To construct a 95% confidence interval for the mean interest rate of the sample of lenders, we'll follow these steps:
- Calculate the sample mean (\( \bar{x} \)).
- Calculate the sample standard deviation (\( s \)).
- Determine the appropriate critical value from the t-distribution.
- Calculate the margin of error.
- Construct the confidence interval.
Step 1: Calculate the Sample Mean
First, we compute the mean (\( \bar{x} \)) of the interest rates.
\[ \bar{x} = \frac{\sum x_i}{n} = \frac{4.751 + 4.372 + 4.175 + 4.678 + 4.421 + 4.229 + 4.122 + 4.254 + 3.952 + 4.196}{10} \]
Calculating the sum:
\[ 4.751 + 4.372 + 4.175 + 4.678 + 4.421 + 4.229 + 4.122 + 4.254 + 3.952 + 4.196 = 43.176 \]
Now, divide by the number of observations \( n = 10 \):
\[ \bar{x} = \frac{43.176}{10} = 4.3176 \]
Step 2: Calculate the Sample Standard Deviation
Now we calculate the standard deviation (\( s \)) using the formula:
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
Calculating \( \sum (x_i - \bar{x})^2 \):
| \( x_i \) | \( x_i - \bar{x} \) | \( (x_i - \bar{x})^2 \) | |------------|---------------------|--------------------------| | 4.751 | 0.4334 | 0.1875 | | 4.372 | 0.0544 | 0.00296 | | 4.175 | -0.1426 | 0.0203 | | 4.678 | 0.3604 | 0.1295 | | 4.421 | 0.1034 | 0.0107 | | 4.229 | -0.0886 | 0.0078 | | 4.122 | -0.1956 | 0.0383 | | 4.254 | -0.0636 | 0.0040 | | 3.952 | -0.3656 | 0.1334 | | 4.196 | -0.1216 | 0.0148 |
Now sum the squared differences:
\[ \sum (x_i - \bar{x})^2 \approx 0.1875 + 0.00296 + 0.0203 + 0.1295 + 0.0107 + 0.0078 + 0.0383 + 0.0040 + 0.1334 + 0.0148 \approx 0.5151 \]
Then plug it into the standard deviation formula:
\[ s = \sqrt{\frac{0.5151}{10 - 1}} = \sqrt{\frac{0.5151}{9}} \approx \sqrt{0.0572} \approx 0.2392 \]
Step 3: Determine the t Critical Value
For a 95% confidence interval with \( n - 1 = 9 \) degrees of freedom, we find the t critical value. Using a t-table, the critical value for \( t_{0.025, 9} \approx 2.262 \).
Step 4: Calculate the Margin of Error
The margin of error (ME) is calculated as follows:
\[ ME = t_{critical} \times \frac{s}{\sqrt{n}} = 2.262 \times \frac{0.2392}{\sqrt{10}} \approx 2.262 \times 0.0756 \approx 0.1716 \]
Step 5: Construct the Confidence Interval
The confidence interval is given by:
\[ \bar{x} \pm ME \]
Calculating the lower and upper limits:
\[ \text{Lower limit} = 4.3176 - 0.1716 \approx 4.1460 \] \[ \text{Upper limit} = 4.3176 + 0.1716 \approx 4.4892 \]
Final Results
Thus, the 95% confidence interval for the mean interest rate is approximately:
\[ (4.1460, 4.4892) \]