Section 3C

Dealing with Uncertainty

Pages 168-178

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.

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Dealing with Uncertainty - OverviewSignificant DigitsUnderstanding Error

Type – Random and Systematic Size – Absolute and Relative Accuracy and Precision

Combining Measured Numbers

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Significant Digits –how we state measurements

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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 133.00 pounds?

133.00 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.

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5

When are digits significant?3-C

Type of Digit SignificanceNonzero 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

Counting Significant Digits

Examples:96.2 km/hr

= 9.62×10 km/hr3 significant digits

(implies a measurement to the nearest .1 km/hr)

100.020 seconds = 1.00020 x 102 seconds

6 significant digits (implies a measurement to the nearest .001 sec.)

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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)

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Counting Significant Digits

Examples:300,000

=3×105

1 significant digit(implies a measurement to the nearest

hundred thousand)

3.0000 x 105 = 300000

5 significant digits (implies a measurement to the nearest ten)

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Rounding with Significant Digits

Examples:1452 x 9076.7; round to 2 significant digits

= 13,179,368.4with 2 significant digits :

13,000,000

1452 x 9076.7; round to 4 significant digits = 13,179,368.4

with 4 significant digits: 13,180,000

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Ever been to a math party?

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Understanding Error3-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 Error3-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 Error3-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.

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Math parties are FUN!

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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 183.5 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 1.5/183.55 = .0082 = .82% for Steve.

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absolute error = measured value – true value

relative error

Size of Error – Absolute vs Relative

absolute error

true value

measured value - true value

true value

Absolute Error vs. Relative Error3-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

absolute error= true value

measured value true value=

true value

relative error

-1 billion= = .02 = 2%

50 billion

Absolute Error vs. Relative Error3-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

absoluteerror

= 100%truevalue

relative error

2 lbs100% 11.1%

18lbs

Accuracy vs. Precision

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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

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Your true height is 62.50 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 62.90 inches.

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Tape measure: read to nearest 1/8”Laser device: read to nearest .05” = 5/100” = 1/20”

Precision

Tape measure: 62⅜ inches = 62.375”(absolute error = 62.375 – 62.50 = -.125”)

Laser device: 62.90 inches(absolute error = 62.90 – 62.5 = .4” )

Accuracy

The tape measure is more accurate.

Actual Height = 62.50 inches

The laser device is more precise.

Math parties are REALLY FUN!

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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 = 300,000 + 1

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Combining Measured Numbers3-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 Numbers3-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 2007-1962 = 45 years old. We round to the nearest one year.

The ruins are 2000 + 45 = 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.

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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 = $234.1666 per person

112.4 millions has 4 significant digits480,000 has 2 significant digits

So we round our answer to 2 significant digits.

$234.1666 rounds to $230 per person.

Combining Measured Numbers

Homework:

Pages 178-180

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