Final answer:
To construct the 99% confidence interval for the mean price of all homes using the provided data, the appropriate Z-value for 99% confidence is used, and the sample mean, population standard deviation, and sample size are plugged into the formula. However, none of the given options match the calculated confidence interval, which may indicate an error in the provided options or question itself.
Step-by-step explanation:
The question involves constructing a 99% confidence interval for the mean price of all homes in a state, given a sample mean and the population standard deviation. We can use the formula for a confidence interval for the mean when the population standard deviation is known:
Confidence Interval = Mean ± (Z-value)(σ/√n)
Where σ is the population standard deviation, n is the sample size, and the Z-value corresponds to the desired confidence level. For a 99% confidence level, the Z-value is approximately 2.576.
Using the given figures:
- Mean = $299,270
- σ = $68,650
- n = 1500
We calculate the margin of error (ME):
ME = 2.576 * (σ/√n) = 2.576 * ($68,650/√1500) ≈ $4557.36
Therefore, the 99% confidence interval is:
Confidence Interval = $299,270 ± $4557.36 = ($294,712.64, $303,827.36)
Comparing our calculated interval with the options, none of them match our findings. It suggests there may be an error in the provided options or the question details.