If the true mean is 0.5 inches, estimate the uncertainty interval at a probability of 70%. Also estimate the uncertainty interval at a probability of 90%.
The correct answer and explanation is:
To estimate uncertainty intervals (also called confidence intervals), we need the true mean, a measure of spread (like standard deviation), and sample size. However, since only the true mean (0.5 inches) is given, we’ll assume:
- The distribution is normal
- A standard deviation (σ) of 1 inch for demonstration
- A sample size (n) large enough to use z-scores directly
Step 1: Use Z-scores for the desired confidence levels
For a 70% confidence level, the critical z-value is approximately ±1.04
For a 90% confidence level, the critical z-value is approximately ±1.645
Step 2: Compute uncertainty intervals
Since no sample size or standard error is given, assume the population distribution is used:
Uncertainty interval formula:
Mean ± z × σ
For 70% probability:
0.5 ± (1.04 × 1) = 0.5 ± 1.04
Interval: (-0.54, 1.54) inches
For 90% probability:
0.5 ± (1.645 × 1) = 0.5 ± 1.645
Interval: (-1.145, 2.145) inches
Explanation:
An uncertainty interval provides a range around the mean where the true value is expected to fall with a certain probability. A 70% interval gives a narrower range, reflecting less certainty, while a 90% interval is wider and shows more confidence in capturing the true value.
The width of the interval depends on both the standard deviation and the z-value. A higher probability requires a larger z-value, which widens the interval to accommodate more possible outcomes.
These intervals are useful in scientific and engineering measurements, where understanding variability and confidence is critical. The mean remains fixed at 0.5 inches, but the uncertainty interval shows how far actual observations might reasonably vary from that mean.