Business Statistics Assignment
Question 1 -
a. If P(A ∩ B) = 0.08, P(A) = 0.2 and P(B) = 0.4, then
i. Calculate P(A| B).
ii. State whether the events A and B are independent or not.
iii. Justify the decision you made in ii.
b. A coffee shop manager randomly selects 40 of the patrons of his coffee shop. He surveys them about their coffee and tea drinking habits. Twenty of them drink coffee, fifteen of them drink tea and five drink both.
i. How many of them drink neither tea nor coffee?
If C is defined as the event that a patron drinks coffee and T as the event that a patron drinks tea,
ii. Calculate P(C).
iii. Calculate P(T|C) and write a sentence that describes this probability in terms of the question.
iv. Calculate P(T-) and write a sentence that describes this probability in terms of the question
c. One of the questions in a recent household survey asks respondents to count the total number of mobile phones owned by members of the household. The table which follows shows the probability distribution for the number of mobile phones per household.
|
X
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8 or more
|
|
p(x)
|
0.00
|
0.02
|
0.18
|
0.30
|
0.20
|
0.15
|
0.10
|
0.05
|
0.00
|
i. Discuss how we can confirm that this is a valid probability distribution.
ii. Calculate the mean number of mobile phones owned per household.
iii. Calculate the standard deviation number of mobile phones owned per household.
iv. What is the probability that for a randomly selected household, the number of phones in this household is at least 3?
Question 2 -
A manufacturing company uses two bottling machines to produce 2 litre bottles of apple juice. Machine A can bottle 200 bottles of juice per hour. Machine B can bottle 300 bottles per hour. Their quality control process has found that if a bottle of juice has been bottled by Machine A, then the probability that it has been under-filled is 4.5%. If a bottle of juice has been bottled by Machine B, however, then the probability that it has been under-filled is 6%.
a. Using letters of the alphabet, define the two simple events described in this problem and also their complements. (4 events altogether). Probability notation is not required to do this.
b. Draw a probability tree to represent the information given in the question using the same letters that you used to define the simple events in part a.
Note: Marks will be deducted if the probabilities do not appear along the branches of the probability tree.
c. Determine the probability that a randomly chosen bottle contained the right amount of juice.
d. Given that a randomly selected bottle was not under-filled, what is the probability that the bottling was done by Machine B?
Question 3 -
A manufacturer vacuum cleaner parts knows that for every 1000 units produced, 4 units are faulty. The manufacturer wants to estimate, using probabilities, the likelihood of faulty units in a selection of 500 randomly selected units.
a. Write down the names (using single letters or Greek symbols) of the two parameters of this binomial distribution and their values.
b. Use the appropriate probability formula to calculate the probability that there are no faulty components among these units.
c. Use Excel to calculate the probability that there are more than 10 faulty units among these units. Express your answer as a decimal correctly rounded to 6 decimal places.
Question 4 -
A shire council has found that historically 20 % of ratepayers choose to pay their rates by instalments. A sample of 25 new ratepayers to the shire were asked if they had paid their current rates by instalment.
a. Write down the name of the distribution being described in this problem and write down the name/s of its parameter/s (using single letters or Greek symbols) and its/their values.
b. Using the tables appropriate for this distribution, calculate the probability that fewer than a quarter of those new ratepayers sampled had paid their rates by instalment.
c. Using Excel, calculate the probability that more than 5 but fewer than 12 had paid their rates by instalment.
Question 5 -
The manager of a group of suburban service stations has noticed a high level of drive-offs at the busiest of these service stations. Using CCTV footage, the manager estimates that the number of drive-offs on a Saturday night is 4 on average. (A drive-off involves a driver driving off without paying for petrol).
a. Write down the name of the probability distribution being described and write down its parameter/s (using letters or Greek symbols) and its/their value/s.
b. What is the probability that there is no more than 1 drive-off on a given Saturday?
c. Use Excel to evaluate the probability that there are more than a total of 20 drive-offs on all Saturdays in the month of December 2016.
Question 6 -
A company pays the staff the award rate. Some of its employees work over-time. For employees employed under a particular award, the employees receive on average $1025 per week with a standard deviation of $250 per week.
a. Write down the names of the parameters of this normal distribution (using Greek symbols) and their values.
b. A randomly chosen employee working under this award is selected. Calculate the probability that that he earned less than $1400 the previous week.
c. Twenty staff employed under this same award are randomly selected. What is the probability that the average weekly pay the previous week for this sample of twenty staff exceeded $1150?
d. A randomly chosen employee working under this award is selected. There is an 80% chance that this employee earns more than $a. What weekly income is represented by the letter 'a'?