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Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions
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Page 1: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

Day 2 Review

Chapters 5 – 7Probability, Random Variables, Sampling

Distributions

Page 2: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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.

Page 3: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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)

Page 4: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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!

Page 5: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 6: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

Expected Value

• The Expected Value, E(X) = μ, is the long-term average value of a Random Variable.

Page 7: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 8: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

Means and Variances of Random Variables

• The following rules are helpful when working with Random Variables.

Page 9: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 10: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 11: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 12: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

Geometric Probabilities

If X is Geometric, the following formulas can be used to calculate the probabilities of events in X.

Page 13: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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:

Page 14: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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

Page 15: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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)

Page 16: Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.

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)


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