You observe a portfolio for five years and determine that its average return is 

12.0112.01​% 

and the standard deviation of its returns is 

19.2819.28​%. 

Would a​ 30% loss next year be outside the​ 95% confidence interval for this​ portfolio?

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Part 1

The low end of the​ 95% prediction interval is 

enter your response here​%. 

 ​(Round to two decimal​ places.)

Part 2

Would a​ 30% loss next year be outside the​ 95% confidence interval for this​ portfolio?  ​(Select the best choice​ below.)

A.

​Yes, you can be confident that the portfolio will not lose more than​ 30% of its value next year. This is because the low end of the prediction interval is less than 

minus−​30%.

B.

​Yes, you can be confident that the portfolio will not lose more than​ 30% of its value next year. This is because the low end of the prediction interval is greater than 

minus−​30%.

C.

​No, you cannot be confident that the portfolio will not lose more than​ 30% of its value next year. This is because the low end of the prediction interval is greater than 

minus−​30%.

D.

​No, you cannot be confident that the portfolio will not lose more than​ 30% of its value next year. This is because the low end of the prediction interval is less than 

minus−​30%.

To answer this question, we need to calculate the 95% confidence interval based on the given information.

### Step 1: Understanding the confidence interval
The 95% confidence interval is given by:
\[
\text{Confidence Interval} = \text{Mean} \pm 1.96 \times \text{Standard Deviation}
\]
Where:
- The **mean return** of the portfolio is 12.01%.
- The **standard deviation** is 19.28%.

### Step 2: Calculate the lower and upper bounds of the 95% confidence interval
\[
\text{Lower Bound} = 12.01\% - 1.96 \times 19.28\%
\]
\[
\text{Upper Bound} = 12.01\% + 1.96 \times 19.28\%
\]

Let me compute that.The 95% confidence interval for the portfolio's returns is:

- **Lower bound:** -25.78%
- **Upper bound:** 49.80%

### Part 1: The lower end of the 95% prediction interval is **-25.78%** (rounded to two decimal places).

### Part 2: Would a 30% loss next year be outside the 95% confidence interval?

A 30% loss means a return of -30%. Since the lower bound of the interval is -25.78%, a -30% return is **lower** than the lower bound. Therefore, it would fall **outside** the 95% confidence interval.

Thus, the correct answer is:

**D.** No, you cannot be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is less than -30%.

---

Would you like more details or further clarification?

Here are five related questions you may find useful:

1. How is a confidence interval constructed in statistics?
2. What does the 95% confidence interval mean in practical terms?
3. How do changes in the standard deviation affect the confidence interval?
4. What is the significance of a 1.96 Z-score in confidence intervals?
5. How can a portfolio's risk be mitigated if the confidence interval suggests potential large losses?

**Tip:** A wide confidence interval suggests high uncertainty in predictions, so diversifying your portfolio might help reduce the risk.

You observe a portfolio for five years and determine that its average return is 12.01% and the standard deviation of its returns is 19.28%. Would a 30% loss next year be outside the 95% confidence interval for this portfolio?

Part 1: The low end of the 95% prediction interval is ______%. (Round to two decimal places.)

Part 2: Would a 30% loss next year be outside the 95% confidence interval for this portfolio?

A. Yes, you can be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is less than -30%.

B. Yes, you can be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is greater than -30%.

C. No, you cannot be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is greater than -30%.

D. No, you cannot be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is less than -30%.

This problem calculates the 95% confidence interval for a portfolio with an average return of 12.01% and a standard deviation of 19.28%. It also examines whether a 30% loss next year falls outside this interval. Key statistical concepts include Z-scores and confidence intervals.

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