Questions: Eat your cereal: Boxes of cereal are labeled as containing 14 ounces. Following are the weights, in ounces, of a sample of 12 boxes. It is reasonable to assume that the population is approximately normal. 13.01, 14.96, 13.10, 13.11, 13.09, 13.01, 13.14, 14.96, 13.04, 13.03, 13.10, 13.11 Part 1 of 2 (a) Construct a 90% confidence interval for the mean weight. Round the answers to at least three decimal places. A 90% confidence interval for the mean weight is < μ < .

Solution Steps

The sample mean \( \bar{x} \) is calculated using the formula:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{160.66}{12} = 13.388 \]

The sample standard deviation \( s \) is computed as follows:

\[ s = \sqrt{\frac{\sum_{i=1}^N (x_i - \bar{x})^2}{N-1}} \approx 0.735 \]

For a 90% confidence level, we use the t-distribution. The critical value \( t \) for \( n-1 = 11 \) degrees of freedom is approximately \( 1.796 \).

The confidence interval is calculated using the formula:

\[ \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} = 13.388 \pm 1.796 \cdot \frac{0.735}{\sqrt{12}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 1.796 \cdot \frac{0.735}{\sqrt{12}} \approx 0.381 \]

Thus, the confidence interval is:

\[ (13.388 - 0.381, 13.388 + 0.381) = (13.007, 13.769) \]

Final Answer