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Chapter 6 - Probability - Coggle Diagram
Chapter 6 - Probability
6-1 Introduction to Probability
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 sample data is used as a basis for drawing conclusions about populations,
Defining Probability
Probability
- a fraction or proportion of all possible outcomes
used to predict future, uncertain outcomes
proportion
- part of a whole
expressed as fraction or decimal
all possible probability values are contained in a limited range
zero = 0% - an event never occurs
1 = 100% - even always occurs
Random Sampling
- process in which it is required that each individual in a population has an equal chance of being selected
simple random sample
- sample obtained via random sampling process
independent random sampling
- procedure that requires that each individual has an equal chance of being selected AND that the probability of being selected stays constant from one selection to the next if more than one is selected
independent random sample (random sample)
- sample obtained via independent random sampling
Consequences of the 2 requirements of random sampling:
1st requirement assures that there is no bias in the selection process; prohibits application of probability definition to situations in which the possible outcomes are not equally likely
2nd requirement for random samples demands sampling with replacement
sampling with replacement
- process to keep probabilities from changing from one selection to the next, necessary to return each individual to the population before making the selection
Probability and Frequency Distributions
Whenever a population is presented in a frequency distribution graph, it is possible to represent probabilities as proportions on the graph
6-2 Probability and the Normal Distribution
Normal Distribution
exact shape of the normal distribution is specified by equation:
a distribution is normal if and only if it has all the right proportions
Normal distribution following a z-score transformation:
The Unit Normal Table
- table that lists proportions of the normal distribution for a full range of possible z-score values
Has a 4 column formation: Column A - first column; lists z-score values corresponding to different positions in a normal distribution. Columns B and C identify the proportion of the distribution in each of the two sections (body - column B and tail -column C). Column D identifies the proportion of the distribution located between the mean and the z-scores.
Remember:
Body always corresponds to the larger part of the distribution (regardless of left or right side). Tail is always smaller section.
Proportions on right are exactly the same as corresponding proportions on the left
proportions are always positive (even though z-score values change sings)
Probabilities, Proportions, and z-Scores
- use Unit Normal Table (in appendix b) to find probabilities and proportions corresponding to specific z-score values
6-3 Probabilities and Proportions for scores from a Normal Distribution
In some situations, it is necessary to find probabilities for specific X values - problem is asking for a specific probability or proportion of a normal distribution
To answer probability questions about scores (X values) from a normal distribution, follow two-step procedure:
Transform the X values into z-scores.
Use the unit normal table to look up the proportions corresponding to the z-score values
To find proportions/probabilities located between 2 scores:
can be solved using proportions of columns B and C (body and tail)
easier to solve with the proportions listed in column D.
determine the z-score corresponding to the x value at each end of the interval
6-4 Percentiles and Percentile Ranks
percentile rank
- the percentage of individuals in the distribution with scores at or below that particular score
refers to a percentage of the distribution
percentile
- the particular score associated with a percentile rank
refers to a score
Finding Percentiles
Find z-score using unit normal table corresponding to given
convert the z-score to an X value
Quartiles
Percentiles divide the distribution in to 100 equal parts, each corresponding to 1% of the distribution.
Area in a distribution can also be divided into four equal parts called
quartiles
, each corresponding to 25%
1st quartile (Q1) is the score that separates the lowest 25% of the distribution from the rest; thus it is the 25th percentile
2nd quartile (Q2) is the score that has 50% (2 quarters) of the distribution below it; the median or 50th percentile of the distribution
3rd quartile (Q3) is the X value that has 75% (3 quarters) of the distribution below it; also the 75th percentile.
For a normal distribution
1st quartile always corresponds to z = -0.67
2nd quartile corresponds to z= 0.00 (mean)
3rd quartile corresponds to z = +0.67
Interquartile Range
distance between the first and third quartile
(this formula is only used for normally distributed population.)
