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Probability - Coggle Diagram
Probability
Probability & Inferential Statistics
Probability is used to predict the type of samples that are likely to be obtained from a population
Inferential Statistics rely on the connections between population & samples as a basis for making conclusions about populations
Probability:
for any specific outcome is defined as a fraction or a proportion of all possible outcomes
Probability of A
= number of outcomes classified as A divided by total number of possible outcomes
Necessary that the outcomes be obtained by random sampling
1. Random Sample:
requires each individual in the population has an equal chance of being selected
Sample then becomes a
simple random sample
2. Independent Random Sample:
requires each individual has an equal chance of being selected and the probability of being selected stays constant from one selection to the next if more than one individual is selected
3. Sampling with replacement:
a sampling technique that returns the current selection to the population before the next selection is made
Probability & Normal Distributions
To find probability within a normal distribution:
1. Probability questions is translated into a proportion question
"Out of all the possible SAT scores, what proportion is greater than 700"
Mean = 500 & Standard Deviation = 100
p (X greater than 700) = ?
2. Set of "all possible SAT scores" = population distribution
Mean = 500 & X=700
3. Identify the exact position of X = 700 by computing the z-score
z = (X-M) / SD
z = (700 - 500) / 100
z = +2.00
4. Proportion we are trying to find may now be expressed in terms of its z-score
p (z greater than 2.00) = ?
p ( X is greater than 700) = p ( z is greater than +2.00) = 2.28%
The Unit Normal Table
: a table listing proportions corresponding to each z-score location in a normal distribution
Table is structured in a 4 column format
Column A
- lists z-scores values corresponding to different positions in a normal distribution
Column B
- identifies proportions in the body
Column C
- identifies proportions in the tail
Column D
- proportions of the distribution that is located between the mean & z-score
Remember:
Body
= larger part of the distribution on either side
Tail
= smaller portion on either side
Normal distributions are symmetrical
proportions on the right side are exactly the same as the corresponding proportions on the left side
Z-score values change the signs (+ & -) from one side to the other but the proportions are always positive
Probability & Binomial Distributions
Binomial Distribution:
show probability associated with each value of X from X = 0 to X = n
Two categories are identified as A &B
probabilities (or proportions) associated with each category are identified as
p = p(A) = probability of A
q = p(B) = probability of B
p + q = 1.00 because A & B are the only two possible outcomes
Number of individuals or observations in the sample is identified by n
The Normal Approximation to the Binomial Distribution
Value of 10 fro pn or qn is the general guide
Mean
= pn
Standard Deviation
= Square Root of (npq)
Z-score = (X - Mean) / by the Standard Deviation
Z-score = (X - pn) / the square root of (npq)
Probability & Frequency Distributions
graph representing the entire population, then different portions of the graph represent different portions of the populations
probabilities & proportions are equivalent, therefor a particular portion of the graph corresponds to a particular probability in the population