remember the honor code!

the width of a room is measured 10 times by an ultrasonic rule and the following measurements are obtained (units of meters): 4.292, 4.295, 4.296, 4.293, 4.292, 4.294, 4.293, 4.290, 4.294, and 4.291.

the width of the same room is then measured by a calibrated steel tape that gives a reading of 4.276 m, which can be taken as the correct value for the width of the room.

(a) what is the measurement precision of the ultrasonic rule?
(b) what is the maximum measurement inaccuracy of the ultrasonic rule?

Question

remember the honor code!

the width of a room is measured 10 times by an ultrasonic rule and the following measurements are obtained (units of meters): 4.292, 4.295, 4.296, 4.293, 4.292, 4.294, 4.293, 4.290, 4.294, and 4.291.

the width of the same room is then measured by a calibrated steel tape that gives a reading of 4.276 m, which can be taken as the correct value for the width of the room.

(a) what is the measurement precision of the ultrasonic rule?
(b) what is the maximum measurement inaccuracy of the ultrasonic rule?

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Measurement Precision", "Measurement Inaccuracy" ], "current_concepts": [ "Measurement Precision", "Measurement Inaccuracy" ] } ## Calculate the mean of the ultrasonic measurements $$ \begin{aligned} x_i &= \{4.292, 4.295, 4.296, 4.293, 4.292, 4.294, 4.293, 4.290, 4.294, 4.291\} \ N &= 10 \ \bar{x} &= \frac{\sum_{i=1}^{10} x_i}{10} = \frac{42.930}{10} = 4.293\text{ m} \end{aligned} $$ ## Calculate the measurement precision (standard deviation) $$ \begin{aligned} s &= \sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N - 1}} \ \sum_{i=1}^{10} (x_i - \bar{x})^2 &= (-0.001)^2 + (0.002)^2 + (0.003)^2 + (0.000)^2 + (-0.001)^2 \ &+ (0.001)^2 + (0.000)^2 + (-0.003)^2 + (0.001)^2 + (-0.002)^2 \ &= (1 + 4 + 9 + 0 + 1 + 1 + 0 + 9 + 1 + 4) \times 10^{-6} = 30 \times 10^{-6}\text{ m}^2 \ s &= \sqrt{\frac{30 \times 10^{-6}}{9}} \approx 0.00183\text{ m} \end{aligned} $$ ## Calculate the maximum measurement inaccuracy $$ \begin{aligned} x_{\text{true}} &= 4.276\text{ m} \ \text{Inaccuracy}_i &= |x_i - x_{\text{true}}| \ \text{Max Inaccuracy} &= \max(|x_i - 4.276|) \ &= 4.296 - 4.276 = 0.020\text{ m} \end{aligned} $$ ## Question a $0.0018\text{ m}$ (or $1.8\text{ mm}$) ## Question b $0.020\text{ m}$ (or $20\text{ mm}$) { "subject": "Natural Science", "question_type": "Multi-part", "knowledge_point": [ "Natural Science", "Physics", "Measurement and Uncertainty" ] }

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Question

Explanation:

Calculate the mean of the ultrasonic measurements

Calculate the measurement precision (standard deviation)

Calculate the maximum measurement inaccuracy

Answer:

Question a

Question b