Q1: In each of the following scenarios, define the type of measurement scale. And state the reason for your choice of type? (2 marks)
a. A kindergarten teacher marks whether each student is a boy or a girl.
b. A ski resort records the daily temperature during the month of January.
c. A restaurant surveys its customers about the quality of its waiting staff on a scale of 1 to 4, where 1 is poor and 4 is exceallent.
d. An investor collects data on the weekly closing price of gold throughout the year.
e. An analyst assigns a sample of bond issues to one of the following credit ratings, given in descending order of credit quality (increasing probability of default): AAA, AA, BBB, BB, CC, D.
f. The dean of the business school at a local university categorizes students by major (i.e., accounting, finance, marketing, etc.) to help in determining class offerings in the future.
g. A meteorologist records the amount of monthly rainfall over the past year.
h. A sociologist notes the birth year of 50 individuals.
Q2: Using 3,000 observations, the following histogram summarises Variable X. (4 marks)
a. Is the distribution symmetric? If not, is it positively or negatively skewed?
b. What proportion of the observations are less than or equal to 20?
c. How many of the observations are greater than 20?
Q3: The following table shows the annual returns (in %) for Mutual Fund 1 and
Mutual Fund 2 over the past 37 years. (4 marks)
|
Year |
MF1 |
MF2 |
|
1 |
56.32 |
-12.16 |
|
2 |
52.47 |
20.27 |
|
3 |
-16.89 |
2.37 |
|
4 |
7.54 |
18.00 |
|
5 |
-7.47 |
5.49 |
|
6 |
-11.78 |
-1.80 |
|
7 |
-2.70 |
15.94 |
|
8 |
16.99 |
42.83 |
|
9 |
10.50 |
-4.49 |
|
10 |
58.97 |
0.04 |
|
11 |
8.72 |
-2.39 |
|
12 |
28.65 |
19.15 |
|
13 |
11.13 |
0.41 |
|
14 |
43.81 |
21.38 |
|
15 |
15.82 |
32.47 |
|
16 |
10.33 |
10.28 |
|
17 |
74.16 |
-14.74 |
|
18 |
131.75 |
34.23 |
|
19 |
-32.30 |
31.77 |
|
20 |
-31.70 |
-11.97 |
|
21 |
-37.79 |
-11.48 |
|
22 |
59.39 |
22.87 |
|
23 |
0.43 |
31.70 |
|
24 |
4.92 |
52.02 |
|
25 |
7.51 |
14.25 |
|
26 |
19.78 |
45.53 |
|
27 |
-51.09 |
-54.00 |
|
28 |
90.29 |
47.09 |
|
29 |
26.69 |
18.99 |
|
30 |
-9.56 |
-4.85 |
|
31 |
17.16 |
4.64 |
|
32 |
31.76 |
24.21 |
|
33 |
10.65 |
-12.64 |
|
34 |
7.40 |
-20.53 |
|
35 |
11.94 |
33.84 |
|
36 |
49.86 |
-2.64 |
|
37 |
-8.79 |
-24.92 |
a. Construct a boxplot for Mutual Fund 1. Does the boxplot suggest that outliers exist?
b. Use z-scores to determine if there are any outliers for Mutual Fund 1. Are your results consistent with part a? Explain why or why not.
c. Construct a boxplot for Mutual Fund 2. Does the boxplot suggest that outliers exist?
d. Use z-scores to determine if there are any outliers for Mutual Fund 2. Are your results consistent with part c? Explain why or why not.
Q4: More and more households are struggling to pay utility bills given high heating costs. Particularly hard hit are households with homes heated with propane or heating oil. Many of these households are spending twice as much to stay warm this winter compared to those who heat with natural gas or electricity. A representative sample of 500 households was taken to investigate if the type of heating influences whether a household is delinquent in paying its utility bill. The following table reports the results. (5 marks)
a. What Is the probability that a randomly selected household uses heating oil?
b. What is the probability that a randomly selected household is delinquent in paying its utility bill?
c. What is the probability that a randomly selected household uses heating oil and is delinquent in paying its utility bill?
d. Given that a household uses heating oil, what is the probability that it is delinquent in paying its utility bill?
e. Given that a household is delinquent in paying its utility bill, what is the probability that the household uses electricity?
f. Are the events “Heating Oil” and “Delinquent in Payment” independent?
Explain using probabilities.