Question

Carol works in 2 positions . For her daytime job at a brokerage, she is paid $15,000 per month, plus a monthly commission. Those commissions are normally distributed, with mean of $10,000 and standard deviation of $2,000. She also works at night as a restaurant hostess, for which her monthly income is normally distributed with mean of $1,000 and standard deviation of $300. Carol's income levels from these two sources are independent of each other.

For a given month, what is the probability that Carol's commission from the brokerage is less than $13,000? 0.9331

For a given month, what is the probability that Carol's commission from the brokerage is no more than $8,000? 0.1586

For a given month, what is the probability that Carol's commission from the brokerage is between $11,000 and $12,000? 0.1499

For a given month, what is the probability that Carol's commission from the brokerage is more than $9,500? 0.5987

The probability is 0.75 that Carol's commission from the brokerage is less than how much in a given month? (Round this answer to 2 decimals.) $

The probability is 0.95 that Carol's commission from the brokerage is at least how much in a given month? (Round this answer to 2 decimals.) $

Can you Check the math on the first 4, the last two, I just cant remember!

Answer 1

The probability is 0.75 that Carol's commission from the brokerage is less than how much in a given month? (Round this answer to 2 decimals.) $

Using z table, we get 0.75 probability corresponds to a zscore = 0.67449
$zscore = \frac{X-10000}{2000}\\ 0.67449 = \frac{X-10000}{2000}\\ X= (0.67449 \times 2000)+10000 = 11348.98$

The probability is 0.95 that Carol's commission from the brokerage is at least how much in a given month? (Round this answer to 2 decimals.) $
Since it at least 0.95, we need to used 1-0.95= 0.05

At least indicate is can be X or more than X amount, hence we need to find the z score corresponding to 0.05.

Using z table, we get 0.05 probability corresponds to a zscore = -1.64485
$zscore = \frac{X-10000}{2000}\\ -1.64485= \frac{X-10000}{2000}\\ X= (-1.64485 \times 2000)+10000 = 6710.293$