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Chapter 1: Problem 39

Three students \((A, B,\) and \(C)\) are asked to determine the volume of a sample of ethanol. Each student measures the volume three times with a graduated cylinder. The results in milliliters are: \(\mathrm{A}(87.1,88.2,\) 87.6)\(; \mathrm{B}(86.9,87.1,87.2) ; \mathrm{C}(87.6,87.8,87.9) .\) The true volume is \(87.0 \mathrm{~mL}\). Comment on the precision and the accuracy of each student's results.

Short Answer

Expert verified
Student B's measurements are the most accurate as their average is closest to the true value, and both students B and C are the most precise as their measurements have the smallest range.

Step by step solution

01

Calculate the Average for Each Student

Calculate the average of each students' measurements. For student A, it's \((87.1+88.2+87.6) / 3 = 87.63\) mL. For student B, it's \((86.9+87.1+87.2) / 3 = 87.07\) mL. For student C, it's \((87.6+87.8+87.9) / 3 = 87.77\) mL.
02

Evaluate the Accuracy

Now compare each average to the true value. The closer the average is to the true value, the more accurate the measurements are. Student A's average is 0.63 mL above the true value, student B's average is 0.07 mL above the true value, and student C's average is 0.77 mL above the true value. Student B's measurements are the most accurate as the difference is least.
03

Evaluate the Precision

Precision involves looking at how close the measurements are to each other. Measure the range (difference between the highest and the lowest measurement) for each student. For student A, it's \(88.2-87.1=1.1\) mL. For student B, it's \(87.2-86.9=0.3\) mL. For student C, it's \(87.9-87.6=0.3\) mL. Both students B and C are the most precise as the range is smallest.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Calculation
Calculating the average is a helpful way to summarize multiple measurements. Imagine you have several measurements and want to find a central value, or average, to represent them. Here's how it's done:

  • Add all the individual measurements together.
  • Divide that total by the number of measurements you have.
For student A, the measurements were 87.1, 88.2, and 87.6 mL. Adding these gives us 262.9. Dividing by 3, we get an average of 87.63 mL.

Similarly, for student B, the average is 87.07 mL, and for student C, it is 87.77 mL. This average value gives a way to compare how close each student's results are to the true volume. Understanding this helps evaluate measurement accuracy.
Experimental Error
Experimental error is the difference between a measured value and the true value. It's a natural part of many experiments and helps us understand how accurate our measurements are.

To find the error, you subtract the true value from your average measurement. Smaller differences indicate more accurate measurements.

  • For student A, the error is 0.63 mL (87.63 - 87.0).
  • For student B, the error is 0.07 mL.
  • For student C, the error is 0.77 mL.
Thus, student B's measurements are more accurate because they have the smallest error. Recognizing and calculating experimental error can significantly improve the reliability of your measurements.
Measurement Analysis
Measurement analysis involves evaluating both the accuracy and precision of data. While accuracy refers to how close a measurement is to the true value, precision is about consistency in repeated measurements.

Precision is assessed by looking at the range of measurements, which is the difference between the highest and lowest measurement:

  • Student A's range is 1.1 mL (88.2 - 87.1).
  • Student B's range is 0.3 mL.
  • Student C's range is 0.3 mL.
Both students B and C show high precision with a range of only 0.3 mL.

By combining both accuracy and precision, you get a complete picture of measurement quality. This analysis helps not only in science but in any field where decision-making is based on data.

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Most popular questions from this chapter

Suppose that a new temperature scale has been devised on which the melting point of ethanol \(\left(-117.3^{\circ} \mathrm{C}\right)\) and the boiling point of ethanol \(\left(78.3^{\circ} \mathrm{C}\right)\) are taken as \(0^{\circ} \mathrm{S}\) and \(100^{\circ} \mathrm{S},\) respectively where \(S\) is the symbol for the new temperature scale. Derive an equation relating a reading on this scale to a reading on the Celsius scale. What would this thermometer read at \(25^{\circ} \mathrm{C}\) ?

A bank teller is asked to assemble "one-dollar" sets of coins for his clients. Each set is made of three quarters, one nickel, and two dimes. The masses of the coins are: quarter: 5.645 g; nickel: \(4.967 \mathrm{~g}\) dime: 2.316 g. What is the maximum number of sets that can be assembled from \(33.871 \mathrm{~kg}\) of quarters, \(10.432 \mathrm{~kg}\) of nickels, and \(7.990 \mathrm{~kg}\) of dimes? What is the total mass (in g) of the assembled sets of coins?

A \(250-\mathrm{mL}\) glass bottle was filled with \(242 \mathrm{~mL}\) of water at \(20^{\circ} \mathrm{C}\) and tightly capped. It was then left outdoors overnight, where the average temperature was \(-5^{\circ} \mathrm{C}\). Predict what would happen. The density of water at \(20^{\circ} \mathrm{C}\) is \(0.998 \mathrm{~g} / \mathrm{cm}^{3}\) and that of ice at \(-5^{\circ} \mathrm{C}\) is $0.916 \mathrm{~g} / \mathrm{cm}^{3}.

Carry out the following conversions: (a) \(22.6 \mathrm{~m}\) to decimeters, (b) \(25.4 \mathrm{mg}\) to kilograms, (c) \(556 \mathrm{~mL}\) to liters, (d) \(10.6 \mathrm{~kg} / \mathrm{m}^{3}\) to \(\mathrm{g} / \mathrm{cm}^{3}\).

A human brain weighs about \(1 \mathrm{~kg}\) and contains about \(10^{11}\) cells. Assuming that each cell is com pletely filled with water (density \(=1 \mathrm{~g} / \mathrm{mL}\) ), calculate the length of one side of such a cell if it were a cube. If the cells are spread out in a thin layer that is a single cell thick, what is the surface area in square meters?
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