Questions: Capital - Winter 2024-2025 Conf Int for Pop Mean: Herminia Castellanos Question 8, 6.1.46-T Part 2 of 4 HW Score: 87.5%, 7 of 8 points Interpret the results and compare the widths of the confidence intervals. 17.52 19.54 16.07 19.26 16.15 20.63 18.93 21.26 16.29 20.36 22.82 20.16 19.09 20.34 17.57 The 90% confidence interval is (18.04^dagger, 19.79^circ). (Round to two decimal places as needed.) The 99% confidence interval is (, ). (Round to two decimal places as needed.)

Solution Steps

The mean \( \mu \) of the given data is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{285.99}{15} = 19.07 \]

The standard deviation \( \sigma \) of the data is found to be:

\[ \sigma \approx 2.00 \]

The Z-Score \( Z \) for a 99% confidence level is:

\[ Z = 2.5758 \]

The margin of error \( E \) is calculated using the formula:

\[ E = \frac{Z \times \sigma}{\sqrt{n}} = \frac{2.5758 \times 2.00}{\sqrt{15}} \approx 1.3295 \]

The 99% confidence interval is given by:

\[ \left( \mu - E, \mu + E \right) = \left( 19.07 - 1.3295, 19.07 + 1.3295 \right) \approx (17.7405, 20.3995) \]

Final Answer