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Probability - Coggle Diagram
Probability
Probabilities and Proportions for Scores from a Normal Distribution
To find a probability for a specific score (X), first convert the score to a z-score, then use the standard normal (z) table. You cannot go directly from an X value to a probability; the z-score is the required intermediate step.
To find the probability between two scores, convert both scores to z-scores and use the areas in the z-table. The probability is the area under the normal curve between the two z-scores.
This method only works for normal distributions. Converting scores to z-scores does not make a non-normal distribution normal
To find a score corresponding to a given probability or percentile, work backward: use the z-table to find the appropriate z-score, then convert the z-score back to an X value using the mean and standard deviation. For example, the top 10% of a distribution corresponds to a z-score of about +1.28.
Probability and the Normal Distribution
Normal distributions are symmetrical with a single mode in the middle
-exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency)
The Unit Normal Table
Compete listing of z-scores and proportions
Table is structured into a four-column format
A. list of z-score values corresponding to different positions in a normal distribution
a vertical line separates the distribution into two sections: larger section called the body and smaller section called the tail
B. & C. identify the proportion of the distribution into two sections
B= proportion the body (larger portion)
C= proportion of the tail
D. identifies the proportion of the distribution that's located between the mean and z-score
Bc normal distribution is always symmetrical, the proportions on the right and the same as the left side
For a negative z-score, the tail of the distribution is on the left side and the body is on the right... for positive it's opposite
Although z-scores change signs (+ and -) from one side to another, the proportions are always positive
Probabilities, Porportions, and z-Scores
unit normal table lists relationships between z-score locations and proportions in normal distribution
Percentiles and Percentile Ranks
Percentile rank is defined as the percentage of individuals in the distribution with scores at or below that particular score... the score associated with a percentile rank is called percentile
Finding Percentiles
Quartiles
The first quartile (Q1) is the score that separates the lowest 25% of the distribution from the rest
The second quartile (Q2) is the score that has 50% of the distribution below it
The third quartile (Q3) is the value that has 75% of the distribution below it
A percentile is the score (X value) below which a certain percentage of data falls. For example, the 34th percentile means 34% of scores are below that value.
Step 1: Convert the percentile to a z-score using the standard normal table. You locate the proportion in the table and identify the matching z-score (including whether it is above or below the mean).
Step 2: Convert the z-score into an X value using the mean and standard deviation. This gives the actual score that corresponds to the percentile.
Introduction to Probability
Probability & Frequency Distribution
Random Sampling
Defining probability
A notation system is used to simplify the discussion of probability
The probability of a specific outcome is expressed with a p (for probability), followed by the specific outcome in parentheses
Ex: probability of obtaining heads in coin toss is written as p(heads)
We're usually concerned with the the probability of obtaining a certain sample from a population...
the terminology of sample and population will not change the basic definition of probability
Role of probability in inferential statistics
Probability is used to predict the type of samples that are likely to be obtained from a population
Probability establishes a connection between samples and populations
Inferential statistics rely on this connection when using sample data as the basis for making conclusions about populations
Typically, we use proportions to summarize previous observations and probability to predict future, uncertain outcomes
For a situation where different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all possible outcomes. If the possible outcomes are identified as A, B, C, D,...
Probability of A= number of outcomes classified as A/total number of possible outcomes
Random sampling requires each individual in the population an equal chance of being selected... a sample obtained in this way is called a
simple random sample
Independent random sampling: the probability of selecting any particular individual isn't influenced by the individuals already selected for the sample
always assume this method is being used
Sampling with replacement: to keep probabilities from changing from one selection to the next it's necessary to return each individual to the population before making the selection
Sampling with replacement, random sapling with replacement, random sampling without replacement
When a population is presented in a frequency distribution graph, it will be possible to represent probabilities as portions of the graph