I’m studying for an exam and came across a multiple choice question that asked:
Which of the following about confidence intervals is true?
Apparently, the answer is: For a two sided test, with $H_{0}: \beta_{1} =0$, if $H_{0}$ is rejected at $\alpha=.05,$ then the $95$ percent confidence interval for $\beta_{1}$ would not contain $0.$
This makes sense to me.
However, another option in this question is: Given that $(5.1,9.8)$ is the $95$ percent confidence interval for $\beta_{1},$ the probability that $(5.1,9.8)$ will cover $\beta_{1}$ is $.95.$
Since there is only one answer, the latter is false, but I am having trouble understanding why. It looks perfectly correct to me.
Can anyone provide me some insight? Thanks in advance!