Download MAT 110 Vocabulary Unit 2

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
𝑛!
from a set of n objects is denoted 𝑃(𝑛, 𝑟). 𝑃(𝑛, 𝑟) = (𝑛−𝑟)!
FACTORIAL NOTATION: 𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) ×∙∙∙× 3 × 2 × 1. By definition 0! = 1.
COMBINATIONS: choosing r objects from a set of n objects (order does not matter). Notation: 𝐶(𝑛, 𝑟).
𝑛!
Read as “n choose r.” 𝐶(𝑛, 𝑟) = 𝑟!(𝑛−𝑟)!.
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.
𝑷(𝑬): read as “the probability of event E.” 0 ≤ 𝑃(𝐸) ≤ 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.
𝑷(𝑬) =
𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝑬 𝒐𝒄𝒄𝒖𝒓𝒔
𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒕𝒉𝒆 𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕 𝒊𝒔 𝒑𝒆𝒓𝒇𝒐𝒓𝒎𝒆𝒅
Theoretical Data: what should happen based on combination formulas
Equally likely events: every event, E, in the sample space, S, has the same probability
𝑃(𝐸) =
𝑛(𝐸)
𝑛(𝑆)
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐸
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑆
UNION: If S is a sample space and E and F are events, then the union of E or F, denoted 𝐸 ∪ 𝐹, 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
𝐸 ∩ 𝐹, 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 ≤ 𝑃(𝐸) ≤ 1
2. 𝑃(∅) = 0
3. 𝑃(𝑆) = 1
4. 𝑃 ( E ) = 1 − P(E)
5. 𝑃(𝐸⋃𝐹) = 𝑃(𝐸) + 𝑃(𝐹) − 𝑃(𝐸 ∩ 𝐹)
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 – 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 𝑥̅ 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 - 𝑥̅ )
STANDARD DEVIATION : a measure based on the distance each data value is from the mean (s or σ)
∑(𝑥− 𝑥̅ )2
𝑠= √
Formula:
1.
2.
3.
4.
5.
𝑛−1
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 – 1 (the value at this point is called VARIANCE)
Calculate the square root of the answer to step 4
𝑠
COEFFICIENT OF VARIATION: CV = 𝑥̅ ∙ 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 – 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 σ then you can convert a
𝑥− 𝜇
data value, x, to a z-score with the formula: 𝑧 = 𝜎
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, 𝝁, OF A NORMAL POPULATION
x z*

n
, where the margin of error is z *

n