Statistics and Probability MATH 2160 4th Exam Review Statistics and Probability

Problem Solving – Polya’s 4 Steps Understand the problem What does this mean? How do you understand? Devise a plan What goes into this step? Why is it important? Carry out the plan What happens here? What belongs in this step? Look back What does this step imply? How do you show you did this?

Problem Solving Polya’s 4 Steps Which step is most important? Understand the problem Devise a plan Carry out the problem Look back Which step is most important? Why is the order important? How has learning problem solving skills helped you in this or another course?

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

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

Statistics Measures of Central Tendency Measures of dispersion Mean Median Mode Measures of dispersion Range

Statistics Experimental Error Absolute value of the difference between an experimental or estimated value and a theoretical or known value divided by the theoretical or known value

Statistics Types of Data Interval/ratio – temperatures, prices, test scores, etc. Ordinal – low, medium, high Nominal – red, blue, green Discrete – people, keys, desks, etc. Ordinal Nominal Interval/ratio Continuous - temperatures, prices, test scores, etc.

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

Statistics Line graph Sometimes does a better job of showing fluctuation in data and emphasizing changes Uses and reports same information as bar graph

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 Line Plot Keys in Pockets: 1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5 Line Plot

Examples Bar Graph Keys in Pockets: 1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5 Bar Graph

Examples Line Graph Keys in Pockets: 1, 2, 2, 3, 5, 6, 8, 5, 2, 2, 4, 1, 1, 3, 5 Line Graph

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

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