26134 Business Statistics Mahrita.Harahap@uts.edu.au

Maths Study Centre CB04.03.331Tutorial 8: Probability Distribution

Key concepts in this tutorial are listed below 1. REVISION

2. Random Variables3. Identifying Normal Distribution

4. Normal Distribution: Z-Score, Converting non-standard normal distributions into standard normal distributions,

Calculating Probabilities using normal distribution applying the complement rule and/or symmetry rule

and/or interval rule5. Applications of Normal Distribution in calculating

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The topics to be tested in Quiz 2 are:THRESHOLD 3: Relating variables and analyzing relationships between variables.The specific topics for this threshold are: 1.) Lecture Topic: Simple Linear Regression 2.) Lecture Topic: Multiple Linear Regression 3.) Lecture Topic: Issues with Regression THRESHOLD 4: Theoretical foundation of statistical inference-Understanding events and using data to calculate the probability of occurrence of an event.The specific topics for this threshold are: 1.) Lecture Topic: ProbabilityThe sample quiz will be uploaded one week prior to the quiz.

REVISION: ProbabilityTo find certain probabilities, first construct a contingency table (frequency table/remember to add TOTAL columns and rows), then construct a probability table (relative frequency table):• Marginal Probability (row or column total probability) P(A) • Intersection Event/Joint Event (A AND B) P( ∩ )𝐴 𝐵• Union Event (A OR B) P(A U B) = P(A) + P(B) – P(A ∩ B)• Complement Event (NOT A) P(A’)=1− (A)𝑃• Mutual Exclusive Events (cannot occur at the same time) P( ∩ )=0𝐴 𝐵• Conditional Probability (A GIVEN B) P(A|B)= P(A ∩ B)/P(B)• Independent Events P( ∩ )=P(A)*P(B)𝐴 𝐵

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REVISION: Regression• Regression Equation

Interpretation of coefficient: As “Predictor1” increases by 1 “unit” then “Response” increases/decreases by β1”units”, holding other variables as constant.

• Significance of variables: Use t-test• Significance of regression model: Use F-test• The adjusted R2 recalculates the R2 based on the number of

independent variables in the model and the sample size. Calculates the model performance.

• To predict the response based on given values of predictors, substitute those values into the regression equation to find the predicted value.

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....)2(Pr)1(PrRe

.....ˆ

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edictoredictorsponse

XXy

Random Variables and Normal Distribution

• A random variable X is defined as a unique numerical value associated with every outcome of an experiment.

• If X follows a normal distribution, then it is denoted as X~N(μ,σ)

• To find probabilities under the normal distribution, random variable X must be converted to random variable Z that follows a standard normal distribution denoted as Z~N(μ=0,σ=1). We need to do this to standardise the distribution.

• To convert random variable X to random variable Z, we calculate the z-score=(x- μ)/ σ

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Calculating Probabilities using normal distribution applying the complement rule and/or symmetry rule and/or interval rule

• Complement Rule P(Z>z)=1-P(Z<z)

• Symmetry Rule P(Z<-z)=P(Z>z)

• Interval Rule P(-z<Z<z)=P(Z<z)-P(Z<-z)

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