A certain brokerage house wants to estimate the mean daily return on a certain stock. A random sample of 15 days yields the following return percentages.

-2.57, -0.74, 2.38, -0.76, -2.79, 2.05, -1.11, 1.83, -1.6, 2.31, 0.44, 2.56, 1.81, 0.91, -1.5

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If we assume that the returns are normally distributed, find a 90% confidence interval for the mean daily return on this stock. Give the lower limit and upper limit of the 90% confidence interval.

Carry your intermediate computations to at least three decimal places, Round your answers to one decimal place. (If necessary, consult a list of formulas.)

Lower limit:

Upper limit:

To find the 90% confidence interval for the mean daily return on the stock, we can proceed with the following steps:

### Step 1: Calculate the Sample Mean (\(\bar{x}\))
First, let's calculate the mean of the sample data.

Given dataTo find the 90% confidence interval for the mean daily return on the stock, we can proceed with the following steps:

### Step 1: Calculate the Sample Mean (\(\bar{x}\))
First, let's calculate the mean of the sample data.

Given 
\[
-2.57, -0.74, 2.38, -0.76, -2.79, 2.05, -1.11, 1.83, -1.6, 2.31, 0.44, 2.56, 1.81, 0.91, -1.5
\]

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where:
- \( \sum x_i \) is the sum of all the data points.
- \( n = 15 \) is the number of data points.

### Step 2: Calculate the Sample Standard Deviation (s)
The formula for the sample standard deviation \(s\) is:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]

### Step 3: Find the Critical Value (t)
Since the sample size is small (\(n = 15\)) and the population standard deviation is unknown, we will use the t-distribution. For a 90% confidence interval with \(n - 1 = 14\) degrees of freedom, the critical value \(t_{\alpha/2}\) can be found using a t-table or calculator.

### Step 4: Calculate the Margin of Error (ME)
The margin of error is calculated using:

\[
ME = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

### Step 5: Calculate the Confidence Interval
Finally, calculate the confidence interval using:

\[
\text{Lower limit} = \bar{x} - ME
\]
\[
\text{Upper limit} = \bar{x} + ME
\]

Let me compute these steps for you.The 90% confidence interval for the mean daily return on the stock is:

- **Lower limit:** \(-0.6\%\)
- **Upper limit:** \(1.1\%\)

Would you like more details or have any other questions? Here are some related questions you might find useful:

1. How does the confidence level affect the confidence interval?
2. Why is the t-distribution used instead of the Z-distribution for small samples?
3. How can you interpret this confidence interval in the context of stock returns?
4. What assumptions must hold for this confidence interval to be valid?
5. How would the confidence interval change with a larger sample size?

**Tip:** When interpreting confidence intervals, remember that a 90% confidence interval means we expect 90 out of 100 such intervals to contain the true population mean.

Learn how to calculate the 90% confidence interval for the mean daily return of a stock using sample data. Step-by-step solution includes calculating sample mean, sample standard deviation, critical values from the t-distribution, margin of error, and interpreting the results. Suitable for college-level students and professionals.

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