1. "MagTek" electronics has developed a smart phone that does things that no other phone yet released into the market-place will do. The marketing department is planning to demonstrate this new phone to a group of potential customers, but is worried about some initial technical problems which have resulted in 4 percent of all the phone malfunctioning. The marketing executive is planning on randomly selecting 50 phones for use in the demonstration but is worried because it is very important that every single one functions OK during the demonstration. The executive believes that whether or not any one phone malfunctions is independent of whether or not any other phone malfunctions and is convinced that the probability that any one phone will malfunction is definitely 0.04. Assuming the marketing executive randomly selects 50 phones for use in the demonstration:
(a) What is the probability that no phone will malfunction? [If you use any probability distribution/s, you are required justify the requirements for particular distributions are satisfied]
(b) What is the probability that at most two phones will malfunction?
(c) The executive has decided that unless the probability of there being no malfunctions is greater than 90%, he will cancel the demonstration. Should he cancel the demonstration or not? Explain your answer.
2. A market researcher selects 20 students at random to participate in a wine-tasting test. Each student is blindfolded and asked to take a drink out of each of two glasses, one containing an expensive wine and the other containing a cheap wine. The students are then asked to identify the more expensive wine. If the students have no ability whatsoever to discern the more expensive wine, what is the probability that the more expensive wine will be correctly identified by:
a. none of the students?
b. more than half of the students?
c. minimum of 15 students?
d. eight of the students?
3. Phone calls arrive at the rate of 60 per hour at the reservation desk for a hotel.
(a) State the appropriate probability distribution suitable to calculate probabilities in parts (c) and (d).
(b) What are the properties/assumptions of the probability distribution mentioned in part (a).
(c) Find the probability of receiving at most two calls in a five-minute interval of time.
(d) Find the probability of receiving exactly eight calls in 15 minutes.
4. An advertising executive receives an average of 10 telephone calls each afternoon between 2 and 4pm. The calls occur randomly and independently of one another.
a. Find the probability that the executive will receive 13 calls between 2 and 4pm on a particular afternoon.
b. Find the probability that the executive will receive seven calls between 2 and 3pm on a particular afternoon.
c. Find the probability that the executive will receive at least five calls between 2 and 4pm on a particular afternoon.
5. During a annual heating season, the average gas bill for customers in a suburb heating their homes with gas was $457. Assuming a normal distribution and a standard deviation of $80,
(a) What proportion of homes heating with gas had a gas bill over $457?
(b) What proportion of homes heating with gas had a gas bill over $382?
(c) What proportion of homes heating with gas had a gas bill between $497 and $537?
(d) What amount was exceeded by only 2.5% of the homes heating with gas?
(e) What amount was exceeded by 95% of the homes heating with gas?
6. The Victorian Department of Transport estimates that the cost of running a small car is $150 per week, based on driving 15,000 km per year. Assuming that the cost is normally distributed with a standard deviation of $10 per week, answer the following questions.
a. What proportion of small cars cost more than $170 per week to run?
b. What proportion of small cars cost between $120 and $180 per week to run?
c. You want to be 95% certain that the weekly cost of running your small car will not exceed your budgeted amount. How much should you budget?