The number of diners at a restaurant each day is recorded an

The number of diners at a restaurant each day is recorded and a daily average is calculated every month (assume 30 days in a month). The number of diners each day has a mean of 142 and a standard deviation of 58, but does not necessarily follow a normal distribution. The probability that a daily average over a given month is greater than x is 2.5%. Calculate x. You may find standard normal table useful. Give your answer to 3 decimal places.

Solution

By central limit theorem, the sampling distribution of the means will be approximately normally distirbuted.

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.975      
          
Then, using table or technology,          
          
z =    1.959963985      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    142      
z = the critical z score =    1.959963985      
s = standard deviation =    58      
n = sample size =    30      
Then          
          
x = critical value =    162.7546521   [ANSWER]