Q1:
year rate cor(year,rate)
fit1 = lm(rate~year)
summary(fit1)
year2 = rep(year, 100)
rate2 = rep(rate, 100)
cor(year2,rate2)
fit2 = lm(rate2~year2)
summary(fit2)
> summary(fit1)
Call:
lm(formula = rate ~ year)
Residuals:
1 2 3 4 5
0.132 -0.003 -0.178 -0.163 0.212
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1419.20800 126.94957 11.18 0.00153 **
year -0.70500 0.06341 -11.12 0.00156 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2005 on 3 degrees of freedom
Multiple R-squared: 0.9763, Adjusted R-squared: 0.9684
F-statistic: 123.6 on 1 and 3 DF, p-value: 0.001559
> summary(fit2)
Call:
lm(formula = rate2 ~ year2)
Residuals:
Min 1Q Median 3Q Max
-0.178 -0.163 -0.003 0.132 0.212
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.419e+03 9.853e+00 144.0 <2e-16 ***
year2 -7.050e-01 4.922e-03 -143.2 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1556 on 498 degrees of freedom
Multiple R-squared: 0.9763, Adjusted R-squared: 0.9763
F-statistic: 2.052e+04 on 1 and 498 DF, p-value: < 2.2e-16
So after copying the data for 100 times, the estimates remain the same, but
the std.error drops dramatically. And t value is much large, t-test is much
more significant.