To construct a 99% confidence interval for the true population mean watermelon weight, we can use the formula for the confidence interval based on the population standard deviation:
\[ CI = \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \(\bar{x}\) is the sample mean.
- \(z^*\) is the z-value that corresponds to the desired confidence level (99% in this case).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Step 1: Identify the values
- Sample mean (\(\bar{x}\)) = 60 ounces
- Population standard deviation (\(\sigma\)) = 5 ounces
- Sample size (\(n\)) = 49
Step 2: Find the z-value for 99% confidence level
For a 99% confidence level, the z-value (from the z-table) is approximately:
\[ z^* \approx 2.576 \]
Step 3: Calculate the standard error of the mean (SEM)
The standard error of the mean is calculated as follows:
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{49}} = \frac{5}{7} \approx 0.7143 \]
Step 4: Calculate the margin of error
The margin of error (ME) is given by:
\[ ME = z^* \cdot SE = 2.576 \cdot 0.7143 \approx 1.8387 \]
Step 5: Calculate the confidence interval
Now we can calculate the confidence interval:
\[ CI = \bar{x} \pm ME = 60 \pm 1.8387 \]
Calculating the lower and upper bounds:
- Lower bound: \(60 - 1.8387 \approx 58.16\)
- Upper bound: \(60 + 1.8387 \approx 61.84\)
Final Result
Thus, the 99% confidence interval for the true population mean watermelon weight is:
\[ \boxed{(58.16, 61.84)} \]