Question 1: An agent for a residential real estate company in a large city would like to be able to predict the weekly rental cost for apartments based on the size of the apartment as defined by the number of square metres in area. A sample of 25 apartments was selected from across the city, and the information gathered recorded in the following table:
|
Apartment
|
Weekly Rent
($)
|
Size (square
metres)
|
Apartment
|
Weekly Rent
($)
|
Size (square
metres)
|
|
1
|
219
|
79
|
14
|
415
|
127
|
|
2
|
369
|
135
|
15
|
323
|
109
|
|
3
|
277
|
101
|
16
|
335
|
114
|
|
4
|
346
|
114
|
17
|
254
|
116
|
|
5
|
219
|
67
|
18
|
392
|
117
|
|
6
|
392
|
138
|
19
|
277
|
107
|
|
7
|
381
|
106
|
20
|
265
|
83
|
|
8
|
216
|
67
|
21
|
369
|
126
|
|
9
|
202
|
65
|
22
|
381
|
97
|
|
10
|
265
|
89
|
23
|
277
|
70
|
|
11
|
323
|
102
|
24
|
185
|
93
|
|
12
|
381
|
119
|
25
|
404
|
111
|
|
13
|
531
|
184
|
|
|
|
(a) What are the dependent and independent variables in this problem? Explain.
(b) Use the least-squares method to find the regression coefficients (show workings). State the simple linear regression equation.
(c) Interpret the meaning of both regression coefficients in this problem.
(d) Predict the weekly rental cost for an apartment that has an area of 100 square metres.
(e) Would it be appropriate to use the model to predict the weekly rental for an apartment that has an area of 50 square metres? Explain.
(f) Your friends are considering signing a lease for an apartment in this city. They are trying to decide between two apartments, one with an area of 100 square metres for a weekly rent of $294 and the other with an area of 120 square metres for a weekly rent of
$329. What would you recommend to them? Why?
Question 2: Refer to the tabulated sample data in Question 1 relating to weekly rental cost for apartments based on the size of the apartment as defined by the number of square metres in area.
(a) Calculate the coefficient of linear correlation (show workings). What does this value indicate regarding the relationship between size of apartment and weekly rental cost?
(b) Showing all workings, compute the estimated standard error of regression (slope).
(c) At the 5% level of significance, test for evidence of a linear relationship between the size of the apartment and the weekly rent.
(d) Compare and comment on the results obtained in parts (a) and (c).
Question 3: In a certain jurisdiction, savings banks are allowed to sell a form of life insurance to their customers. The approval process consists of underwriting, which includes a review of the application, a medical information statement check, possible requests for additional medical information and medical examinations, and a policy compilation stage where the policy pages are produced and sent to the bank for delivery. The ability to deliver approved policies to customers in a timely manner is critical to the profitability of this service to the bank. During a period of 1 month, a random sample of 27 approved policies was selected and the total processing time in days was recorded with the following results.
|
73
|
19
|
16
|
64
|
28
|
28
|
31
|
90
|
60
|
|
56
|
31
|
56
|
22
|
18
|
39
|
48
|
17
|
17
|
|
17
|
91
|
92
|
63
|
50
|
51
|
69
|
16
|
17
|
(a) In the past, suppose that the mean processing time averaged 45 days. At the 5% level of significance, is there evidence that the mean processing time has changed from 45 days?
(b) What assumption about the population distribution must be made in part (a)?
(c) Do you think that the assumption made in part (b) has been seriously violated? Explain. (HINT: compute the 5-number summary for the data and analyse the results).
Question 4: The inspection division of a government department is interested in determining whether the proper amount of soft drink has been placed in 2-litre bottles at the local bottling plant of a large nationally known soft drink producing company. The bottling plant has informed the inspection division that the standard deviation of bottle fill for 2-litre bottles is 0.05 litre.
A random sample of 100 2-litre bottles obtained from this bottling plant indicates a sample mean bottle fill of 1.99 litres.
(a) At the 5% level of significance, use the critical value approach to test for evidence that the mean amount of soft drink in the bottles is different from 2.0 litres.
(b) Compute the p-value (probability value) and interpret its meaning.
(c) Set up a 95% confidence interval estimate of the population mean amount of soft drink in the bottles.
(d) Compare the results in parts (a) and (c). What conclusions do you reach from this comparison?
Question 5: During the first half of a recent calendar year the share market was quite volatile and many major share indexes declined. Assume that the returns for funds invested in shares during this time period are normally distributed with a mean of -10.0% (that is, a loss) and a standard deviation of 8.0%.
(a) Find the probability that a share fund lost 18% or more.
(b) Find the probability that a share fund gained in value.
(c) Find the probability that a share fund gained at least 10%.
(d) The return for 80% of share funds was greater than what value?
(e) The return for 90% of share funds was less than what value?
(f) 95% of share funds had returns between what two values symmetrically distributed around the mean?