Transcript Section 3B Putting Numbers in Perspective

Section 3C
Dealing with Uncertainty
Pages 168-178
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.
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
Suppose I measure my weight to be 132 pounds on a scale
that can be read only to the nearest pound.
3
What is wrong with saying that I weigh 132.00 pounds?
5
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-C
When are digits significant?
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) Always significant
or other significant zeros (first zero in 30.0)
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
Not significant
but before the decimal point (40,000 or 210) 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)
100.020 seconds
= 1.00020 x 102 seconds
6 significant digits
(implies a measurement to the nearest .001 sec.)
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)
Counting Significant Digits
Examples:
300,000
=3×105
1 significant digits
(implies a measurement to the nearest hundred
thousand)
3.0000 x 105 = 300,000
5 significant digits
(implies a measurement to the nearest ten)
Ever seen a Julia Set?
very cool!
3-C
Understanding Error
•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.
3-C
Two Types of Measurement Error
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.
Examples – Type of Error
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
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.
Size of Error – Absolute vs Relative
3-C
is the error big enough to be of concern or small enough to be unimportant
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.
Absolute Error = Measured Value – True Value
Relative Error =
Absolute Error
True Value
NOTE: Claimed value is measured value.
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.
3-C
Absolute Error vs. Relative Error
Ex5a/178 My true weight is 125 pounds, but the scale says I
weight 130 pounds.
absolute error
= measured value – true value
= 130 – 125 = 5 pounds
relative error
= absolute error
true value
= measured value  true value
true value
5
=
= .04 = 4%
125
The measured weight is too high by 4%.
3-C
Absolute Error vs. Relative Error
Ex5b/178 The government claims that a program costs $49.0
billion, but an audit shows that the true cost is $50.0 billion
absolute error
= claimed value – true value
= $49.0 billion – $50.0 billion = $-1 billion
relative error
= absolute error
true value
= measured value  true value
true value
-1 billion
=
=  .02 = -2%
50 billion
The claimed cost is too low by 2%.
very, very cool!
3-C
Accuracy vs. Precision
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.
3-C
Example
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?
3-C
65/180 (solution)
Precision
Tape measure: read to nearest 1/8 inch
Laser device: read to nearest .05 = 5/100 = 1/20 inch
Accuracy
Tape measure: 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 )
The laser device is more precise.
The tape measure is more accurate.
awesome!
3-B
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?
NO!
300001 = 300000 + 1
3-C
Combining Measured Numbers
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
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
3-C
Combining Measured Numbers
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.
Homework:
Pages 181-184
28, 32, 34, 41, 59, 62, 64, 66, 70, 74, 76