Section 3C Dealing with Uncertainty

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Section 3CDealing with Uncertainty

• Pages 168-178

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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)! Thats 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 - Overview

• Significant Digits
• Understanding 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
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
dont! The digits in a number that represent
actual measurement and therefore have meaning are
called significant digits.
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When are digits significant?
3-C
Type of Digit Significance
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
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Counting Significant Digits

• Examples
• 17/179 96.2 km/hr
• 9.6210 km/hr
• 3 significant digits
• (implies a measurement to the nearest .1 km/hr)
• 19/179 100.020 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
• 21/179 0.00098 mm
• 9.810(-4)
• 2 significant digits
• (implies a measurement to the nearest .00001 mm)
• 23/179 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 25/179 300,000 3105 1
significant digits (implies a measurement to the
nearest hundred thousand) 27/179 3.0000 x 105
300000 5 significant digits (implies a
measurement to the nearest ten)
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Ever been to a math party?
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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.

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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.
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Examples Type of Error
pg171 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)
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Examples Type of Error
35/179A count of SUVs passing through a busy
intersection during a 20 minute period.
37/179The average income of 25 people found by
checking their tax returns.
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Math parties are FUN!
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3-C
Size of Error Absolute vs Relative
pg 173 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.
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.
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3-C
Size of Error Absolute vs Relative
absolute error
relative error
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Absolute Error vs. Relative Error
3-C
Ex5b/174 The government claims that a program
costs 49.0 billion, but an audit shows that the
true cost is 50.0 billion
absolute error
measured value true value
49.0 billion 50.0 billion -1 billion
relative error
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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.
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Example
3-C
55/180 Your true height is 62.50 inches. A
tape measure that can be read to the nearest ?
inch gives your height as 62? inches. A new
laser device at the doctors office that gives
reading to the nearest 0.05 inches gives your
height as 62.90 inches.
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3-C
55/180 (solution)
Precision
Tape measure read to nearest 1/8 inch Laser
device read to nearest .05 5/100 1/20 inch
Accuracy
Tape measure 62? inches 62.375
inches (absolute error 62.375 62.5 -.125
inches) Laser device 62.90 inches (absolute
error 62.90 62.5 .4 inches )
The laser device is more precise. The tape
measure is more accurate.
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Math parties are REALLY FUN!
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Combining Measured Numbers
3-B

• Pg177 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!

• 300001 300000 1

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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.
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Combining Measured Numbers
3-C
Ex 7/177 A book written in 1962 states that the
oldest Mayan ruins are 2000 years old. How old
are they now (in 2006)? The book is 2006-1962
44 years old. We round to the nearest one
year. The ruins are 2000 44 2044 years
old. 2000 is the least precise (of 2000 and
44). We round our answer to the nearest 1000
years. The ruins are 2000 years old.
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Combining Measured Numbers
3-C
65/180 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 digits 480,000 has 2
significant digits So we round our answer to 2
significant digits. 234.1666 rounds to 230 per
person.
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• Homework for Friday
• Pages 178-180
• 20, 24, 26, 30, 38,40, 52, 54, 56, 58, 63, 66