Download 26134 Business Statistics

26134 Business Statistics
[email protected]
Maths Study Centre 11am to 5pm
CB04.03.331
Tutorial 8: Probability Distribution
Key concepts in this tutorial are listed below
1. REVISION
2. Random Variables
3. 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 rule
5. Applications of Normal Distribution in calculating probabilities.
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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 inferenceUnderstanding events and using data to calculate the probability of
occurrence of an event.
The specific topics for this threshold are:
1.) Lecture Topic: Probability
The sample quiz will be uploaded one week prior to the quiz.
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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)
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REVISION: Regression
• Regression Equation
yˆ   0   1X 1   2X 2  .....
•
•
•
•
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Re sponse   0   1(Pr edictor1)   2(Pr edictor 2)  ....
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
1. Random Variables
2. Identifying Normal Distribution
3. 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
4. Applications of Normal Distribution in calculating probabilities.
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Random Variables and Normal
Distribution
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• 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:
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P[X<x]=
(𝑥−𝜇)
𝑥
1
−
2𝜎2
𝑒
−∞ 2𝜋𝜎
𝑑𝑥
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:
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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)
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X=marks for assignment out of 50
X~N(μ=0.7*50=35, σ=√100/100*50=5)=N(μ=35,σ=5)
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END OF TUTORIAL
NEXT WEEK: CONFIDENCE
INTERVALS
[email protected]
Maths Study Centre 11am to 5pm
CB04.03.331
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