Tutorial 8: Probability Distribution 26134 Business Statistics bstats@uts.edu.au Maths Study Centre 11am to 5pm CB04.03.331 Tutorial 8: Probability Distribution Key concepts in this tutorial are listed below REVISION Random Variables Identifying Normal Distribution 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 rule Applications of Normal Distribution in calculating probabilities.

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.    1.) Lecture Topic: Probability The sample quiz will be uploaded one week prior to the quiz.

REVISION: Probability To 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−𝑃(A) Mutual Exclusive Events (cannot occur at the same time) P(𝐴∩𝐵)=0 Conditional Probability (A GIVEN B) P(A|B)= P(A ∩ B)/P(B) Independent Events P(𝐴∩𝐵)=P(A)*P(B)

REVISION: Regression Regression Equation Interpretation of coefficient: As “Predictor1” increases by 1 “unit” then “Response” “increases/decreases” by β1”units”, on average, holding other variables as constant. Significance of variables: Use t-test (p-values) Significance of regression model: Use F-test (p-values) 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.

Tutorial 8: Probability Distribution Key concepts in this tutorial are listed below Random Variables Identifying Normal Distribution 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 rule Applications of Normal Distribution in calculating probabilities.

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

Why do we standardise the normal distribution to find probabilities? When we are given a problem about a random variable X~N(μ,σ) that follows a normal distribution, to find cumulative probabilities we would have to compute a definite integral: To do this computation easier, we have standard practices where we could convert the normal random variable X~N(μ,σ) into a standard normal random variable Z~N(μ=0,σ=1) (by computing a z-score=(x-μ)/σ), then we can use the z tables to find certain probabilities: P[X<x]= −∞ 𝑥 1 2𝜋 𝜎 𝑒 − (𝑥−𝜇) 2 2 𝜎 2 𝑑𝑥

Calculating Probabilities using normal distribution applying the rules We have these rules because the z-tables only give us cumulative probabilities P(Z<z) 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)

X=marks for assignment out of 50 X~N(μ=0.7*50=35, σ=√100/100*50=5)=N(μ=35,σ=5)

END OF TUTORIAL NEXT WEEK: CONFIDENCE INTERVALS Mahrita.Harahap@uts.edu.au Maths Study Centre 11am to 5pm CB04.03.331