Download MATH 2160 1st Exam Review

MATH 2160 4th Exam
Review
Statistics and Probability
Problem Solving
• Polya’s 4 Steps
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Understand the problem
Devise a plan
Carry out the problem
Look back
Problem Solving
• Strategies for Problem Solving
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–
–
–
–
–
–
–
–
–
Make a chart or table
Draw a picture or diagram
Guess, test, and revise
Form an algebraic model
Look for a pattern
Try a simpler version of the problem
Work backward
Restate the problem a different way
Eliminate impossible situations
Use reasoning
• Mean
Statistics
– Most widely used measure of central
tendency
– Arithmetic mean or average
– Sum the terms and divide by the number of
terms to get the mean
– Good for weights, test scores, and prices
– Effected by extreme values
– Gives equal weight to the value of each
n
measurement
xi

x1  x 2  x3    x n
or
i 1
x

x
n
n
Statistics
• Median
– Put the data in order first
– Odd number of data points choose the
middle term
– Even number of data points take the
average of the middle two terms
– Used when extraordinarily high or low
numbers are included in the data set
instead of mean
– Can be considered to be a positional
average
Statistics
• Mode
– The mode occurs most often. If every
measurement occurs with equal
frequency, then there is no mode. If the
two most common measurements occur
with the same frequency, the set of data
is bimodal. It may be the case that
there are three or more modes.
– Used when the most common
measurement is desired
– Finding the best tasting pizza in town
Statistics
• Range
– The difference of the highest and
lowest terms
– Highest – lowest = range
– Radically effected by a single extreme
value
– Most widely used measure of dispersion
• Line Plot
Statistics
– Useful for organizing
data during data
collection
– Categories must be
distinct and cannot
overlap
– Not beneficial to use
with large data sets
0
1
2
3
4
5
0
1
2
3
1
2
1
1
2
3
-1
2
5
4
3
1
1
1
1
1
1
3
3
4
2
2
2
0
Weird Horse Race
1
2
3
Winning Horses
4
5
6
– Another way of
representing data
from a frequency
line plot
– More convenient
when frequencies
are large
Weird Horse Race
Number of Wins
• Bar graph
Statistics
6
5
4
3
2
1
0
zero
one
two
three
Winning Horse
four
five
– Sometimes does a
better job of
showing fluctuation
in data and
emphasizing changes
– Uses and reports
same information as
bar graph
Weird Horse Race
Number of Wins
• Line graph
Statistics
6
5
4
3
2
1
0
zero
one
two
three
Winning Horse
four
five
Examples
Test scores:
89, 73, 71, 46, 83, 67, 83, 74, 76, 79,
81, 84, 105, 84, 85, 99, 48, 74, 60,
83, 75, 75, 82, 55, 76
Mean
=
Sum of scores/Number of scores
=
1906/25
=
76.25
Examples
Test scores:
46, 48, 55, 60, 67, 71, 73, 74, 74, 75,
75, 76, 76, 79, 81, 82, 83, 83, 83, 84,
84, 85, 89, 99, 105
Median = 76
Mode = 83
Range = 105 – 46 = 59
Examples
Keys in Pockets:
1, 2, 2, 3, 5, 6, 8, 5, 2,
2, 4, 1, 1, 3, 5
9
8
Number of Keys
Line Plot
Keys in Pockets
7
6
5
4
3
2
1
0
Examples
Keys in Pockets:
1, 2, 2, 3, 5, 6, 8, 5,
2, 2, 4, 1, 1, 3, 5
Keys in Pockets
8
7
6
5
Bar Graph
Keys
4
3
2
1
0
1
2
People
3
4
5
Examples
Keys in Pockets:
1, 2, 2, 3, 5, 6, 8, 5,
2, 2, 4, 1, 1, 3, 5
Keys in Pocket
Line Graph
Number of People
5
4
3
2
1
0
1
2
3
4
5
Number of Keys
6
7
8
•
•
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•
•
Probability
Sample space – ALL possible outcomes
Experiment – an observable situation
Outcome – result of an experiment
Event – subset of the sample space
Probability – chance of something
happening
• Cardinality – number of elements in a set
Probability
• 0  P(E)  1
• P() = 0
• P(E) = 0 means the event can NEVER
happen
• P(E) = 1 means the event will
ALWAYS happen
Probability
• P(E’) is the compliment of an event
• P(E) + P(E’) = 1
• P(E’) = 1 – P(E)
Probability
• Experiment Examples
– Sample Spaces
•
•
•
•
One coin tossed: S = {H, T}
Two coins tossed: S = {HH, HT, TH, TT}
One die rolled: S = {1, 2, 3, 4, 5, 6}
One coin tossed and one die rolled:
S=
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
Probability
• Experiment Examples
– Cardinality of Sample Spaces
• One coin tossed: S = {H, T}
– n(S) = 21 = 2
• Two coins tossed: S = {HH, HT, TH, TT}
– n(S) = 22 = 4
• One die rolled: S = {1, 2, 3, 4, 5, 6}
– n(S) = 61 = 6
• One coin tossed and one die rolled:
S=
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
– n(S) = 21 x 61 = 12
Probability
• Probability of Events
– What is the probability of choosing a prime
number from the set of digits?
•
•
•
•
•
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
E = {2, 3, 5, 7}
n(S) = 10 and n(E) = 4
P(E) = n(E)/n(S) = 4/10 = 2/5 = 0.4
The probability of choosing a prime number from
the set of digits is 0.4
Probability
• Probability of Events
– What is the probability of NOT
choosing a prime number from the set
of digits?
•
•
•
•
•
•
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
E = {2, 3, 5, 7}
n(S) = 10 and n(E) = 4
P(E) = n(E)/n(S) = 4/10 = 2/5 = 0.4
P(E’) = 1 – P(E) = 1 – 0.4 = 0.6
The probability of NOT choosing a prime
number from the set of digits is 0.6
I think you all will
probability pass
this test without
any trouble!! 
Just like puttin’
money in the bank!!!

Test Taking Tips
• Get a good nights rest before the
exam
• Prepare materials for exam in
advance (scratch paper, pencil, and
calculator)
• Read questions carefully and ask if
you have a question DURING the
exam
• Remember: If you are prepared, you
need not fear