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 109 and a standard deviation of 53, 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. Give your answer to 3 decimal places.
Accepted Solution
A:
Answer: 212.88Step-by-step explanation:Given : The probability that a daily average over a given month is greater than x = [tex]2.5\%=0.025[/tex]The probability that corresponds to 0.025 from a Normal distribution table is 1.96. Mean : [tex]\mu = 109[/tex]Standard deviation : [tex]\sigma = 53[/tex]The formula for z-score : -[tex]z=\dfrac{x-\mu}{\sigma}[/tex][tex]\Rightarrow\ 1.96=\dfrac{x-109}{53}\\\\\Rightarrow\ x=53\times1.96+109\\\\\Rightarrow\ x=212.88[/tex]