Question

John has two jobs. For daytime work at a jewelry store he is paid

$15,000 per month, plus a commission. His monthly commission is

normally distributed with mean $10,000 and standard deviation

$2000. At night he works occasionally as a waiter, for which his

monthly income is normally distributed with mean $1,000 and

standard deviation $300. John's income levels from these two

sources are independent of each other. For a given month, what is

the probability that John's commission from the jewelry store is

between $9,000 and $11,000?

Answer 1

Given Information:

John's mean monthly commission = μ = $10,000

Standard deviation of monthly commission = σ = $2,000

Answer:

`P(9,000 < X < 11,000) = 0.383\\\\P(9,000 < X < 11,000) = 38.3 \%`

The probability that John's commission from the jewelry store is between $9,000 and $11,000 is 38.3%

Explanation:

What is Normal Distribution?

We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.

We want to find out the probability that John's commission from the jewelry store is between $9,000 and $11,000?

`P(9,000 < X < 11,000) = P( (x - \mu)/(\sigma) < Z < (x - \mu)/(\sigma) )\\\\P(9,000 < X < 11,000) = P( (9,000 - 10,000)/(2,000) < Z < (11,000 - 10,000)/(2,000) )\\\\P(9,000 < X < 11,000) = P( (-1,000)/(2,000) < Z < (1,000)/(2,000) )\\\\P(9,000 < X < 11,000) = P( -0.5 < Z < 0.5 )\\\\P(9,000 < X < 11,000) = P( Z < 0.5 ) - P( Z < -0.5 ) \\\\`

The z-score corresponding to 0.50 is 0.6915

The z-score corresponding to -0.50 is 0.3085

`P(9,000 < X < 11,000) = 0.6915 - 0.3085 \\\\P(9,000 < X < 11,000) = 0.383\\\\P(9,000 < X < 11,000) = 38.3 \%`

Therefore, the probability that John's commission from the jewelry store is between $9,000 and $11,000 is 38.3%

How to use z-table?

Step 1:

In the z-table, find the two-digit number on the left side corresponding to your z-score. (e.g 1.4, 2.2, 0.5 etc.)

Step 2:

Then look up at the top of z-table to find the remaining decimal point in the range of 0.00 to 0.09. (e.g. if you are looking for 0.50 then go for 0.00 column)

Step 3:

Finally, find the corresponding probability from the z-table at the intersection of step 1 and step 2.