day 2 review chapters 5 – 7 probability, random variables, sampling distributions
TRANSCRIPT
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Day 2 Review
Chapters 5 – 7Probability, Random Variables, Sampling
Distributions
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Probability
• A measurement of the likelihood of an event. It represents the proportion of times we’d expect to see an outcome in a long series of repetitions.
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Formulas
The following facts/formulas are helpful in calculating
and interpreting the probability of an event:
• 0 ≤ P(A) ≤ 1
• P(SampleSpace) = 1
• P(AC) = 1 - P(A)
• P(A or B) = P(A) + P(B) – P(A and B)
• P(A and B) = P(A) P(B|A)
• A and B are independent iff P(B) = P(B|A)
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Hints and Tricks
• When calculating probabilities, it helps to consider the Sample Space.
• List all outcomes if possible.
• Draw a tree diagram or Venn diagram
• Use the Multiplication Counting Principle
• Sometimes it is easier to use common sense rather than memorizing formulas!
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Random Variables
• A Random Variable, X, is a variable whose outcome is unpredictable in the short-term, but shows a predictable pattern in the long run.
• Discrete vs. Continuous
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Expected Value
• The Expected Value, E(X) = μ, is the long-term average value of a Random Variable.
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Variance of Expected Value
• The Variance, Var(X) = ,is the amount of variability from μ that we expect to see in X.
• The Standard Deviation of X,
• Var(X) for a Discrete X
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Means and Variances of Random Variables
• The following rules are helpful when working with Random Variables.
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Binomial Settings
Some Random Variables are the result of events that have only two outcomes (success and failure).
We define a Binomial Setting to have the following features
• Two Outcomes - success/failure
• Fixed number of trials - n Independent trials
• Equal P(success) for each trial
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Binomial Probabilities
If conditions are met, a binomial situation may be approximated by a normal distribution
If np ≥10 and n(1-p)≥10, then B(n,p) ~ Normal
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Geometric Setting
Some Random Variables are the result of events that have only two outcomes (success and failure), but have no fixed number of trials. We define a Geometric Setting to have the following features
• Two Outcomes - success/failure
• No Fixed number of trials (go until you succeed)
• Independent trials
• Equal P(success) for each trial
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Geometric Probabilities
If X is Geometric, the following formulas can be used to calculate the probabilities of events in X.
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Parameters and Statistics
Our goal in statistics to to gain information about the population by collecting data from a sample.
• Parameter Population Characteristic: μ, p (or π)
• Statistic Sample Characteristic:
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Sampling Distributions
• When we take a sample, we are not guaranteed the statistic we measure is equal to the parameter in question.
• Repeated sampling may result in different statistic values.
• Bias and Variability
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Proportions
The distribution of p-hat is approximately normal if:
• If the population proportion is p or (π) andnp≥10, n(1-p)≥10
• population>10n (10% rule)
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Means
The sampling distribution of x-bar is approximately normal if:
• If the population mean is μ (and we know )
• If the population is Normal OR n≥30 (Central Limit Theorem)