Question 4.5: JSE Companies – Sector and Size Study (a) What is the probab......
JSE Companies – Sector and Size Study
(a) What is the probability that a randomly selected JSE-listed company is either a large company or a financial company, or both?
(b) What is the probability that a randomly selected JSE-listed company is either a mining company or a service company?
(c) What is the probability of selecting a small retail company from the JSE-listed sample of companies?
(d) Is company size statistically independent of sector in the JSE-listed sample of companies?
Table 4.5 Cross-tabulation table – JSE companies by sector and size
| Sector | Company size | Row total | ||
| Small | Medium | Large | ||
| Mining | 3 | 8 | 30 | 41 |
| Financial | 9 | 21 | 42 | 72 |
| Service | 10 | 6 | 8 | 24 |
| Retail | 14 | 13 | 6 | 33 |
| Column total | 36 | 48 | 86 | 170 |
(a) Let A = event (large company).
Let B = event (financial company).
Events A and B are not mutually exclusive as they can occur simultaneously (i.e. a company can be both large and financial).
From Table 4.5, the following marginal and joint probabilities can be derived:
P(A) = P(large) =\frac{86}{170} = 0.5059
P(B) = P(financial) = \frac{72}{170} = 0.4235
P(A ∩ B) = P(large and financial) = \frac{42}{170} = 0.2471
Then P(A ∪ B) = P (either large or financial or both)
= P(large) + P( financial) − P(large and financial)
= 0.5059 + 0.4235 – 0.2471 = 0.682 (68.2%)
There is a 68.2% chance that a randomly selected JSE-listed company will be either a large company or a financial company, or both (i.e. a large financial company).
(b) Let A = event (mining company).
Let B = event (service company).
Events A and B are mutually exclusive as they cannot occur simultaneously (a company cannot be both a mining company and a service company), so P(A ∩ B) = 0. From Table 4.5, the following marginal probabilities can be derived:
P(A) = P(mining) = \frac{41}{170} = 0.241 (24.1%)
P(B) = P(service) = \frac{24}{170} = 0.141 (14.1%)
P(A ∩ B) = 0
Then P(A ∪ B) = P(either mining or service)
= P(mining) + P(service) = 0.241 + 0.141 = 0.382 (38.2%)
There is a 38.2% chance that a randomly selected JSE-listed company will be either a mining company or a service company, but not both.
(c) Let A = event (small company).
Let B = event (retail company).
Intuitively, from Table 4.5, P(A ∩ B) = P(small and retail) = \frac{14}{170} = 0.082 (8.2%). Alternatively, this probability can be calculated from the multiplication rule Formula 4.5.
P(B) = P(retail) = \frac{33}{170} = 0.1942
P(A|B) = P(small|retail) = \frac{14}{170} = 0.4242
Then P(A ∩ B) = P(A|B) × P(B) = P(small|retail) × P(retail) = 0.4242 × 0.1942 = 0.082 (8.2%)
There is only an 8.2% chance that a randomly selected JSE-listed company will be a small retail company.
(d) To test for statistical independence, select one outcome from each event A and B and apply the decision rule in Formula 4.7 above.
Let A = event (medium-sized company).
Let B = event (mining company).
Then P(A) = P(medium-sized company) = \frac{48}{170} = 0.2824 (28.24%)
P(A|B) = P(medium-sized company|mining) = \frac{8}{41} = 0.1951 (19.51%)
Since the two probabilities are not equal (i.e. P(A|B) ≠ P(A)), the empirical evidence indicates that company size and sector are statistically dependent (i.e. they are related).