BMGT230 Final Exam Info PDF

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BMGT230B Final Exam – DIAGNOSTICS ON Chapter 6 – Correlation & Linear Regression • Scatterplot: plots one quantitative variable against another; explanatory x-axis, response y-axis o Direction – positive, negative, no direction o Strength – how much variation or scatter there (weak, strong, somewhat strong…) o Form – linear, curved, clustered, no pattern o Outliers – data that has low probability of occurrence (unusual/unexpected) • Standardizing using Z scores !!! OR (value-mean)/SD o 𝑍 =! !

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• • • •

(!!!)(!!!) !! !!

!! !!

Correlation Coefficient 𝑟 = !

= !!! !!! o between -1 and +1; measures direction and strength; 0 is no relationship o calculated using mean and SD of both x and y variable; only 2 quantitative variables o no units of measurement; sensitive to unusual observations Ecological Correlations – based on rates/averages, tend to overstate strength of associations Correlation is NOT causation – not enough evidence that change in x causes a change in y Linear Model – equation of a straight line through the data o Residual: 𝑒 = 𝑦 − 𝑦 or the distances from each point to the LSRL and yhat is the predicted value Least Squares Regression Line/Line of Best Fit 𝑦 = 𝑏! + 𝑏! 𝑥 o Sum of residuals always 0 !! o b1 is the slope of the line or estimated change in y with one unit change of x: 𝑏! = 𝑟 !!

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o b0 is the y intercept, average value of y when x is 0: 𝑏! = 𝑦 − 𝑏! 𝑥 Correlation determination, r2, is the percentage of variance in y that can be explained by changes in x Residual Plot – if CURVED, your relationship is not linear; change in variability across plot (ex. Increasing spread) is a warning o BEWARE of lurking, AVOID extrapolating, REMEMBER correlation does not mean causation, LOOK for outliers

Chapter 11 – Confidence Intervals & Hypothesis Tests for Means ! • Sampling Distribution SD !, Mean is 𝑋, SE(𝑋) •

Central Limit Theorem: When randomly sampling from any population with mean 𝜇 and SD s and when n is large enough, ! 𝑤ℎ𝑒𝑟𝑒!𝑛 ≥ 40!𝑜𝑟!25, larger sample = better approximations the sampling distribution is approximately normal. 𝑁 𝜇,

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∗ Confidence Interval: 𝑋 ± 𝑡!!! ∗ 𝑆𝐸 𝑌 !𝑜𝑟!𝑋 ± 𝑀𝐸

!

!

o

Standard Deviation 𝑠 =

o

Standard Error of Mean!!!

(𝑥! − 𝑥)! STAT-Edit-L1 then STAT-CALC-1-VarStats

!!! ! !

∗ and Margin of error: 𝑚 = 𝑡!!!

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T distributions

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Hypothesis: 𝐻! : 𝜇! = 𝑛𝑜!𝑒𝑓𝑓𝑒𝑐𝑡!𝑎𝑛𝑑!𝐻 ! : 𝜇 ≠≤≥ 𝑒𝑓𝑓𝑒𝑐𝑡! ! !∗! ∗ Sample Size: 𝑀𝐸 = 𝑍 ∗ or 𝑛 = ( )! ; if n is LESS THAN 60, 𝑀𝐸 = 𝑡!!!

! !

!

, use tcdf(p/-e^99, p/e^99, df)

o

!

!"

! !

∗ !!!! ! ! ) !"

!or!𝑛 = (

Chapter 12 – Comparing Two Groups - 𝑝!∝!𝑓𝑎𝑖𝑙!𝑡𝑜!𝑟𝑒𝑗𝑒𝑐𝑡!𝑡h𝑒!𝑛𝑢𝑙𝑙!h𝑦𝑝𝑜𝑡h𝑒𝑠𝑖𝑠! • Two Sample Z o

•

𝑍=

!! !!! !(!! !!! ) ! !! !!!! !! !!

; 𝑆𝐸 =

!!! !!

+

!!! !!

and df is equal to the smallest of (𝑛! − 1, 𝑛! − 1)

Two Sample T o Hypothesis: 𝐻! : 𝜇! − 𝜇! = 0!𝑎𝑛𝑑!𝐻! : 𝜇 ≠≤≥ 0! o

o

𝑡!" =

!! !!! !! ! ! !! !! ! !! !!

, use tcdf, with df being the smallest of (𝑛! − 1, 𝑛! − 1)

∗ Confidence Interval: (𝑥! − 𝑥! ) ± 𝑡!"

!!!

!! ! !(! !!)! ! !! !! !! ! !

!!

+ !!

!

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Pooled Standard Deviation 𝑠!! =

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USE POOLED WHEN IT SAYS “Normally distributed with equal variances” o

CI: 𝑥! − 𝑥! ± 𝑡 ∗ 𝑠!

!! !!!!! !

!!

+

! !!

where df = 𝑛! + 𝑛! − 2

and hypothesis test 𝑡!" =

(!! !!!) !!

! ! ! !! !!

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Paired T Test o Hypothesis: 𝐻! : 𝜇! = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒!𝑜𝑟!𝑛𝑜!𝑒𝑓𝑓𝑒𝑐𝑡!𝑎𝑛𝑑!𝐻 ! :!𝜇 ≠≤≥ 𝑒𝑓𝑓𝑒𝑐𝑡; Reject null means there’s an effect o o o

! !!!!!

Mean Paired Difference: 𝑑 = Confidence Interval: 𝑑

!

!! ∗ ± 𝑡!!! ! !!!!

Hypothesis Testing: 𝑡 =

! (! !!)! !!! !

where 𝑠! =

!!!

!! !

Chapter 14 – Inference for Regression (degrees of freedom for critical t n-2) • 𝐹𝐼𝑇:!𝜇! = 𝑏! + 𝑏! 𝑥 and Data = fit + residual (ei) •

Regression Standard error, 𝒔𝒆 =

(𝒚𝒊 !𝒚𝒊 )𝟐 𝒏!𝟐

, granted it’s independent, linear,

normal variation, SD same for all values of x !

!!

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CI for b0, 𝑏! ± 𝑡 ∗ 𝑆𝐸(𝑏! ) where 𝑆𝐸 𝑏! = 𝑠!

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CI for b1, 𝑏! ± 𝑡 ∗ 𝑆𝐸(𝑏! ) where 𝑆𝐸 𝑏! =

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Significance test for the slope o We can test the hypothesis H0: b1 = 0 versus a 1 or 2 sided alternative. o We calculate t = b1 / SE(b 1) which has the t (n – 2) distribution to find the p-value of the test. Testing the Hypothesis of no relationship o test the hypothesis that the regression slope parameter β is equal to zero. o H0: β1 = 0 vs. H0: β1 ≠ 0 IF YOU REJECT THE NULL, It means it is significant o Testing H0: β1 = 0 also allows to test the hypothesis of no correlation between x and y in the population. o Slope b1 = r (s(y) / s(x)) Confidence interval for µy = y ± tn − 2 * SE(µ) The standard error of the mean response µy is:

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SE(µˆ ) = SE 2 (b1 )(x * −x )2 +

o •

!! (!! !!)!

(!! !!)! !!

=

!! !!!

1 (x * −x )2 se2 + = se n n ∑(xi − x )2

The standard error for predicting an individual response ŷ is:

ˆ = SE 2 (b1 )(x * −x )2 + SE(y)

o • • •

+

!

1 (x * −x )2 se2 2 + se = se 1+ + n n ∑(xi − x )2

CI used to measure accuracy of mean response of all individuals in a population (Parameter) Prediction Intervals – measure accuracy of single individual’s predicted value (single random variable) Outlier – large residual/distance, influential if it gives regression model different slope without it

Chapter 15 – Multiple Regression • ANOVA table, k is the number of predictor variables df Sum of squares Mean of squares F P value Model (regression) k SSR (explained) MSR=SSR/k MSR/MSE Error (residual) n-k-1 SSError MSError=SSError/(n(unexplained) k-1) Total n-1 SST=SSError+SSR MST=SST/(n-1) !!" !!" =1− • 𝑅! = tells what percentage of the variation in the response is explained by the linear relationship between y !!"

!!"

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and predictions !"# decreases when you add predictors that aren’t significant Adjusted 𝑅 ! = 1 −

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Testing the Model 𝐻! : 𝐵! = 𝐵! … 𝐵! , !𝐻! : 𝐴𝑡!𝑙𝑒𝑎𝑠𝑡!1!𝛽!𝑖𝑠!𝑛𝑜𝑡!0; REJECT NULL = LINEAR RELATIONSHIP

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∗ CI: 𝑏! ± 𝑡!!!!! 𝑆𝐸 𝑏! !𝑤ℎ𝑒𝑟𝑒!𝑆𝐸!𝑖𝑠! !

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𝑒 =𝑦−𝑦

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𝑆! =

!"#

!!

! !!!

!!"##$# !!!!!

and 𝑆! =

(!!!)! !!!

𝐻! : 𝐵! = 0, !𝐻! :!𝐵! ≠ 0

𝑡=

!! !"(!!)

!𝑜𝑟! 𝑀𝑆𝐸𝑟𝑟𝑜𝑟!

EXCESS • If 0 is within the confidence interval, it means THERE IS NO SIGNIFICANCE IN PROPORTIONS BETWEEN X AND Y...