Question

Mean height of plants obtained from a random sample of size 100 is 64 inches. The population standard deviation of the plants is 3 inches. If the plant heights are distributed normally, then the 99% confidence limits of the mean population height of plants, are:

• A. (63.2, 64.8)
• B. (62, 64.8)
• C. (63.2, 65)
• D. (62.2, 65.8)

Hint:

Memorize the common Z-values for confidence intervals: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. When the population standard deviation \(\sigma\) is known, always use the Z-test, regardless of the sample size (as long as the population is normal).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks for a 99% confidence interval for the population mean height. Since the population standard deviation (\(\sigma\)) is known and the sample size is large (n>30), we can use the Z-distribution to construct the interval.

Step 2: Key Formula or Approach:
The formula for a confidence interval for the population mean (\(\mu\)) when \(\sigma\) is known is: \[ \text{CI} = \bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \] Where: - \( \bar{x} \) is the sample mean.
- \( \sigma \) is the population standard deviation.
- \( n \) is the sample size.
- \( Z_{\alpha/2} \) is the critical Z-value for the desired confidence level.

Step 3: Detailed Explanation:
First, identify the given values: - Sample size, \( n = 100 \).
- Sample mean, \( \bar{x} = 64 \) inches.
- Population standard deviation, \( \sigma = 3 \) inches.
- Confidence level = 99%, which means \( \alpha = 1 - 0.99 = 0.01 \).
Next, find the critical Z-value, \( Z_{\alpha/2} \). - \( \alpha/2 = 0.01 / 2 = 0.005 \).
- We need the Z-value such that the area in the upper tail of the standard normal distribution is 0.005.
- From the Z-table or standard statistical knowledge, \( Z_{0.005} \approx 2.576 \).
Now, calculate 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, construct 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 confidence interval is approximately (63.2, 64.8).

Step 4: Final Answer:
The 99% confidence limits for the mean population height of plants are (63.2, 64.8).