To determine which range estimate involves the least risk under the assumption that the ends of a range of estimates are +/- 3 sigma from the mean, follow these steps:
1. Understand the relationship between risk and standard deviation (sigma):
- The risk is related to the variability of the estimate, which is captured by the standard deviation.
- A smaller standard deviation means less variability and less risk.
2. Calculate the total range for each estimate:
- The total range is given by [tex]\( 6 \times \sigma \)[/tex] because the range extends from [tex]\(-3\sigma\)[/tex] to [tex]\(+3\sigma\)[/tex].
3. Consider the given standard deviation (sigma) values:
- Assume we have four standard deviations to compare: 1, 2, 3, and 4.
4. Calculate the ranges:
- For [tex]\(\sigma = 1\)[/tex]: The range is [tex]\( 6 \times 1 = 6 \)[/tex]
- For [tex]\(\sigma = 2\)[/tex]: The range is [tex]\( 6 \times 2 = 12 \)[/tex]
- For [tex]\(\sigma = 3\)[/tex]: The range is [tex]\( 6 \times 3 = 18 \)[/tex]
- For [tex]\(\sigma = 4\)[/tex]: The range is [tex]\( 6 \times 4 = 24 \)[/tex]
5. Identify the range with the smallest value:
- The ranges calculated are 6, 12, 18, and 24.
6. Determine which range corresponds to the least risk:
- The smallest range is 6, which corresponds to [tex]\(\sigma = 1\)[/tex].
Therefore, the range estimate that involves the least risk is the one with a standard deviation of [tex]\(\sigma = 1\)[/tex], resulting in a total range of 6. This means that the estimate with [tex]\(\sigma = 1\)[/tex] involves the least risk.
