4 The stop light at the corner of Grand and 19th street stay

4) The stop light at the corner of Grand and 19^th street stays red for a random amount of time distributed something bizarre with mean 180 seconds and variance 361 seconds.

I swear that light still hates me. I have timed it every day for the past 31 days, and my average was 184 seconds. How often does that (or a longer average) happen?

What two times mark the most likely (middle) 95% of averages for 30 days?

Solution

A)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    184      
u = mean =    180      
n = sample size =    31      
s = standard deviation =    19      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.172160918      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.172160918   ) =    0.120566228

B)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    180          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    19          
n = sample size =    30          
              
Thus,              
              
Lower bound =    173.2010623          
Upper bound =    186.7989377          
              
Thus, the confidence interval is              
              
(   173.2010623   ,   186.7989377   ) [ANSWER]