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MATH 2160 4MATH 2160 4thth Exam Exam ReviewReview
MATH 2160 4MATH 2160 4thth Exam Exam ReviewReview
Statistics and ProbabilityStatistics and Probability
Problem Solving• Polya’s 4 Steps
– Understand the problem– Devise a plan– Carry out the problem– Look back
Problem Solving• Strategies for Problem Solving
– 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
Statistics• Mean
– 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
measurement or
n
xxxxx n
321
n
x
x
n
ii
1
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
0 21 52 43 34 15 10 11 12 13 11 32 31 41 22 23 2
Weird Horse Race
-1 0 1 2 3 4 5 6
Winning Horses
Statistics• Line Plot
– Useful for organizing data during data collection
– Categories must be distinct and cannot overlap
– Not beneficial to use with large data sets
Statistics• Bar graph
– Another way of representing data from a frequency line plot
– More convenient when frequencies are large
Weird Horse Race
01
23
45
6
zero one two three four five
Winning Horse
Nu
mb
er
of
Win
s
Statistics• Line graph
– Sometimes does a better job of showing fluctuation in data and emphasizing changes
– Uses and reports same information as bar graph
Weird Horse Race
01
23
45
6
zero one two three four five
Winning Horse
Nu
mb
er o
f W
ins
ExamplesTest 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
ExamplesTest 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 = 76Mode = 83Range = 105 – 46 = 59
Examples
Keys in Pockets: 1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5
Line Plot
Keys in Pockets
0
1
2
3
4
5
6
7
8
9
Num
ber o
f Key
s
Examples
Keys in Pockets: 1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5
Bar Graph
Keys in Pockets
0 1 2 3 4 5
1
2
3
4
5
6
7
8
Keys
People
ExamplesKeys in Pockets:
1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5
Line Graph
Keys in Pocket
0
1
2
3
4
5
1 2 3 4 5 6 7 8
Number of Keys
Num
ber o
f Peo
ple
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