Question

The net weekly sales for a small business follows a normal distribution with a mean of 588 dollars and standard deviation of 107 dollars.

Calculate each of the following.

In each case, round your response to at least 4 decimal places.

a) What is the probability that in a randomly selected day, the business will do no more than 674 dollars in net sales?

b) Suppose 16 days are randomly selected. What is the probability that the average net sales the business will do is at least 568 dollars?

c) What is the probability that for 16 randomly selected days, the business will have a mean net sales between 573 dollars and 643 dollars?

Answer 1

a)

X ~ N ( = 588 , = 107 )

We convert this to standard normal as

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

P ( ( X < 674 ) = P ( Z < 674 - 588 ) / 107 )

= P ( Z < 0.80 )

P ( X < 674 ) = 0.7881 (from Z table)

b)

Given,

= 588 , = 107

Using central limit theorem,

P( < x) = P ( Z < ( X - ) / ( / sqrt(n) ) )

So,

P( > 568) - P(Z > ( 568 - 588) / ( 107 / sqrt(16) ) )

= P(Z > -0.75)

= P(Z < 0.75)

= 0.7734 (From Z table)

c)

P( 573 < < 643) = P( < 643) - P( < 573)

= P(Z < ( 643 - 588) / ( 107 / sqrt(16) ) ) - P(Z > ( 573 - 588) / ( 107 / sqrt(16) ) )

= P(Z < 2.06) - P( Z < -0.56)  

= 0.9803 - 0.2877 (From Z table)

= 0.6926