Question

The analysis of sales receipts at Jones Department Store shows the dollar value of sales are normally distributed with a mean of $200 and a standard deviation of $30.

A. Find the probability that the dollar value of sales will be between $180 and $205.

B. Find the probability that the dollar value of sales will be less than $170.

C. Jones Department Store wants to give a discount to shoppers who purchase a large dollar volume. What dollar volume should be established, if Jones only wants to give the discount to the largest 10% of dollar volume?

Answer 1

Given,

= 200 , = 30

We convert this to standard normal as

P(X < x) = P(Z < (x - ) / )  

a)

P(180 < X < 205) = P(X < 205) - P(X < 180)

= P(Z < ( 205 - 200) / 30) - P(Z < ( 180 - 200) / 30)

= P(Z < 0.17) - P(Z < -0.67)

= 0.5675 - 0.2514 (From Z table)

= 0.3161

b)

P(X < 170) = P(Z < (171 -200) / 30)

= P(Z < -1)

= 0.1587

c)

We have to calculate x such that P(X > x) = 0.10

That is

P(X < x) = 0.90

P(Z < ( x - ) / ) = 0.90

From Z table, z-score for the probability of 0.90 is 1.2816

(x -  ) / = 1.2816

( x - 200) / 30 = 1.2816

Solve for x

x = 238.45