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Let's go through the questions provided:

### Part a: Compute the expected value of the proposal.

1. **Problem Setup**: 
   - You are given a failure probability of **2%** (which implies that **98%** of the products do not fail).
   - The replacement cost is **$1700**.
   - The company charges **$29** for a 2-year warranty.

   To calculate the expected value, we define the following:

   - $ X $: The amount the company profits or loses per warranty.
     - If the product **does not fail**, the company profits **$29** (the full warranty price).
     - If the product **fails**, the company loses **$1700** for the replacement, but earns **$29** from the warranty, resulting in a net loss of **$1700 - 29 = -$1671**.

   - $ P(X) $: The probability associated with each outcome.
     - The probability that the product does not fail is **98%** or **0.98**.
     - The probability that the product fails is **2%** or **0.02**.

   **Expected Value Formula**:
   \[
   E(X) = P(\text{no failure}) \times \text{profit if no failure} + P(\text{failure}) \times \text{loss if failure}
   \]
   Plugging in the values:
   \[
   E(X) = (0.98 \times 29) + (0.02 \times (-1671))
   \]
   Let me calculate the expected value.

### Part b: Interpretation of the expected value.
Once we have the expected value, we will interpret it in terms of the company's average profit or loss for each warranty sold.

### Part c: Review of the proposal.
This part involves evaluating whether the company benefits from this proposal based on the computed expected value. If the expected value is positive, the company profits on average. If negative, the company loses money on average. I will also discuss how the new warranty price was chosen and how it impacts the company's financial outlook.

Let me compute the expected value for you.The expected value $ E(X) $ of the proposal is approximately **-5**. This means that, on average, the company would lose **$5** for every 2-year warranty sold under the current proposal.

### Part b: Interpretation of the Expected Value
The expected value of **-5** indicates that for every warranty sold at a price of **$29**, the company expects to lose **$5** on average. This is because the potential cost of replacing failed products outweighs the income generated from selling the warranties.

### Part c: Review of the Proposal
Given the computed expected value, the proposal would not benefit the company. Here's why:

- The expected value of **-5** means that, over time, the company will incur a net loss for each warranty sold.
- The company is charging **$29** for a 2-year warranty, but the potential replacement cost of **$1700** for the **2%** of products that fail leads to significant losses.
  
To make this proposal profitable, the company would either need to:
1. **Increase the warranty price** above $29, or
2. **Reduce the replacement cost** or the failure rate of the product.

In conclusion, without adjustments to the warranty price or failure management, this proposal would not be financially beneficial for the company.

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Do you have any questions, or would you like more details?

Here are 5 related questions to think about:
1. What is the warranty price that would make the expected value zero or positive?
2. How does the failure rate affect the warranty price and expected value?
3. Could the company offer a partial refund instead of full replacement to mitigate losses?
4. How does increasing the warranty price impact customer demand for warranties?
5. What strategies could the company adopt to lower the product failure rate?

**Tip**: When calculating expected value, remember that even small probabilities of large losses can significantly affect the result.

Suppose you work for a company that manufactures electronics. The development analysts estimate that 2% of their flagship product will fail within 2 years of the purchase date, with a replacement cost of $1700. A newly hired associate at the company proposes to charge $29 for a 2-year warranty. Compute the expected value of this proposal, interpret the expected value, and write a review of the proposal addressing its benefits to the company.

Analyze the expected value of charging $29 for a 2-year warranty on an electronic product with a 2% failure rate and a replacement cost of $1700. This example illustrates the application of probability and expected value calculations in business decision-making.

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