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