Measurement Uncertainty Flashcards: Absolute & Percentage Practice

Question 1

Classroom linear interval model; all combinations of distinct input values are allowed. A length is 6.40 ± 0.08 cm. What interval does this specify?

Accepted Answer

6.32 to 6.48 cm.

Subtract and add the supplied half-width: 6.40 − 0.08 and 6.40 + 0.08.

Question 2

In the classroom interval model, how do you turn an absolute half-width Δx into a percentage for a positive nominal value x?

Accepted Answer

100 × Δx / x percent.

Use the same unit for Δx and x. The ratio Δx/x is the relative amount before multiplying by 100.

Question 3

Classroom linear interval model; all combinations of distinct input values are allowed. For a difference a − b, which expression gives its exact absolute half-width?

Accepted Answer

Δa + Δb.

The largest difference pairs the upper value of a with the lower value of b; the smallest pairs the opposite endpoints.

Question 4

Classroom linear interval model; all combinations of distinct input values are allowed. A quantity is multiplied by an exact positive factor k. What happens to its absolute half-width?

Accepted Answer

Multiply it by k.

Both endpoints scale by k.

Question 5

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. For z = ab with positive a and b and small relative half-widths, what relative amount do you use for z?

Accepted Answer

Δa/a + Δb/b, approximately.

Multiplying the nominal product by this sum gives its first-order absolute amount. The omitted product ΔaΔb prevents this from being a general exact symmetric bound.

Question 6

A measurement is printed as 17.2 s, with no uncertainty statement. Can its uncertainty be determined from those digits alone?

Accepted Answer

No.

The displayed digits do not specify an interval, a standard deviation, or a coverage probability. An uncertainty statement and its meaning are needed.

Question 7

Classroom linear interval model; all combinations of distinct input values are allowed. Add 9.60 ± 0.06 mm and 3.40 ± 0.02 mm. What is the total’s exact absolute half-width?

Accepted Answer

0.08 mm.

Add the widths: 0.06 + 0.02. The nominal total is 13.00 mm.

Question 8

Classroom linear interval model; all combinations of distinct input values are allowed. A nominal mass of 72 g has a 1.5% interval half-width. What is its absolute half-width?

Accepted Answer

1.08 g.

72 × 1.5/100 = 1.08. This converts the supplied percentage; it does not estimate an instrument’s uncertainty.

Question 9

Classroom linear interval model; all combinations of distinct input values are allowed. A positive quantity has a 1.2% half-width. For its fourth power, what is the first-order percentage amount?

Accepted Answer

Approximately 4.8%.

Multiply 1.2% by the magnitude of the exact exponent: 4 × 1.2%. This is a linear estimate, not an exact power interval.

Question 10

Classroom linear interval model; all combinations of distinct input values are allowed. A model length 9.60 ± 0.24 mm is divided by exactly 12. What is the result’s exact absolute half-width?

Accepted Answer

0.020 mm.

0.24/12 = 0.020. The nominal result is 0.80 mm; the exact divisor adds no uncertainty contribution.

Question 11

Classroom linear interval model; all combinations of distinct input values are allowed. Subtract 12.10 ± 0.02 cm from 18.70 ± 0.03 cm. What is the difference’s exact absolute half-width?

Accepted Answer

0.05 cm.

0.03 + 0.02 = 0.05. The nominal difference is 6.60 cm; subtracting nominal values does not subtract the interval widths.

Question 12

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. Multiply lengths 4.00 ± 0.04 cm and 9.00 ± 0.18 cm. What is the product’s approximate percentage amount?

Accepted Answer

Approximately 3.0%.

The contributions are 0.04/4.00 = 1.0% and 0.18/9.00 = 2.0%. Add them for the first-order product estimate.

Question 13

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. Divide 84.0 ± 0.42 m by 7.00 ± 0.14 s. What is the quotient’s approximate absolute amount?

Accepted Answer

Approximately 0.30 m/s.

The nominal quotient is 12 m/s. Its relative amount is 0.42/84.0 + 0.14/7.00 = 0.025, so 12 × 0.025 = 0.30 m/s.

Question 14

Classroom linear interval model; all combinations of distinct input values are allowed. A calculated force is 42.0 ± 0.21 N. What is its percentage half-width?

Accepted Answer

0.50%.

100 × 0.21/42.0 = 0.50. Use the calculated quantity’s own nominal value in the denominator.

Question 15

Classroom linear interval model; all combinations of distinct input values are allowed. A positive length has a 0.75% half-width. What is the first-order percentage amount for its square?

Accepted Answer

Approximately 1.50%.

The exact exponent is 2, so multiply the relative amount by 2. Do not square the percentage.

Question 16

Classroom linear interval model; all combinations of distinct input values are allowed. A permitted length interval runs from 9.90 to 10.10 cm. What is its absolute half-width?

Accepted Answer

0.10 cm.

Half the interval’s full width is (10.10 − 9.90)/2 = 0.10 cm. Its midpoint is 10.00 cm.

Question 17

Classroom linear interval model; all combinations of distinct input values are allowed. Convert 0.875 ± 0.003 m to centimeters using the exact factor 100 cm per meter. What is the absolute half-width in centimeters?

Accepted Answer

0.30 cm.

100 × 0.003 m = 0.30 cm. Convert the width along with the nominal value, which becomes 87.5 cm.

Question 18

Classroom linear interval model; all combinations of distinct input values are allowed. Two time values are 22.40 ± 0.08 s and 16.40 ± 0.04 s. What percentage half-width belongs to their positive difference before a later ratio?

Accepted Answer

2.0%.

The difference is 6.00 s with half-width 0.08 + 0.04 = 0.12 s. Then 100 × 0.12/6.00 = 2.0%.

Question 19

A student says that writing 7.80 g establishes an interval half-width of 0.01 g. Is that conclusion supported by the notation alone?

Accepted Answer

No.

The final decimal place is not an uncertainty model. Ask for the supplied uncertainty and whether it is a bound, a standard uncertainty, or another stated quantity.

Question 20

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. A calculated sum is S = 36.0 ± 0.18 cm. Multiply it by a distinct length 8.00 ± 0.04 cm. What is the area’s approximate absolute amount?

Accepted Answer

Approximately 2.88 cm².

The nominal area is 288 cm². The relative contributions are 0.18/36.0 = 0.005 and 0.04/8.00 = 0.005, so 288 × 0.010 = 2.88 cm².

Question 21

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. For a quotient of positive quantities, the numerator’s contribution is 2% and the denominator’s is 3%. A student subtracts them. What approximate percentage should replace that step?

Accepted Answer

Approximately 5%.

The linear interval estimate adds the magnitudes of the relative contributions for a quotient: 2% + 3%.

Question 22

Classroom linear interval model; all combinations of distinct input values are allowed. A is 80.0 ± 0.4 cm and B is 30.0 ± 0.3 cm. Which has the larger percentage half-width?

Accepted Answer

B.

A has 100 × 0.4/80.0 = 0.5%; B has 100 × 0.3/30.0 = 1.0%. A larger absolute amount need not be larger in percentage terms.

Question 23

Classroom linear interval model; all combinations of distinct input values are allowed. A positive quantity has a 1.6% half-width. What is the first-order percentage amount for its square root?

Accepted Answer

Approximately 0.8%.

The exact exponent is 1/2. Multiply 1.6% by 1/2; do not take the square root of the percentage.

Question 24

Classroom linear interval model; all combinations of distinct input values are allowed. Add masses 7.80 ± 0.03 g and 4.20 ± 0.05 g, then divide by exactly 4. What is the final exact absolute half-width?

Accepted Answer

0.020 g.

Add the widths first, then scale: (0.03 + 0.05)/4 = 0.020 g. The nominal final value is 3.00 g.

Question 25

Classroom linear interval model; all combinations of distinct input values are allowed. A rectangle has sides 2.00 ± 0.02 cm and 3.00 ± 0.03 cm. What is its exact permitted area interval?

Accepted Answer

5.8806 to 6.1206 cm².

Both sides stay positive. Multiply the lower endpoints for the minimum: 1.98 × 2.97. Multiply the upper endpoints for the maximum: 2.02 × 3.03.

Question 26

Classroom linear interval model; all combinations of distinct input values are allowed. A length 7.00 ± 0.07 m is multiplied by exactly 5. A student keeps 0.07 m as the result’s half-width. What should replace it?

Accepted Answer

0.35 m.

The exact factor scales both nominal value and half-width: 5 × 0.07 = 0.35 m. The nominal result is 35.00 m.

Question 27

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. Evaluate q = (a − b)/t using a = 46.0 ± 0.12 cm, b = 34.0 ± 0.06 cm, and t = 3.00 ± 0.03 s. What is q’s approximate percentage amount?

Accepted Answer

Approximately 2.5%.

The difference is 12.0 cm with half-width 0.18 cm, giving 1.5%. Add the time’s 1.0% contribution for the later quotient.

Question 28

A worksheet supplies ± amounts but never says whether they are interval half-widths or standard uncertainties. Can you choose addition of widths or root-sum-square from that notation alone?

Accepted Answer

No.

First identify what the amounts mean and how the inputs relate. Root-sum-square combines suitable standard-uncertainty contributions under its stated assumptions; it does not turn interval half-widths into standard deviations.

Question 29

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. Two positive factors have percentage contributions of 1% and 4%. A student multiplies the percentages. What approximate percentage belongs to the product?

Accepted Answer

Approximately 5%.

Add the first-order relative contributions: 1% + 4%. Multiplying percentages does not give the linear interval estimate.

Question 30

Classroom linear interval model; all combinations of distinct input values are allowed. A rectangle has sides 2.00 ± 0.02 cm and 3.00 ± 0.03 cm. Its first-order area estimate is 6.00 ± 0.12 cm². Does that symmetric interval contain every permitted area?

Accepted Answer

No.

The exact upper endpoint is 2.02 × 3.03 = 6.1206 cm², beyond 6.12 cm². The omitted cross term is 0.02 × 0.03 = 0.0006 cm².

Question 31

Classroom linear interval model; all combinations of distinct input values are allowed. A positive quantity x has a 1.0% half-width. For x raised to the exact power −3, what is the first-order percentage amount?

Accepted Answer

Approximately 3.0%.

Use the magnitude of the exponent: 3 × 1.0% = 3.0%. The reciprocal power does not create a negative uncertainty.

Question 32

In a classroom linear interval calculation, a computed force has nominal value 60 N and a first-order percentage amount of 2.4%. What approximate absolute amount corresponds to it?

Accepted Answer

Approximately 1.44 N.

60 × 2.4/100 = 1.44 N. Convert the final quantity’s percentage using its own nominal value.

Question 33

Classroom linear interval model; all combinations of distinct input values are allowed. Add 40.0 ± 0.4 g and 60.0 ± 0.3 g. A student adds the input percentages and reports 1.5%. What percentage half-width should the sum have?

Accepted Answer

0.70%.

The sum is 100.0 g with absolute half-width 0.4 + 0.3 = 0.7 g. Convert after adding: 100 × 0.7/100.0 = 0.70%.

Question 34

Before using a classroom first-order quotient estimate with denominator y ± Δy and positive y, what inequality keeps its entire denominator interval above zero?

Accepted Answer

y > Δy.

The lower endpoint y − Δy must be positive. A finite quotient interval cannot be assumed when the denominator interval reaches zero.

Question 35

Classroom linear interval model; all combinations of distinct input values are allowed. A length 8.00 ± 0.08 m is divided by exactly 100. What is the result’s percentage half-width?

Accepted Answer

1.0%.

The result is 0.0800 ± 0.0008 m. Both quantities scale together, so 100 × 0.0008/0.0800 remains 1.0%.

Question 36

In NIST’s standard-uncertainty model, what does a standard uncertainty represent?

Accepted Answer

An estimated standard deviation.

It is a different input quantity from a permitted-interval half-width. The same ± notation does not make the two interchangeable.

Question 37

A classroom interval model specifies 11.0 ± 0.2 cm as permitted values, with no probability distribution. Does this justify a 95% coverage claim?

Accepted Answer

No.

A permitted interval in this exercise model does not assign probabilities. A coverage claim needs a justified probabilistic interpretation.

Question 38

Classroom linear interval model; all combinations of distinct input values are allowed. Use first-order summed relative contributions. Form ab + c with a = 6.00 ± 0.06 cm, b = 5.00 ± 0.05 cm, and distinct c = 20.00 ± 0.10 cm². What is the final first-order absolute amount?

Accepted Answer

Approximately 0.70 cm².

The product’s nominal value is 30 cm² and its relative amount is 1% + 1% = 2%, giving 0.60 cm². For the later addition, add absolute amounts: 0.60 + 0.10 = 0.70 cm².

Question 39

A classroom first-order interval calculation carries nominal value 13.846 cm and approximate absolute amount 0.274 cm. Report the amount to two significant figures and the nominal value to the same decimal place. What report follows?

Accepted Answer

Approximately (13.85 ± 0.27) cm.

0.274 rounds to 0.27 cm; its last place is hundredths. Round 13.846 to that place. This uses the supplied reporting convention, not a universal rule.

Question 40

For combined standard uncertainty, can the covariance terms always be dropped just because there are two input measurements?

Accepted Answer

No.

Uncorrelated input estimates justify dropping the covariance contributions. Otherwise their effect must be checked; a count of inputs does not establish the relationship.

Question 41

Classroom linear interval model; all combinations of distinct input values are allowed. For positive quantities a = 6.00 ± 0.06 and b = 2.00 ± 0.02, what is the exact interval for a/b? Give exact fractions.

Accepted Answer

297/101 to 101/33.

The minimum is 5.94/2.02 = 297/101; the maximum is 6.06/1.98 = 101/33. They are about 2.940594 and 3.060606, so the deviations from the nominal 3 are asymmetric.

Question 42

Classroom linear interval model; all combinations of distinct input values are allowed. Subtract 98.0 ± 0.10 m from 100.0 ± 0.10 m. What is the difference’s exact percentage half-width?

Accepted Answer

10%.

The nominal difference is 2.0 m and its half-width is 0.20 m. Thus 100 × 0.20/2.0 = 10%; a small difference can have a large relative amount.

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Measurement Uncertainty Flashcards: Absolute & Percentage Practice