1. Section 3C Dealing with Uncertainty Pages 168-178

2. Motivating Story (page 172) In 2001, government economists projected a cumulative surplus of $5.6 trillion in the US federal budget for the coming 10 years (through 2011)! That’s $20,000 for every man, woman and child in the US. A mere two years later, the projected surplus had completely vanished. What happened? Assumptions included highly uncertain predictions about the future economy, future tax rates, and future spending. These uncertainties were diligently reported by the economists but not by the news media. Understanding the nature of uncertainty will make you better equipped to assess the reliability of numbers in the news.

3. Dealing with Uncertainty - Overview Significant Digits Understanding Error  Type – Random and Systematic  Size – Absolute and Relative  Accuracy and Precision Combining Measured Numbers

4. Significant Digits –how we state measurements 3-C Suppose I measure my weight to be 132 pounds on a scale that can be read only to the nearest pound. What is wrong with saying that I weigh 132.00 pounds? 132.00 incorrectly implies that I measured (and therefore know) my weight to the nearest one hundredth of a pound and I don’t! The digits in a number that represent actual measurement and therefore have meaning are called significant digits. 3 5

5. When are digits significant? 3-C Type of DigitSignificance Nonzero digit (123.457)Always significant Zeros that follow a nonzero digit and lie to the right of the decimal point (4.20 or 3.00) Always significant Zeros between nonzero digits (4002 or 3.06) or other significant zeros (first zero in 30.0) Always significant Zeros to the left of the first nonzero digit (0.006 or 0.00052) Never significant Zeros to the right of the last nonzero digit but before the decimal point (40,000 or 210) Not significant unless stated otherwise

6. Counting Significant Digits Examples: 96.2 km/hr = 9.62×10 km/hr 3 significant digits (implies a measurement to the nearest.1 km/hr) 100.020 seconds = 1.00020 x 10 2 seconds 6 significant digits (implies a measurement to the nearest.001 sec.)

7. Counting Significant Digits Examples: 0.00098 mm =9.8×10 (-4) 2 significant digits (implies a measurement to the nearest.00001 mm) 0.0002020 meter =2.020 x 10 (-4) 4 significant digits (implies a measurement to the nearest.0000001 m)

8. Counting Significant Digits Examples: 300,000 =3×10 5 1 significant digits (implies a measurement to the nearest hundred thousand) 3.0000 x 10 5 = 300,000 5 significant digits (implies a measurement to the nearest ten)

9. Ever seen a Julia Set?

10. very cool!

11. Understanding Error 3-C Errors can occur in many ways, but generally can be classified as one of two basic types: random or systematic errors. Whatever the source of an error, its size can be described in two different ways: as an absolute error, or as a relative error. Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.

12. Two Types of Measurement Error 3-C 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.

13. pg175 weighing babies in a pediatricians office Shaking and crying baby introduces random error because a measurement could be “shaky” and easily misread. A miscalibrated scale introduces systematic error because all measurements would be off by the same amount. (adjustable) Examples – Type of Error

14. 45/183 A count of SUVs passing through a busy intersection during a 20 minute period. 47/183 The average income of 25 people found by checking their tax returns.

15. 3-C pg 177 You ask for 6 pounds of hamburger and receive 4 pounds. A car manual gives the car weight as 3132 pounds but it really weighs 3130 pounds. Size of Error – Absolute vs Relative is the error big enough to be of concern or small enough to be unimportant Absolute Error in both cases is 2 pounds Relative Error is 2/4 =.5 = 50% for hamburger. Relative Error is 2/3130 =.0003194 =.03% for car. Absolute Error = Measured Value – True Value Relative Error = Absolute Error True Value NOTE: Claimed value is measured value.

16. Absolute Error vs. Relative Error 3-C absolute error= measured value – true value Ex5a/178 My true weight is 125 pounds, but the scale says I weight 130 pounds. = 130 – 125 = 5 pounds relative error The measured weight is too high by 4%.

17. Absolute Error vs. Relative Error 3-C absolute error= claimed value – true value Ex5b/178 The government claims that a program costs $49.0 billion, but an audit shows that the true cost is $50.0 billion = $49.0 billion – $50.0 billion = $-1 billion relative error The claimed cost is too low by 2%.

18. very, very cool!

19. Accuracy vs. Precision 3-C 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.

20. Example 3-C 65/180 Your true height is 70.50 inches. A tape measure that can be read to the nearest ⅛ inch gives your height as 70⅜ inches. A new laser device at the doctor’s office that gives reading to the nearest 0.05 inches gives your height as 70.90 inches. Which device is more accurate? Which is more precise?

21. 3-C Tape measure: read to nearest 1/8 inch Laser device: read to nearest.05 = 5/100 = 1/20 inch Precision Tape measure: 70⅜ inches = 70.375 inches (absolute error = 70.375 – 70.5 = -.125 inches) Laser device: 70.90 inches (absolute error = 70.90 – 70.5 =.4 inches ) Accuracy The laser device is more precise. The tape measure is more accurate. 65/180 (solution)

22. awesome!

23. Combining Measured Numbers Pg180 The population of your city is reported as 300,000 people. Your best friend moves to your city to share an apartment. Is the new population 300,001? 3-B NO! 300001 = 300000 + 1

24. Combining Measured Numbers 3-C Rounding rule for addition or subtraction: Round your answer to the same precision as the least precise number in the problem. Rounding rule for multiplication or division: Round your answer to the same number of significant digits as the measurement with the fewest significant digits. Note: You should do the rounding only after completing all the operations – NOT during the intermediate steps!!! We round 300,001 to the same precision as 300,000. So, we round to the hundred thousands to get 300,000.

25. Combining Measured Numbers 69/184 Subtract 1.45 hours from 60 hours 60 - 1.45 = 58.55 = (round to least precise) = 60 hours 71/184 Multiply 62.5 km/hr by 2.4 hours. 62.5 x 2.4 = 150 (round to fewest sig. digits) = 150 km 73/184 A freeway sign tells you that it is 36 miles to downtown. Your destination is 2.2 miles beyond downtown. How much farther do you have to drive? 36 + 2.2 = 38.2 (round to least precise) = 38 miles

26. 3-C 75/184 What is the per capita cost of $2.1 million recreation center in a city with 120,342 people? $2,100,000 ÷ 120,342 people = $17.45026675 per person 2.1 million has 2 significant digits 120,342 has 6 significant digits So we round our answer to 2 significant digits. $17.45026675 rounds to $17 per person. Combining Measured Numbers

27. Homework: Pages 181-184 28, 32, 34, 41, 59, 62, 64, 66, 70, 74, 76