- Calculate the standard error: \(SE = \frac{5.71}{\sqrt{50}} \approx 0.81\).
- Find the critical Z-score for \(98\%\) confidence: \(z_{0.01} \approx 2.33\).
- Determine the margin of error: \(ME = 2.33 \times 0.81 \approx 1.88\).
- Calculate the confidence interval: \(\bar{x} \pm ME = 18.84 \pm 1.88\), yielding \(\boxed{\$16.96} < \mu < \boxed{\$20.72}\).
Explanation
- Understand the problem and provided data
We have a sample of \(n = 50\) cars with a sample mean bill of \(\bar{x} = \$18.84\). The population standard deviation is known to be \(\sigma = \$5.71\). Our objective is to estimate the population mean bill \(\mu\) using a \(98\%\) confidence interval. Since the population standard deviation is known, we use the Z-distribution to calculate the confidence interval.
- Determine the critical Z-score
The confidence level is \(98\%\), so the significance level is \(\alpha = 1 - 0.98 = 0.02\). We need to find the critical Z-score that corresponds to the area of \(1 - \frac{\alpha}{2} = 1 - 0.01 = 0.99\) in the standard normal distribution table. The Z-score for this level of confidence is approximately \(z_{0.01} = 2.33\).
- Calculate the standard error
The standard error of the mean (SE) is calculated using the formula \(SE = \frac{\sigma}{\sqrt{n}}\).
Substituting the given values:
\[SE = \frac{5.71}{\sqrt{50}} \approx \frac{5.71}{7.071} \approx 0.81\]
Rounding to two decimal places, the standard error is \(\$0.81\).
- Calculate the margin of error
The margin of error (ME) is the product of the critical Z-score and the standard error:
\[ME = z_{\alpha/2} \times SE = 2.33 \times 0.81 \approx 1.8873\]
Rounding to two decimal places, the margin of error is \(\$1.89\). Using more precise values before rounding, we find \(ME = 2.3263 \times 0.8075 \approx 1.8786\), which rounds to \(\$1.88\). Let's use \(\$1.88\) for the margin of error.
- Construct the confidence interval
The confidence interval is calculated as \(\bar{x} \pm ME\):
\(\text{Lower Limit} = 18.84 - 1.88 = 16.96\)
\(\text{Upper Limit} = 18.84 + 1.88 = 20.72\)
Thus, we are \(98\%\) confident that the population mean bill \(\mu\) lies between \(\$16.96\) and \(\$20.72\).
Examples
This statistical technique is widely used in business management. For instance, if you manage a fast-food franchise, you can use confidence intervals to predict the average revenue per customer. By taking a sample of 50 transactions, you can calculate an interval for the true population mean, which helps you plan inventory, staff shifts, and set pricing strategies effectively. It provides a reliable range rather than just a single estimate, allowing for better risk management.