Questions: Day Care Tuition A random sample of 47 four-year-olds attending day care centers provided a yearly tuition average of 3966 and the population standard deviation of 636. Part 1 of 2 Find the 99% confidence interval of the true mean. Round your answers to the nearest whole number. 3727<mu<4205 Part: 1 / 2 Part 2 of 2 If a day care center were starting up and wanted to keep tuition low, what would be a reasonable amount to charge? Round your answer to the nearest hundred. would be a reasonable amount to charge.

Solution Steps

To find the \(99\%\) confidence interval for the true mean tuition, we use the formula:

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

Where:

• \(\bar{x} = 3966\) (sample mean)
• \(\sigma = 636\) (population standard deviation)
• \(n = 47\) (sample size)
• \(z\) is the z-score corresponding to a \(99\%\) confidence level, which is approximately \(3.0\).

Calculating the margin of error:

\[ \text{Margin of Error} = z \frac{\sigma}{\sqrt{n}} = 3.0 \cdot \frac{636}{\sqrt{47}} \approx 239.0 \]

Thus, the confidence interval is:

\[ (3966 - 239.0, 3966 + 239.0) = (3727.0, 4205.0) \]

A reasonable amount to charge for tuition would be the lower bound of the confidence interval, which is:

\[ 3727.0 \]

Rounding this to the nearest hundred gives:

\[ \text{Reasonable amount to charge} = 3700.0 \]

Final Answer