Question

Milk Demand

A supermarket has determined that daily demand for milk containers has an approximate bell shaped distribution, with a mean of 55 containers and a standard deviation of six containers.
What percentage of the time will the number of containers of milk sold within 2 standard deviations from the mean?

Answer 1

Solution:

Given, an approximate bell shaped distribution (Normal distribution) with,

   = 55

= 6

* Using Empirical Rule

According to the empirical rule , 68% of the data lie within 1 standard deviations from the mean ,

95% of the data lie within 2 standard deviations from the mean ,  99.7% of the data lie within 3 standard deviations from the mean.

So , answer is 95%

* Using Z transformation

P(within 2 standard deviations)

= P[ ( - 2) < X < ( + 2) ]

= P[ (55 - 2*6) < X < (55 + 2*6​​​​​​​)]

= P[ (55 - 12) < X < (55 + 12)]

= P[43 < X < 67]

= P(X < 67) - P(X < 43)

=  P[(X - )/ <  (67 - 55)/6] -   P[(X - )/ <  (43 - 55)/6]

= P[Z < 2.00] - P[Z < -2.00]

= 0.9772 - 0.0228..Use z table

= 0.9544

= 95.44% or nearly 95%