https://file.aiquickdraw.com/mathbot/file/17297150446669pd8nbk9.jpg

Let's analyze the table and the provided statements to solve the question.

The table shows the classification of a portfolio of 200 stocks by two criteria: **Industry Sector** and **Investment Objective**. The total number of stocks in each category is summarized, and we are tasked with determining which of the given statements is true based on this data.

The table provides the following information:

| Investment Objective | Electronics | Airlines | Healthcare | Total |
|----------------------|-------------|----------|------------|-------|
| **Growth**            | 84          | 21       | 35         | 140   |
| **Income**            | 36          | 9        | 15         | 60    |
| **Total**             | 120         | 30       | 50         | 200   |

Now, let's evaluate each statement:

### a) Growth and Healthcare are complementary events.
- Complementary events mean that they together cover all possible outcomes (i.e., their union equals the total number of outcomes). Here, Growth and Healthcare together represent only \(35\) stocks from Healthcare in the Growth category, but there are other stocks in Growth and Healthcare that are not counted, so they are **not complementary**.

### b) Electronics and Growth are independent.
- To check independence, we can use the formula:
  \[
  P(\text{Electronics and Growth}) = P(\text{Electronics}) \times P(\text{Growth})
  \]
  Let's calculate these probabilities:
  - \( P(\text{Electronics}) = \frac{120}{200} = 0.6 \)
  - \( P(\text{Growth}) = \frac{140}{200} = 0.7 \)
  - \( P(\text{Electronics and Growth}) = \frac{84}{200} = 0.42 \)
  
  Now, check if \( 0.6 \times 0.7 = 0.42 \):
  \[
  0.6 \times 0.7 = 0.42
  \]
  Since the equality holds, **Electronics and Growth are independent**. This statement is **true**.

### c) Electronics and Growth are mutually exclusive.
- Mutually exclusive events cannot happen together. Since 84 stocks fall under both Electronics and Growth, they are not mutually exclusive. This statement is **false**.

### d) Airlines and Healthcare are collectively exhaustive.
- Collectively exhaustive means that their union covers all 200 stocks. However, Airlines and Healthcare together account for \(30 + 50 = 80\) stocks, which is not equal to the total of 200 stocks. Hence, this statement is **false**.

### e) Electronics and Healthcare are collectively exhaustive.
- Electronics and Healthcare account for \(120 + 50 = 170\) stocks, which is not the entire 200, so they are **not collectively exhaustive**. This statement is **false**.

### Conclusion:
The correct answer is **b) Electronics and Growth are independent**.

Would you like further explanation on any part? Here are five related questions to deepen your understanding:

1. How do you calculate if two events are independent in probability?
2. What does it mean for two events to be mutually exclusive?
3. How would you determine if events are collectively exhaustive in a set of data?
4. Can two events be independent and mutually exclusive simultaneously?
5. How would you apply this table analysis to other types of investment objectives?

**Tip:** Always remember to check the conditions of independence using probability formulas.

Meagan Debian manages a portfolio of 200 common stocks. Her staff classified the portfolio stocks by 'industry sector' and 'investment objective.' Based on the table data, determine which of the following statements is true:

(a) Growth and Healthcare are complementary events.
(b) Electronics and Growth are independent.
(c) Electronics and Growth are mutually exclusive.
(d) Airlines and Healthcare are collectively exhaustive.
(e) Electronics and Healthcare are collectively exhaustive.

Analyze a portfolio of 200 stocks to determine which of the given statements about investment objectives and industry sectors is true. Understand concepts like independence, mutual exclusivity, and complementary events in probability. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation