STAT20029 (T2,2020) Questions:
Week 5
Activity 5.1
Activity 5.1: For the following the probability distribution
x |
0 |
1 |
2 |
3 |
4 |
P(x) |
0.10 |
0.45 |
0.30 |
0.10 |
0.05 |
(a) Determine the mean and variance of X
(b) Sketch the probability distribution for X. How would you describe the shape?
(c) Find P(X ≤ 2)
Practice problem 5.1
PP5-1: Determine the mean, variance and standard deviation of the following discrete distribution. Interpret the mean value.
x |
p(x) |
1.00 |
0.268 |
2.00 |
0.314 |
3.00 |
0.256 |
4.00 |
0.167 |
5.00 |
0.124 |
Practice problem 5.4
PP5-4: Suppose 20% of the people in a city choose to travel to work by bus. If a random sample of six people is chosen, the number of bus commuters could range from 0 to 6. Shown here are the possible numbers of bus commuters in a sample of six people and the probability of each number of bus commuters occurring in the sample. Use the data to determine the mean number of bus commuters in a sample of six people in the city and also compute the standard deviation. Explain the results.
x |
P(x) |
0 |
0.270 |
1 |
0.346 |
2 |
0.259 |
3 |
0.104 |
4 |
0.017 |
5 |
0.004 |
6 |
0 |
Activity 5.2
Activity 5.2: The probability distribution of the number of children under 15 years in a West Australian indigenous family is given in the table below based on ABS census data. What is the expected number of children under 15 years in a West Australian indigenous family?
No. of children |
Probability |
0 |
0.011 |
1 |
0.323 |
2 |
0.297 |
3 |
0.189 |
4 |
0.105 |
5 |
0.068 |
6 |
0.007 |
Activity 5.3
Activity 5-3: An accountant works mainly with personal income tax cases. He knows from past experience that the probability of a customer being satisfied with the service he offers is 0.8. Given that he sees 8 clients today, determine the probability that:
(a) all 8 customers are satisfied
(b) at least 6 customers are satisfied.
(c) fewer than 5 customers are satisfied.
(d) What are the mean and standard deviation of the probability distribution?
Review Problem 5.5
RP5-5: Suppose that 20% of all share market investors are retired people. Suppose a random sample of 25 share market investors is taken.
(a) What is the probability that exactly seven are retired people?
(b) What is the probability that 10 or more are retired people?
(c) How many retired people would you expect to find in a random sample of 25 share market investors?
Practice problem 5.9
PP5-9: The 2 most popular majors in a business degree are marketing and international business, with 30% of students enrolled in marketing, and 18% enrolled in international business. Suppose 20 students are selected at random. (a) What is the probability that:
(i) at least half of them are studying a marketing major
Activity 5.4
Activity 5.4: According to the New Zealand Department of Labour approximately 4% of all workers in Auckland are unemployed. What is the probability of selecting two or fewer unemployed workers in a sample of 20?
Activity 5.5
Activity 5.5: The ratio of boys to girls at birth in China is 1.15:1. In the nursery of the Maternity Ward in a hospital in China, there are 20 newborns. What is the probability that exactly 7 (seven) of them are girls?
Activity 5.6
Activity 5.6: Research shows that 17% of the cars being driven in Melbourne CBD during office hours are looking for a street parking space. Suppose that you are driving in the city and you spot a street parking space vacant at a distance, and there are 5 cars between you and the parking space. What is the probability that no one before you will park and you get the parking space?
Activity 5.7
Activity 5.7: According to Cancer Council Australia, the survival rate of stage IV lung cancer patients beyond two years is 20 percent. If a hospital is treating 12 stage IV lung cancer patients, what is the probability that at least 2 of those patients will survive beyond two years? Assume survival rate is constant and independent from patient to patient.
Practice problem 5.15
PP5-15: Find the following values by using table A.2.
(a) P(x = 6|λ = 3.8)
(c) P(3 ≤ x ≤ 9|λ = 4.2)
Practice problem 5.18
PP5-18: A restaurant manager is interested in predicting customer load and has been advised that a statistical approach can be applied in this case. She begins the process by gathering data. Restaurant staff count customers every 5 minutes from 7.00 pm until 8.00 pm every Saturday night for three weeks. The data are shown below. Assume that customer arrivals have a Poisson distribution with mean λ for a 5-minute interval. After the data are gathered, the manager computes λ using the data from all three weeks as a single data set.
week 1 |
week 2 |
week 3 |
3 |
1 |
5 |
6 |
2 |
3 |
4 |
4 |
5 |
6 |
0 |
3 |
2 |
2 |
5 |
3 |
6 |
4 |
1 |
5 |
7 |
5 |
4 |
3 |
1 |
2 |
4 |
0 |
5 |
8 |
3 |
3 |
1 |
3 |
4 |
3 |
(a) What value of λ did she find?
(b) What is the probability that six or more customers arrive during any given 5-minute interval?
(c) What is the probability that during a 10-minute interval fewer than four customers arrive?
Review Problem 5.7
RP5-7: Suppose that for every lot of 100 computer chips a company produces, an average of 1.4 are defective. Another company buys many lots of these chips at a time, from which one lot is selected randomly and tested for defects. If the tested lot contains more than three defects, the buyer will reject all the lots sent in that batch. What is the probability that the buyer will accept the lots? Assume that the defects per lot are Poisson distributed.
Activity 5.6
Activity 5.6: John has found a date, but before inviting her to his house, he wants to cover his dining table with a sheet of glass. The number of air bubbles in a particular type of glass that John wants to buy follows a Poisson distribution with a mean rate of 0.006 per square metre. If John’s dining table is 2metre × 1metre, what is the probability of fewer than 2 air bubbles in the glass sheet on his dining table?
Activity 5.7
Activity 5.7: The number of arrivals at a greengrocer’s between 9:00am and 11:00am has a Poisson distribution with a mean of 10. Find the probability of there being exactly 3 arrivals between 10:00am and 11:00am.
Activity 5.8
Activity 5.8: The number of new patient arrivals late night in the emergency department of a hospital between 9:00pm and midnight has a Poisson distribution with a mean of 18. Find the probability of there being more than 3 new patient arrivals between 11:00pm and midnight.
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