Confidence Intervals

What does it mean by 95% confidence interval?  This is something related to the Type I error and, therefore, the $\alpha$. If we set the $\alpha$ to 0.05, the confidence interval is (1-$\alpha$)*100 %. The 95% is the area under a distribution curve (0.95), between $z_{\alpha/2}$ and $1- z_{\alpha/2}$.

>> z2=norminv(0.025,0,1)
z2 =
   -1.9600
>> normspec([z2,-z2],0,1)
ans =
    0.9500

The shade area is computed from
 \[P({z^{\alpha /2}} < Z < {z^{1-\alpha /2}}) = 1-\alpha \]
where
 \[Z = \frac{{\bar X - \mu }}{{\sigma /\sqrt n }}\]
therefore
 \[P({z^{\alpha /2}} < \frac{{\bar X-\mu }}{{\sigma /\sqrt n }} < {z^{1-\alpha /2}}) = 1-\alpha \]
and some rearrangements
\[P(\bar X - {z^{1 - \alpha /2}}\frac{\sigma }{{\sqrt n }} < \mu  < \bar X - {z^{\alpha /2}}\frac{\sigma }{{\sqrt n }}) = 1-\alpha \]
Thus the lower limit is \[\bar X - {z^{1 - \alpha /2}}\frac{\sigma }{{\sqrt n }}\] and the upper limit is \[\bar X - {z^{1 - \alpha /2}}\frac{\sigma }{{\sqrt n }}\]
For the travel time problem ($\bar X=47.2$,$\sigma /\sqrt n=1.5$), if we set the $\alpha$ to be 0.05, the confidence interval is:

 >> interval = 47.2 - [-z2 z2]*1.5
interval =
   44.2601   50.1399
 
That means we are 95% confidence that the real average travel time is between 44.26 to 50.14 minutes, which is actually correct since the compared average travel time is 45 minutes.