrees of freedom
## Multiple R-squared: 0.841, Adjusted R-squared: 0.8376
## F-statistic: 253.2 on 8 and 383 DF, p-value: < 2.2e-16There appear to be many significant interactions. Year and weight, year and horsepower, acceleration and horsepower all have significant interaction effects. Year and origin did not have an interaction effect.
Problem 9f
lm_auto7 <- lm(mpg~.-name+log(horsepower), data=Auto)
summary(lm_auto7)##
## Call:
## lm(formula = mpg ~ . - name + log(horsepower), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5777 -1.6623 -0.1213 1.4913 12.0230
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.674e+01 1.106e+01 7.839 4.54e-14 ***
## cylinders -5.530e-02 2.907e-01 -0.190 0.849230
## displacement -4.607e-03 7.108e-03 -0.648 0.517291
## horsepower 1.764e-01 2.269e-02 7.775 7.05e-14 ***
## weight -3.366e-03 6.561e-04 -5.130 4.62e-07 ***
## acceleration -3.277e-01 9.670e-02 -3.388 0.000776 ***
## year 7.421e-01 4.534e-02 16.368 < 2e-16 ***
## origin 8.976e-01 2.528e-01 3.551 0.000432 ***
## log(horsepower) -2.685e+01 2.652e+00 -10.127 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.959 on 383 degrees of freedom
## Multiple R-squared: 0.8592, Adjusted R-squared: 0.8562
## F-statistic: 292.1 on 8 and 383 DF, p-value: < 2.2e-16The residual standard error of all previous attempts was over 3. Adding the log of horsepower reduced the RSE to just under 3, a slight improvement of fit.
lm_auto8 <- lm(mpg~.-name+sqrt(horsepower), data=Auto)
summary(lm_auto8)##
## Call:
## lm(formula = mpg ~ . - name + sqrt(horsepower), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5402 -1.6717 -0.0778 1.4861 11.9754
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.299e+01 7.251e+00 5.929 6.82e-09 ***
## cylinders 6.037e-02 2.928e-01 0.206 0.836748
## displacement -5.870e-03 7.156e-03 -0.820 0.412560
## horsepower 4.239e-01 4.532e-02 9.353 < 2e-16 ***
## weight -3.285e-03 6.604e-04 -4.975 9.87e-07 ***
## acceleration -3.342e-01 9.705e-02 -3.443 0.000638 ***
## year 7.398e-01 4.536e-02 16.308 < 2e-16 ***
## origin 9.159e-01 2.526e-01 3.626 0.000326 ***
## sqrt(horsepower) -1.050e+01 1.039e+00 -10.104 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.961 on 383 degrees of freedom
## Multiple R-squared: 0.8591, Adjusted R-squared: 0.8561
## F-statistic: 291.8 on 8 and 383 DF, p-value: < 2.2e-16Adding the square root of horsepower did not fit as well as log horsepower, but did improve the fit from the original attempts
lm_auto9 <- lm(mpg~.-name+I(horsepower^2), data=Auto)
summary(lm_auto9)##
## Call:
## lm(formula = mpg ~ . - name + I(horsepower^2), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5497 -1.7311 -0.2236 1.5877 11.9955
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.3236564 4.6247696 0.286 0.774872
## cylinders 0.3489063 0.3048310 1.145 0.253094
## displacement -0.0075649 0.0073733 -1.026 0.305550
## horsepower -0.3194633 0.0343447 -9.302 < 2e-16 ***
## weight -0.0032712 0.0006787 -4.820 2.07e-06 ***
## acceleration -0.3305981 0.0991849 -3.333 0.000942 ***
## year 0.7353414 0.0459918 15.989 < 2e-16 ***
## origin 1.0144130 0.2545545 3.985 8.08e-05 ***
## I(horsepower^2) 0.0010060 0.0001065 9.449 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.001 on 383 degrees of freedom
## Multiple R-squared: 0.8552, Adjusted R-squared: 0.8522
## F-statistic: 282.8 on 8 and 383 DF, p-value: < 2.2e-16Adding the square of horsepower produced a model with a slightly lower fit as compared to the log and square root versions.
lm_auto10 <- lm(mpg~.-name+log(horsepower)+horsepower:year, data=Auto)
summary(lm_auto10)##
## Call:
## lm(formula = mpg ~ . - name + log(horsepower) + horsepower:year,
## data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2717 -1.5815 -0.1233 1.4498 11.4355
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.280e+01 1.890e+01 0.677 0.498708
## cylinders 1.715e-01 2.868e-01 0.598 0.550167
## displacement -6.998e-03 6.933e-03 -1.009 0.313408
## horsepower 6.132e-01 9.437e-02 6.498 2.54e-10 ***
## weight -3.464e-03 6.386e-04 -5.424 1.04e-07 ***
## acceleration -2.874e-01 9.446e-02 -3.043 0.002503 **
## year 1.381e+00 1.413e-01 9.773 < 2e-16 ***
## origin 8.867e-01 2.459e-01 3.605 0.000353 ***
## log(horsepower) -1.996e+01 2.959e+00 -6.745 5.69e-11 ***
## horsepower:year -6.727e-03 1.413e-03 -4.761 2.74e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.879 on 382 degrees of freedom
## Multiple R-squared: 0.8671, Adjusted R-squared: 0.8639
## F-statistic: 276.9 on 9 and 382 DF, p-value: < 2.2e-16Adding the log of horsepower and the interaction of horsepower and year produced the best model so far with a lower RSE of 2.879 and a higher R-squared of 0.8671
Problem 10a
carseats1 <- lm(Sales~Price+Urban+US, data = Carseats)
summary(carseats1)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16
Problem 10b
The Price coefficient was -0.054459. This suggests that, holding all other predictors constant, as price goes up by one unit, the predicted sales decrease by 0.054459 (54 car seats). The Urban Coefficient was -0.021916 for the yes dummy variable. This suggests that sales in urban areas would be predicted to be 0.021916 units less as compared to non-urban areas. However, the urban predictor was not statistically significant, so the coefficient effectively is 0 and there is no difference between urban and non-urban areas. The US coefficient is 1.200573. This suggests that, holding all other predictors constant, sales in the US would be predicted to be 1.200573 (1200 carseats) higher in the US as compared to not in the US.
Problem 10c
The model in equation form would be Sales = 13.04 - 0.054459 X Price - 0.021916 X Urban {yes:1, no:0} + 0.021916 X US {Yes:1, no:0}
Problem 10d
Out of the above model, the predictors in which the null hypothesis can be rejected are Price (p < 2e-16) and US (p = 4.86e-06)
Problem 10e
carseats2 <- lm(Sales~Price+US, data = Carseats)
summary(carseats2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16
Problem 10f
For model a, the RSE was 2.472 and the R-squared was 0.2393. Therefore, only 23.93% of the variation in the data can be explained by the model. In model e, the RSE went down slightly to 2.469, implying a slight improvement in fit. The R-squared was still 0.2393 - so still not much of the variation in the data is explained by the model. Including more variables would likely improve the fit.
Problem 10g
confint(carseats2)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632
Problem 10g
par(mfrow=c(2,2))
plot(carseats2)
There does appear to be one high-leverage point and a few named points that could possibly be outliers.
rstudent(carseats2)[which(rstudent(carseats2)>=3)]## named numeric(0)rstudent(carseats2)[order(rstudent(carseats2), decreasing = T)][1:3]## 377 69 26
## 2.891521 2.642364 2.599652The highest rstduent values are still under 3, meaning that they are likely not outliers.
hatvalues(carseats2)[which(hatvalues(carseats2)>=0.04)]## 43
## 0.04333766hatvalues(carseats2)[order(hatvalues(carseats2), decreasing = T)][1]## 43
## 0.04333766Observation 43 appears to be a high-leverage point.
Problem 12a
The inverse function would still have the same numerator. Therefore, the only way the Y~X and X~Y would be the same is if the denominators were the same. In order for the denominators to be the same, the sum of squares would have to be the same for both x and y.
Problem 12b
set.seed(118899)
x <- rnorm(100)
head(x)## [1] 0.8603984 -0.9067076 -0.9196529 -1.3193413 0.7636590 -1.0923197set.seed(998811)
y1 <- 3.4*x + rnorm(100,0,1)
head(y1)## [1] 2.0556230 -2.5141623 -3.8314505 -5.3120112 0.9522532 -3.4434890lmy1x <- lm(y1~x)
summary (lmy1x)##
## Call:
## lm(formula = y1 ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.60344 -0.67417 0.03659 0.62670 2.68131
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.10978 0.10242 1.072 0.286
## x 3.28433 0.09939 33.046 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.024 on 98 degrees of freedom
## Multiple R-squared: 0.9176, Adjusted R-squared: 0.9168
## F-statistic: 1092 on 1 and 98 DF, p-value: < 2.2e-16lmxy1<- lm(x~y1)
summary(lmxy1)##
## Call:
## lm(formula = x ~ y1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.98527 -0.17722 0.02145 0.16763 0.69975
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.031487 0.029878 -1.054 0.295
## y1 0.279402 0.008455 33.046 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2987 on 98 degrees of freedom
## Multiple R-squared: 0.9176, Adjusted R-squared: 0.9168
## F-statistic: 1092 on 1 and 98 DF, p-value: < 2.2e-16
Problem 12c
y2 <- sample(x,100)
head(y2)## [1] 0.01073568 -0.61294029 1.42286511 -1.10156519 -0.07048174 -1.07413785lmy2x <- lm(y2~x)
summary (lmy2x)##
## Call:
## lm(formula = y2 ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.86658 -0.60622 -0.07885 0.63862 2.70095
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0098744 0.1040965 -0.095 0.925
## x -0.0004983 0.1010152 -0.005 0.996
##
## Residual standard error: 1.041 on 98 degrees of freedom
## Multiple R-squared: 2.483e-07, Adjusted R-squared: -0.0102
## F-statistic: 2.434e-05 on 1 and 98 DF, p-value: 0.9961lmxy2 <- lm(x~y2)
summary (lmxy2)##
## Call:
## lm(formula = x ~ y2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.86634 -0.60607 -0.07908 0.63794 2.70023
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0098744 0.1040965 -0.095 0.925
## y2 -0.0004983 0.1010152 -0.005 0.996
##
## Residual standard error: 1.041 on 98 degrees of freedom
## Multiple R-squared: 2.483e-07, Adjusted R-squared: -0.0102
## F-statistic: 2.434e-05 on 1 and 98 DF, p-value: 0.9961