Velocity $= X$, distance to stop $= Y$
$\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$
$\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$
degrees of freedom $=48$
(a) is there a linear relationship?
(b) Test $H_0:\beta_0 = -15$ vs $H_1: \beta_0 \neq -15$ at $\alpha = 0.05$
My solution below.
My $T$ test value shows that $T=-2.5791$
p-value $= 2\cdot\mathrm{pt}(-2.5791, 48)= 0.01302575$
since my p-value $<\alpha$ I reject $H_0$
Now my confidence interval for $\beta_0 = [-31.1635, -3.9947]$ and I feel i should not reject $H_0$
Does this mean that the answer for (a) is the relationship is not linear? or am I doing something incorrectly.
Thanks in Advance
If a $1-\alpha$ confidence interval for $\beta_0$ contains $-15$, then you would not reject that null hypothesis. Your t-statistic should be a fraction whose numerator is the least-squares estimate of $\beta_0$ (which you say is $-17.5791$) minus $-15$, thus $-17.5791-(-15) = 02.5791$. The denominator of the t-statistic should depend on the standard error. It appears that you neglect that part. One doesn't use the t-statistic unless there is a denominator that includes an estimate of a standard deviation.