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MAT 110 Vocabulary Unit 2 Probability and Counting SYSTEMATIC COUNTING: counting a set by listing its elements TREE DIAGRAM: displays all possible outcomes of an event FUNDAMENTAL COUNTING PRINCIPAL (FCP): to count the total number of different ways a series of tasks can be performed, multiply together the number of ways each task can be performed SLOT DIAGRAM: a pictorial representation to organize a counting problem PERMUTATION: an ordering of distinct objects in a straight line. The number of ways to order r objects π! from a set of n objects is denoted π(π, π). π(π, π) = (πβπ)! FACTORIAL NOTATION: π! = π × (π β 1) × (π β 2) ×βββ× 3 × 2 × 1. By definition 0! = 1. COMBINATIONS: choosing r objects from a set of n objects (order does not matter). Notation: πΆ(π, π). π! Read as βn choose r.β πΆ(π, π) = π!(πβπ)!. RANDOM: a phenomenon where individual outcomes are uncertain EXPERIMENT: any observations of a random phenomenon OUTCOME: a possible result of an experiment SAMPLE SPACE: the set of all possible outcomes PROBABILITY OF AN OUTCOME OF AN EXPERIMENT: a proportion of the number of times an outcome occurs in a very large number of repetitions. EVENT: a subset of a sample space. π·(π¬): read as βthe probability of event E.β 0 β€ π(πΈ) β€ 1. Also, the sum of the probabilities of all the outcomes in a sample space must be 1. Emperical Data: data resulting from an experiment. π·(π¬) = πππ ππππππ ππ πππππ π¬ ππππππ πππ ππππππ ππ πππππ πππ ππππππππππ ππ πππππππππ Theoretical Data: what should happen based on combination formulas Equally likely events: every event, E, in the sample space, S, has the same probability π(πΈ) = π(πΈ) π(π) = π‘βπ ππ’ππππ ππ ππ’π‘πππππ ππ πΈ π‘βπ ππ’ππππ ππ ππ’π‘πππππ ππ π UNION: If S is a sample space and E and F are events, then the union of E or F, denoted πΈ βͺ πΉ, is the set of all outcomes in E or F. (Include all elements that are present in one or the other or both.) MAT 110 Vocabulary Unit 2 INTERSECTION: If S is a sample space and E and F are events, then the intersection of E and F, denoted πΈ β© πΉ, is the set of all outcomes in E and F. (Only elements that the sets have in common.) COMPLEMENT: If S is a sample space and E is an event, then the complement of E, denoted E , is the set of all outcomes in S that are not in E. PROPERTIES OF PROBABILITY: 1. 0 β€ π(πΈ) β€ 1 2. π(β ) = 0 3. π(π) = 1 4. π ( E ) = 1 β P(E) 5. π(πΈβπΉ) = π(πΈ) + π(πΉ) β π(πΈ β© πΉ) Statistics STATISTICS: an area of mathematics in which we gather, organize, analyze and make predictions from numerical information called DATA or a DISTRIBUTION SAMPLE: a subset of a population BIASED: does not accurately reflect the population as a whole β 2 types: SELECTION and LEADINGQUESTION BIAS FREQUENCY DISTIBUTION: a set of data listed with their frequencies RELATIVE FREQUENCY DISTRIBTION: showing the percent of time that each item occurs in a frequency distribution (usually expressed as a decimal) REPRESENTATION OF DATA: See pages 716-722 for specific examples: FREQUENCY TABLE, BAR GRAPH, HISTOGRAM, and STEM AND LEAF DISPLAY Bin: a range of scores MEASURES OF CENTRAL TENDENCY: mean, median, mode and quartiles MEAN: average of the data (notation π₯Μ or µ) MEDIAN: the middle number in a list of data when arranged in order. If there are two middle numbers, then they must be averaged MODE: the number that appears most frequently. (There can be two modes if two numbers appear the most or no mode if more than two scores appear most frequently.) FIVE-NUMBER SUMMARY: Lowest value, Q1 (Median of the lower half), MEDIAN, Q3 (Median of the upper half), Highest value MAT 110 Vocabulary Unit 2 BOX AND WHISKER PLOT: a graph that represents the five-number summary: Q2 Q1 Min Q3 Max source: ellerbruch.nmu.edu RANGE: the difference between the largest and smallest data values (Largest # minus smallest #) DEVIATION FROM THE MEAN: distance a number is form the mean (negative if below the mean, positive if above) ( x - π₯Μ ) STANDARD DEVIATION : a measure based on the distance each data value is from the mean (s or Ο) β(π₯β π₯Μ )2 π = β Formula: 1. 2. 3. 4. 5. πβ1 which is to: Find the mean for the data Find the deviation from the mean for each Square the values in 2. Add up the squares and then divide by n β 1 (the value at this point is called VARIANCE) Calculate the square root of the answer to step 4 π COEFFICIENT OF VARIATION: CV = π₯Μ β 100 = % NORMAL CURVE or NORMAL DISTRIBUTION: the most common distribution and describes many reallife data sets; Bell-shaped curve; mean median and mode the same; symmetric with respect to the mean; area under the curve equals 1 34% 2.5% 13.5% 34% 13.5% 2.5% Source: www.syque.com Roughly 68% of data is with 1 standard deviation from the mean, 95% are within 2 standard deviations from the mean and 99.7% are within 3 standard deviations of the mean. MAT 110 Vocabulary Unit 2 Z β SCORE: the number of standard deviations a data value is from the mean. If a normal distribution has a mean of µ and a standard deviation of Ο then you can convert a π₯β π data value, x, to a z-score with the formula: π§ = π LEVEL C CONFIDENCE INTERVAL (2 parts) 1. Interval calculated from the data in the form of estimate ± margin of error 2. Confidence level C: the probability the interval will capture the true mean in repeated samples. A confidence level is the rate of success for the method. CONFIDENCE INTERVAL FOR THE MEAN, π, OF A NORMAL POPULATION xο± z* ο³ n , where the margin of error is z * ο³ n