Step 1: Concept Overview:
This problem requires calculating a 99% confidence interval for the population mean height. Because the population standard deviation (\(\sigma\)) is known and the sample size is large (n>30), the Z-distribution is appropriate for constructing the interval.
Step 2: Formula and Approach:
The confidence interval for the population mean (\(\mu\)), given a known \(\sigma\), is calculated as follows:
\[ \text{CI} = \bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]
Where:- \( \bar{x} \) represents the sample mean.
- \( \sigma \) represents the population standard deviation.
- \( n \) represents the sample size.
- \( Z_{\alpha/2} \) represents the critical Z-value for the specified confidence level.
Step 3: Step-by-Step Solution:
First, identify the provided values:- Sample size: \( n = 100 \).
- Sample mean: \( \bar{x} = 64 \) inches.
- Population standard deviation: \( \sigma = 3 \) inches.
- Confidence level: 99%, therefore \( \alpha = 1 - 0.99 = 0.01 \).
Next, determine the critical Z-value, \( Z_{\alpha/2} \).- \( \alpha/2 = 0.01 / 2 = 0.005 \).
- We seek the Z-value corresponding to an upper tail area of 0.005 in the standard normal distribution.
- Using a Z-table or statistical software, \( Z_{0.005} \approx 2.576 \).
Now, compute the margin of error (ME):\[ \text{ME} = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} = 2.576 \times \frac{3}{\sqrt{100}} = 2.576 \times \frac{3}{10} = 0.7728 \]
Finally, establish the confidence interval:\[ \text{CI} = \bar{x} \pm \text{ME} = 64 \pm 0.7728 \]- Lower limit = \( 64 - 0.7728 = 63.2272 \)
- Upper limit = \( 64 + 0.7728 = 64.7728 \)
The resulting confidence interval is approximately (63.2, 64.8).
Step 4: Conclusion:
The 99% confidence interval for the mean population height of plants is (63.2, 64.8).