Questions: There is a 94% probability that the true population mean number of hours is within the boundaries of our confidence interval.

Solution Steps

To determine the confidence interval for the mean of a single population with known variance at a 94% confidence level, we use the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where:

• \(\bar{x} = 50\) (sample mean)
• \(z \approx 1.88\) (z-value for 94% confidence level)
• \(\sigma = 10\) (population standard deviation)
• \(n = 30\) (sample size)

Substituting the values into the formula:

\[ 50 \pm 1.88 \cdot \frac{10}{\sqrt{30}} \]

Calculating the margin of error:

\[ \frac{10}{\sqrt{30}} \approx 1.8257 \]

Thus, the margin of error is:

\[ 1.88 \cdot 1.8257 \approx 3.43 \]

This gives us the confidence interval:

\[ 50 - 3.43 \text{ to } 50 + 3.43 \implies (46.57, 53.43) \]

The statement claims that there is a 94% probability that the true population mean is within the boundaries of our confidence interval. However, this interpretation is incorrect. The correct interpretation of a 94% confidence level is that if we were to take many samples and construct confidence intervals from each, approximately 94% of those intervals would contain the true population mean.

Based on the calculations and interpretations, the statement is false. The confidence level does not imply a probability for a specific interval containing the true mean.

Final Answer