Question

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Explanation:

Step1: Identify the formula for margin of error (E)

The formula for the margin of error when the population standard deviation ($\sigma$) is known is $E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$, where $z_{\alpha/2}$ is the critical value, $\sigma$ is the population standard deviation, and $n$ is the sample size. Here, $\sigma = 2.70$, $n = 400$, and for a 99% confidence interval, $\alpha = 0.01$, so $\alpha/2 = 0.005$ and $z_{0.005} = 2.576$.

Step2: Calculate the standard error

The standard error is $\frac{\sigma}{\sqrt{n}} = \frac{2.70}{\sqrt{400}} = \frac{2.70}{20} = 0.135$.

Step3: Calculate the margin of error (E)

Substitute the values into the margin of error formula: $E = 2.576 \cdot 0.135 = 0.34776$.

Step4: Calculate the confidence interval

The confidence interval is calculated as $\bar{x} \pm E$, where $\bar{x}$ is the sample mean. Here, $\bar{x} = 8.84$. So the lower bound is $8.84 - 0.34776 \approx 8.49$ and the upper bound is $8.84 + 0.34776 \approx 9.19$.

Answer:

The 99% confidence interval for the population mean distance is \((8.49, 9.19)\)