How do I test $H_0 : \beta_1\leq 2$ in R? How do I test $H_0 : \beta_1\leq 2$ in R?
The data is as follows:
x<-c(1,2,3,3,4,5,5)
y<-c(3,7,5,8,11,14,12)

 A: If the regression equation is the simplest one, e.g.
$$Y_i = \beta_0 + \beta_1x_i + \epsilon_i,$$
then to test $H_0:\beta_1\leq 2$ against $H_1:\beta_1>2$ you can do
x<-c(1,2,3,3,4,5,5)
y<-c(3,7,5,8,11,14,12)
n <- length(y)

summary(mod <- lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
       1        2        3        4        5        6        7 
 0.02128  1.57447 -2.87234  0.12766  0.68085  1.23404 -0.76596 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   0.5319     1.5881   0.335  0.75127   
x             2.4468     0.4454   5.494  0.00273 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.632 on 5 degrees of freedom
Multiple R-squared:  0.8579,    Adjusted R-squared:  0.8294 
F-statistic: 30.18 on 1 and 5 DF,  p-value: 0.002729



t_obs <- (2.4468-2)/0.4454 # observed t statistic
pt(t_obs, df=n-2, lower.tail = F) # p-value

A: The general idea of t-testing a regression coefficient $\beta_k$ is that the following follows a t-distribution.
$$
\dfrac{
\hat\beta_k-\beta_{k0}
}{
\widehat{SE}(\hat\beta_k)
}
$$
We typically want to test if the coefficient is nonzero, so we take $\beta_{k0}=0$, but nothing stops us from testing with $\beta_{k0}=2$. We then would do only a one-sided t-test for your exact null hypothesis, though a two-sided test could be performed for an alternative hypothesis of $\beta_k\ne2$.
In software, we get the observed coefficient and standard error from the regression summary. We then calculate the test statistic from the fraction above and calculate the one-side do-value as usual.
Here is an example of how you could implement this in R.
set.seed(2022)
N <- 51
k <- 1 # which variable to test
x <- rnorm(N)
y <- 2*x +rnorm(N)
L <- lm(y~x)
s <- summary(L)$coef
tstat <- (s[k+1,1] - 2)/s[k+1, 2]
dof <- summary(L)$df[2] # degrees of freedom
p.value <- 1 - pt(tstat, dof) 

