Question

Refer to the country in Example 4.11 on p. 91, where household incomes follow Normal distribution with µ = 900 coins and σ = 200 coins. (a) A recent economic reform made households with the income below 640 coins qualify for a free bottle of milk at every breakfast. What portion of the population qualifies for a free bottle of milk? (b) Moreover, households with an income within the lowest 5% of the population are entitled to a free sandwich. What income qualifies a household to receive free sandwiches?

Answer 1

for normal distribution,
mu = 900
sigma = 200

a)

x = 640
z = (x-mu)/sigma //converting to standard normal distribution
z = (640-900)/200 = -1.3
probability of z<-1.3 = 0.0968

this probability is calculated using Q function (gaussian cdf). This is found using tables or online calculators or definite integration

Probability = 0.0968

Percentage of population = 9.68%

---------------------

b)
lowest 5 percent = 0.05 probability
probability of z = 0.05
z = -1.65 (approx from table using interpolation)
z = (x-mu)/sigma
x = z*sigma+mu = 900-1.65*200 = 570
the income qualified for free sandwich is 570 coins (approx) or less

--------------------------

(checkout -1.3 and -1.65 in this table)