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# Probability of Mutual Funds, Generator Shafts and Computers

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Problem 1
Mutual funds have become an increasing popular investment alternative for smaller investors. To help investors decide on the particular fund to invest in, various publications regularly report the average annual rate of return achieved by each of more than 100 mutual funds over a 10 year period. Some publications also indicate each fund's level of risk by classifying the historical variability of each fund's rate of return as high, intermediate, or low.

If the annual (percentage) rates of return over a 10 year period for two leading mutual funds are as shown below, which fund would be your better choice for investment? How do you define better in this case? Describe your answer in detail. Perform a thorough statistical analysis to back up your statements.

Fund A -8.3 -6.2 20.9 -2.7 33.6 42.9 24.4 -5.2 -3.1 30.5
Fund B 12.1 0.8 6.4 12.2 27.8 15.3 18.2 10.7 1.3 11.4

Problem 2:
Following their production, industrial generator shafts are tested for static and dynamic balance, and the necessary weight is added to the pre-drilled holes in order to bring each shaft within balance specifications. From past experience, the amount of weight added to a shaft has been normally distributed, with an average of 35 grams and a standard deviation of 9 grams.
a. What is the probability that a randomly selected shaft will require between 35 and 40 grams of weight for proper balance?
b. What is the probability that a randomly selected shaft will require at least 50 grams of weight for proper balance?
c. What is the probability that a randomly selected shaft will require no more than 26 grams of weight for proper balance?
d. Management has just directed that the "best" 5% of the output be reserved for shipment to aerospace customers. Translating the "best 5%" into an amount of balancing weight, what weight cutoff should be used in deciding which generator shafts to reserve for aerospace customers?

Problem 3:

The life expectancy of computer terminals is normally distributed with a mean of 4.5 years and a standard deviation of 18 months. Suppose you select a sample of size 36 terminals. Use Excel to answer the following questions.
a. What is the probability that the sample mean life expectancy for the terminals will be more than 5 years?
b. What is the probability that the sample mean life expectancy for the terminals will be between 5.25 and 5.75 years?
c. If the manufacturer guarantees the terminals for 4 years (and will replace them if they malfunction), what percentage (of samples) of the terminals will need to be replaced?
d. The top 10% of the sample mean life expectancies (for samples of size 36) for the terminals will be for at least how many years?

Problem 4:

The manager of a local branch of Twentieth Bank and Trust determined that 40% of all depositors have multiple accounts at the bank. You select a random sample of 200 depositors. Use Excel to answer the following questions.
a. What is the probability that the sample proportion of depositors with multiple accounts at the bank is less than .30?
b. What is the probability that the sample proportion of depositors with multiple accounts at the bank is between .34 and .39?
c. What is the probability that the sample proportion of depositors with multiple accounts at the bank is greater than .51?
d. The probability is 90% that the sample percentage is contained within what symmetrical limits of the population percentage?