23.7: C1.07- Interval Part 3

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Precision of end points: The numbers at the ends of the interval of actual values will always have exactly one more decimal place of precision than the given rounded number.

Discussion: Notation.

In the examples below, the same answer is given with several different forms of notation. You do not have to write all of those different ways for every answer. You must understand all the ways, but pick the one you prefer and write your answers with just that one form of notation.

Example 7: A measured number is reported as 0.03724 kilometers

state the implied precision as a number
state the implied precision in words
underline the significant digits
find the interval of possible actual values consistent with this rounded number.

Solution:

The precision here is implied to be 0.00001
The precision is one hundred-thousandth.
The significant digits are underlined: 0.03724 kilometers
The next-smaller number with the same precision is 0.03723 and the next-larger number with the same precision is 0.03725. Write those as 0.037230 and 0.037240 and 0.037250. Then find and label the half-way points.

$1$

The actual number is between 0.037235 and 0.037245 kilometers.

Example 8: A measured number is reported as 1.27 liters.

state the implied precision as a number
state the implied precision in words
underline the significant digits
what interval of possible actual values are consistent with that and what are several ways this might be reported?

Solution:

The rounding precision here is implied to be 0.01.
The precision is one-hundredth.
The significant digits are underlined: 1.27 liters.
The next-smaller number with the same precision is 1.26 and the next-larger number with the same precision is 1.28. Write those as 1.260, 1.270, and 1.280. Then find and label the half-way points.

Or it might be reported as $1.27\pm0.005$ liters

Another method of reporting it would be $1.27_{+0.005}^{-0.005}$ liters, which is usually only used if the distances on the two sides are not equal.

Example 9: If a measured number is reported as 52700 feet, rounded to the nearest hundred feet, underline the significant digits and identify the interval of possible actual values.

Solution: (All numbers here are in feet.) The significant digits are underlined: 52700 feet

$3$