1. The lifetime of a certain brand of tire is normally distributed with a mean of 80,000 km. and standard deviation of 7,000 km. The tire carries a warranty for 68,000 km.
a) What is the probability that the tire you recently purchased will last at most 81,000 km.?
b) What percent of this brand of tire will fail before the warranty expires? % format{~.dd}
c) What should the mileage warranty be so that only 1% of the tires need to be replaced under warranty? km. format{ w}
2. A firm's marketing manager believes that total sales for next year can be represented by a normal distribution, with a mean of $2.4 million and a standard deviation of $350,000. The firm has fixed costs of $1.6 million.
a) What is the probability that the firm's sales will be less than $3.4 million?
b) What is the probability that the firm will have sufficient sales to cover fixed costs?
c) What is the probability that the firm's sales will be within $140,000 of the expected sales?
d) Determine the sales level that has only a 7% chance of being exceeded. $ million. format{~.dd}
3. The owner of a convenience store has copies of the local newspaper delivered early each morning. The demand for papers is normally distributed with a mean of 76 and a standard deviation of 17.
{Assume daily demand is continuous}
a) What is the probability that there will not be enough newspapers to meet demand if the owner orders 68 copies?
b) How many copies should be ordered so that the probability of "selling out" is at most 13% ? papers.
4. An investment broker reports that the annual returns on common stock and municipal bonds are both normally distributed. The stocks have a mean return of 11.4% with a standard deviation of 19.8% . On the other hand the bonds have a mean return of 6.2% with a standard deviation of 10.6% .
a) If you are a conservative investor and just don't like to lose money, which type of investment should you choose? StocksBonds
If you made this selection, what would be your probability of losing money?
b) If you are more ambitious an investor and would like to have the best chance of making more than 3%, which investment should you choose? StocksBonds
If you made this choice, what would be the probability of making more than 3% ?
5. In the movie Forest Gump, the public school required an IQ of at least 70 to be admitted.
a) If IQ test scores are normally distributed with a mean of 100 and a standard deviation of 17, what percent of children would qualify for admittance to the school? % format{~.dd}
b) If the public school wished to have 85% of all children qualify for admittance, what minimum IQ test score should be required for admittance? format{ ~.d}
6. At a certain university the cumulative grade point average (CGPA) of first year students usually averages 2.81 with a standard deviation 0.56 . It has been found that the marks are usually approximately normally distributed.
a) What is the probability that a student will have a CGPA that is between 1 and 3 ?
b) What percent of students will be on probation, i.e. their CGPA is less than 2 ? % format{~.dd}
c) Academic scholarships are awarded to the top 2% of first year students. What minimum CGPA is needed to receive a scholarship?