Question

distinguish between the absolute error and the relative error in a measurement. give an example in which the absolute error is large but the relative error is small and another example in which the absolute error is small but the relative error is large.
d. the relative error describes how far a measured (or claimed) value lies from the absolute error. the absolute error compares the size of the relative error to the true value and is often expressed as a percentage.
give an example in which the absolute error is large but the relative error is small.
a. a chemist has 2.9 mg of substance, but a scale measures 2.1 mg.
b. a census says that the population of a town is 72,453, but the true population is 96,000.
c. a runners true weight is 125 pounds, but a scale says he weighs 130 pounds.
d. a company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.
give an example in which the absolute error is small but the relative error is large.
a. a chemist has 2.9 mg of substance, but a scale measures 2.1 mg.
b. a runners true weight is 125 pounds, but a scale says he weighs 130 pounds.
c. a woman weighs 102.4 pounds. the scale at the gym says she weighs 102.7, but the scale at the doctors office says she weighs 102 1/4.
d. a census says that the population of a town is 72,453, but the true population is 96,000.

Explanation:

1. Definition of absolute and relative error: Absolute error is the difference between the measured value and the true value. Relative error is the ratio of the absolute error to the true value, often expressed as a percentage.
2. Large absolute - small relative error example: In option D, the absolute error is \(|7.32 - 7.30|= 0.02\) billion. The relative error is \(\frac{0.02}{7.32}\approx0.00273\) or \(0.273\%\). A large - scale value (billions) makes the relative error small despite a non - zero absolute error.
3. Small absolute - large relative error example: In option A, the absolute error is \(|2.9 - 2.1| = 0.8\) mg. The relative error is \(\frac{0.8}{2.9}\approx0.276\) or \(27.6\%\). A small true value (mg) makes the relative error large for a relatively small absolute error.

Answer:

1. For large absolute error but small relative error: D. A company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.
2. For small absolute error but large relative error: A. A chemist has 2.9 mg of substance, but a scale measures 2.1 mg.