1. Calculate the 25th, 50th, and 75th percentile of the data:
0, 7, 12, 5, 33, 14, 8, 0, 9, 22.
2. The number of spots turning up when a 6-sided die is tossed is observed. Consider the following events:
A: The number observed is at most 2.
B: The number observed is an even number. C: The number 4 turns up.
Answer the following questions.
a. Define the sample space for this random experiment and assign probabilities to the simple events.
b. Find P(A)c.
c. Find P(A).
d. Are events A and C mutually exclusive?
e. Find P(A ∪ C).
f. Find P(A ∩ B).
g. Find P(A ∪ B).
h. Find P(C|B).
3. A stock market analyst feels that
- The probability that a certain mutual fund will receive increased contributions from investors is 0.6.
- The probability of receiving increased contributions from investors becomes 0.9 if the stock market goes up.
- There is a probability of 0.5 that the stock market rises. The events of interest are:
- A: The stock market rises
- B: The company receives increased contribution Calculate the following probabilities
a. The probability that both A and B will occur, P(A ∩ B) [sharp increase in earnings].
b. The probability that either A or B will occur, P(A ∪ B) [at least moderate increase in earnings].
4. Suppose we are interested in the condition of a machine that produces a particular item.
- From experience it is known that the machine is in good condition 90% of the time.
- When in good condition, the machine produces a defective item 1% of the time.
- When in bad condition, the machine produces a defective 10% of the time.
An item is selected at random from the current production run. What is the probability that the item was found to be defective? With this additional information, what is the probability that the machine is in good condition?
5. The number of cars a dealer is selling daily was recorded over the last 200 days. The data are summarised as follows:
| Daily sales |
Frequency |
| 0 |
10 |
| 1 |
30 |
| 2 |
70 |
| 3 |
50 |
| 4 |
40 |
|
200 |
a. Estimate the probability distribution.
b. State the probability of selling more than 2 cars a day.
c. Determine the expected value and standard deviation of X, the number of cars sold.
d. With the probability distribution of cars sold per week, assume a salesman earns a fixed weekly wage of $150 plus $200 commission for each car sold. What is his expected wage, and the variance of the wage, for the week?
6. The monthly sales at a computer store have a mean of $25,000 and a standard deviation of $4,000. Profits are 30% of the sales less fixed costs of $6,000. Find the mean and standard deviation of the monthly profit.