To calculate the 95 percent confidence interval for the average savings account balance, we can use the following formula for confidence intervals when dealing with a sample mean:
\[ \text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \]
Where:
- \(\bar{x}\) = sample mean
- \(z\) = z-value corresponding to the desired confidence level (for 95%, \(z \approx 1.96\))
- \(s\) = sample standard deviation
- \(n\) = sample size
Given:
- \(\bar{x} = 18412\)
- \(s = 993\)
- \(n = 187\)
First, we calculate the standard error (SE):
\[ SE = \frac{s}{\sqrt{n}} = \frac{993}{\sqrt{187}} \approx \frac{993}{13.645} \approx 72.8 \]
Next, we calculate the margin of error (ME):
\[ ME = z \cdot SE \approx 1.96 \cdot 72.8 \approx 142.81 \]
Now we can find the confidence interval:
\[ \text{Lower limit} = \bar{x} - ME \approx 18412 - 142.81 \approx 18269.19 \] \[ \text{Upper limit} = \bar{x} + ME \approx 18412 + 142.81 \approx 18554.81 \]
Thus, the 95 percent confidence interval is:
\[ (18269.19, 18554.81) \]
Rounding to two decimal places, the final confidence interval is approximately:
\[ (18269.67, 18554.33) \]
So the correct response is:
$18,269.67–$18,554.33