The number of customers entering a certain restaurant in any

The number of customers entering a certain restaurant in any given day is approximately normally distributed with a mean of 40 and standard deviation of 10.

find the probability that during a given day between 35 and 45 customers arrive,

find the probability that the average number of customers, averaged over 7 days, is between 38 and 42.

Solution

So the probability that during a given day between 35 and 45 customers arrive is

P(35<X<45) = P((35-40)/10 <(X-mean)/s <(45-40)/10)

=P(-0.5<Z<0.5) = 0.3829 (from standard normal table)

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So the probability that the average number of customers, averaged over 7 days, is between 38 and 42 is

P(38<xbar<42) = P((38-40)/(10/sqrt(7)) <(xbar-mean)/(s/vn) <(42-40)/(10/sqrt(7)))

=P(-0.53<Z<0.53)

=0.4039 (from standard normal table)