Section 3C Dealing with Uncertainty Pages

Motivating Story (page 168) 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-C

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

Significant Digits –how we state measurements 3-C Suppose I measure my weight to be 133 pounds on a scale that can be read only to the nearest pound. What is wrong with saying that I weigh pounds? 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

When are digits significant? 3-C Type of DigitSignificance Nonzero digit ( )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 ) 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

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) seconds = x 10 2 seconds 6 significant digits (implies a measurement to the nearest.001 sec.) 3-C

Counting Significant Digits Examples: mm =9.8×10 (-4) 2 significant digits (implies a measurement to the nearest mm) meter =2.020 x 10 (-4) 4 significant digits (implies a measurement to the nearest m) 3-C

Counting Significant Digits Examples: 300,000 =3× significant digit (implies a measurement to the nearest hundred thousand) x 10 5 = significant digits (implies a measurement to the nearest ten) 3-C

Rounding with Significant Digits Examples: 1452 x ; round to 2 significant digits = 13,179,368.4 with 2 significant digits : 13,000, x ; round to 4 significant digits = 13,179,368.4 with 4 significant digits: 13,180,000 3-C

Ever been to a math party? 3-C

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.

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.

Example: 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 3-C

Examples- Types of Error A count of SUVs passing through a busy intersection during a 20 minute period. The average income of 25 people found by checking their tax returns. 3-C

Math parties are FUN! 3-C

A scale says Trig weighs 16.5 lbs but he really only weighs 15 lbs. The same scale says my husbands weighs 185lbs, but he really weighs lbs. Size of Error – Absolute vs Relative Absolute Error in both cases is 1.5 lbs pounds Relative Error is 1.5/15 =.1 = 10% for Trig. Relative Error is 1.5/ =.0082 =.82% for Steve.

3-C absolute error = measured value – true value relative error Size of Error – Absolute vs Relative

Absolute Error vs. Relative Error 3-C absolute error= measured value – true value 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

Absolute Error vs. Relative Error 3-C absolute error= measured value – true value Example: The label on a bag of dog food says “20 pounds,” but the true weight is only 18 pounds. = 20 lbs – 18 lbs = 2 lbs relative error

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.

Example 3-C Your true height is inches. A tape measure that can be read to the nearest ⅛ inch gives your height as 62⅜ inches. A new laser device at the doctor’s office that gives reading to the nearest 0.05 inches gives your height as inches.

3-C Tape measure: read to nearest 1/8” Laser device: read to nearest.05” = 5/100” = 1/20” Precision Tape measure: 62⅜ inches = ” (absolute error = – = -.125”) Laser device: inches (absolute error = – 62.5 =.4” ) Accuracy The tape measure is more accurate. Actual Height = inches The laser device is more precise.

Math parties are REALLY FUN! 3-C

Combining Measured Numbers 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? NO! 300,001 = 300, C

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.

Combining Measured Numbers 3-C A book written in 1962 states that the oldest Mayan ruins are 2000 years old. How old are they now (in 2007)? The book is = 45 years old. We round to the nearest one year. The ruins are = 2045 years old. [2000 is the least precise (of 2000 and 45).] We round our answer to the nearest 1000 years. The ruins are 2000 years old.

3-C The government in a city of 480,000 people plans to spend $112.4 million on a transportation project. Assuming all this money must come from taxes, what average amount must the city collect from each resident? $112,400,000 ÷ 480,000 people = $ per person millions has 4 significant digits 480,000 has 2 significant digits So we round our answer to 2 significant digits. $ rounds to $230 per person. Combining Measured Numbers

Homework: Pages # 12, 20, 24, 26, 30, 40, 52, 54, 56, 58, 64