2. Copyright © 2011 Pearson Education, Inc. Numbers in the Real World

3. Copyright © 2011 Pearson Education, Inc. Slide 3-3 Unit 3C Dealing with Uncertainty Understanding the nature of uncertainty will make you better equipped to assess the reliability of numbers as they are typically reported.

4. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-4 Significant Digits Not significant unless stated otherwise Zeros to the right of the last nonzero digit but before the decimal point as in (40,000 or 210) Never significantZeros to the left of the first nonzero digit (as in 0.006 or 0.00052) Always significantZeros between nonzero digits (as in 4002 or 3.06) or other significant zeros (such as the first zero in 30.0) Always significantZeros that follow a nonzero digit and lie to the right of the decimal point (as in 4.20 or 3.00) Always significantNonzero digits SignificanceType of Digit

5. Significant Digits Class Notes 1-4 *Key: how we state measurements! State number of significant digits 1. 11.90 seconds 2..000067 meter 3. population reported as 240,000 4. population reported as 2.40 x 10^5 Copyright © 2011 Pearson Education, Inc. Slide 3-5

6. Rounding with Significant Digits Class Notes 5-6 Round each of the following operations, give your answer with the specified number of significant digits 5. 7.7 mm x 9.92 mm give answer with 2 significant digits. 6. 240,000 x 72,106 Give your answer with 4 significant digits Copyright © 2011 Pearson Education, Inc. Slide 3-6

7. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-7 Random errors occur because of random and inherently unpredictable events in the measurement process. Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by the same amount. Types of Error

8. 3-C Errors in Global Warming Data CN 7 Consider 2 difficulties in trying to interpret historical temperature a. Temps were measured with simple thermometers and the data were recorded by hand. b. Most temps were recorded in near urban areas which tend to be warmer than surrounding areas 7. Do each of these produce random or systematic errors? Copyright © 2011 Pearson Education, Inc. Slide 3-8

9. 3-C The Census CN 8 The Constitution fo the US mandates a census of the population every 10 years. The US Census Bureau conducts the census by distributing household surveys through the mail. 8. Suggest several sources of both random and systematic error in the census. Copyright © 2011 Pearson Education, Inc. Slide 3-9

10. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-10 The absolute error describes how far a measured (or claimed) value lies from the true value. absolute error = measured value – true value The relative error compares the size of the error to the true value. Size of Errors

11. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-11 absolute error = measured value – true value Example: A projected budget surplus of $17 billion turns out to be $25 billion at the end of the fiscal year. = $25 billion – $17 billion = $8 billion = $8 billion / $17 billion ≈ 0.471 = 47.1% relative error Absolute vs. Relative Error

12. 3-C Find the Absolute and Relative Error CN 9-10 9. Your true weight is 125 lbs, but a scale says you weigh 130 lbs. 10. The government claims that a program costs $49.0 billion, but an audit show the true cost is $50.0 billion Copyright © 2011 Pearson Education, Inc. Slide 3-12

13. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-13 Describing Results Accuracy describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value. Precision describes the amount of detail in a measurement.

14. 3-C Accuracy and Precision in your Weight CN 11 10. Suppose your true weight is 102.4 lbs. The scale at the doctor’s office, which can be read only to the nearest quarter pound, says that you weigh 102 ¼ lbs. The scale at the gym, which gives a digital readout to the nearest.1 lb says that you weigh 100.7 lbs. 11. Which scale is more precise? Which is more accurate? Copyright © 2011 Pearson Education, Inc. Slide 3-14

15. 3-C Does the Census Measure the True Pop CN 12 Class Discussion 12. Give your thoughts on measuring the true population through the census. Copyright © 2011 Pearson Education, Inc. Slide 3-15

16. 3-C Copyright © 2011 Pearson Education, Inc. Slide 3-16 Combining Measured Numbers Rounding rule for addition or subtraction: Round the answer to the same precision as the least precise number in the problem. Rounding rule for multiplication or division: Round the answer to the same number of significant digits as the measurement with the fewest significant digits. To avoid errors, round only after completing all the operations, not during the intermediate steps.

17. 3-C Combining CN 13-14 A book written in 1962 states that the oldest Mayan ruins are 2000 years old. 13. How old are they now? The government in a town of 82,000 people plans to spend $41.5 million this year. 14. Assuming all this money must come from taxes, what average amount must the city collect from each resident? Copyright © 2011 Pearson Education, Inc. Slide 3-17