Business Statistics Final Exam Cheat Sheet, Cheat Sheet of Statistics

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COMM 291 – Final Formula Sheet

Chapter 5: Displaying and Describing Quantitative Data

Measures of Centre Excel Mean

𝑥̅ =

∑ 𝑥𝑖

𝑛

=AVERAGE

Median 𝑛 + 1

=MEDIAN

Mode Highest point in the histogram =MODE.SING Measures of Spread Range (^) Maximum value – minimum value

Standard deviation s = √∑(𝑥𝑖−𝑥̅^ )

2

𝑛− 1

=STDEV.S

Percentiles 75 th^ percentile = 3rd^ or upper quartile = Q 25 th^ percentile = 1st^ or lower quartile = Q

=QUARTILE

Interquartile range (IQR) IQR = Q 3 – Q 1 Outliers (^) Greater than: Q3 + 1.5 (IQR) Less than: Q1 – 1.5 (IQR) Extreme Outliers Lower outer fence = Q1 – 3(IQR) Upper outer fence = Q3 + 3(IQR)

Chapter 6: Correlation and Linear Progression

Measure Formula Excel Correlation Coefficient (^) r = 1 𝑛− 1 ∑^ (

𝑥𝑖−𝑥̅ 𝑆𝑥^ )^ (

𝑦𝑖−𝑦̅ 𝑆𝑦^ )^

=CORREL

Residuals 𝑒

𝑖 =^ 𝑦𝑖 −^ 𝑦̂ 𝑖

Observed y – predicted y

Least Square Regression 𝑦̂ = 𝑏

0 +^ 𝑏 1 𝑥^

Data analysis tool pack (‘Regression’) b 1

𝑟

𝑆𝑦

𝑆𝑥

Data analysis tool pack (‘Regression’)

b 0 𝑦̅ − 𝑏 1 𝑥̅ Data analysis tool pack

(‘Regression’)

R-Squared r^2 Data analysis tool pack

(‘Regression’); (=CORREL)^2 Standard deviation of the residuals (^) 𝑠𝑒 = √ ∑^ 𝑒𝑖

2

𝑛 − 2

Chapter 8: Random Variables and Probability Models

COMM290 Review : If X is a random variable:

Measure Formula Excel Expected Value E(X)=∑X*P(X) =SUMPRODUCT E(X) =  = np Variance Var(X)=∑(x-μ)^2 *P(X) ^2 = 𝑛𝑝𝑞, where 𝑞 = 1 − 𝑝 Standard Deviation (^) σ = √Var(X) =STDEV.S

 = √ 𝑛𝑝𝑞

Linear transformation: a + bX

Expected Value

𝐸(𝑎 + 𝑏𝑋) = 𝑎 + 𝑏𝐸(𝑋)

Variance 𝑉𝑎𝑟(𝑎 + 𝑏𝑋) = 𝑏^2 𝑉𝑎𝑟(𝑋)

Standard Deviation

𝑆𝐷(𝑎 + 𝑏𝑋) = |𝑏|𝑆𝐷(𝑋)

Sum of two INDEPENDENT random variables: X+Y

Expected Value

𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌)

(Mean of a sum = sum of the means) Variance 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋)

• 𝑉𝑎𝑟(𝑌)

Standard Deviation

𝑆𝐷(𝑋 + 𝑌) = √(𝑉𝑎𝑟(𝑋) +

𝑉𝑎𝑟(𝑌))

Difference of two INDEPENDENT random variables: X–Y

Expected Value

𝐸(𝑋 − 𝑌) = 𝐸(𝑋) − 𝐸(𝑌)

Variance 𝑉𝑎𝑟(𝑋 − 𝑌) = 𝑉𝑎𝑟(𝑋)

• 𝑉𝑎𝑟(𝑌)

Standard Deviation

𝑆𝐷(𝑋 − 𝑌)

= √𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌)

Linear combination of two INDEPENDENT random variables: aX + bY

Expected Value

𝐸(𝑎𝑋 + 𝑏𝑌) = 𝑎𝐸(𝑋)

+ 𝑏𝐸(𝑌)

Variance 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎^2 𝑉𝑎𝑟(𝑋)

• 𝑏^2 𝑉𝑎𝑟(𝑌)

Standard Deviation

𝑆𝐷(𝑎𝑋 + 𝑏𝑌)

= √𝑎^2 𝑉𝑎𝑟(𝑋) + 𝑏^2 𝑉𝑎𝑟(𝑌)

Linear combination of two DEPENDENT random variables: aX + bY

Expected Value

𝐸(𝑎𝑋 + 𝑏𝑌) = 𝑎𝐸(𝑋)

+ 𝑏𝐸(𝑌)

Variance (^) 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) (^) = 𝑎^2 𝑉𝑎𝑟(𝑋)

• 𝑏^2 𝑉𝑎𝑟(𝑌)

Standard Deviation

𝑆𝐷(𝑎𝑋 + 𝑏𝑌)

= √𝑎^2 𝑉𝑎𝑟(𝑋) + 𝑏^2 𝑉𝑎𝑟(𝑌)

The Empirical Rule

68% of the area under the normal curve lies within μ ± σ

95% of the area under the normal curve lies within μ ± 2σ

99.7% of the area under the normal curve lies within μ ± 3σ

=NORM.DIST(X, μ, σ, TRUE): Given X, find probability.

=NORM.INV: Given probability, find X.

Empirical Rule of Thumb

s ≈ Range/6 (for bell- shaped distributions and large n)

s ≈ Range/4 (for bell- shaped distributions and small n, approx. 20)

Standardization

𝑍 =

𝑋 − 𝜇

𝜎

=NORM.S.DIST: the standardized version; given Z, find probability.

=NORM.S.INV: Given probability, find z.

In our applications, X and Y will always be independent. But, if they are not, the variances cannot always simply be added.

Inference and Sampling Distribution for Proportions

When to use: when a binary predictor is used to measure a binary outcome.

Where p = successes and q = failures,

One sample z test

Excel

Parameter p

Estimate (^) 𝑝̂ = 𝑋 𝑛 (X = number of successes, n = sample) Model One sample z test

SE (estimate of standard dev) SD(𝑝̂^ ) =^ √

𝑝 0 𝑞 0 𝑛

Null Hypothesis H 0 : p = p 0

Alternative Hypothesis

HA: p ≠ p 0

(Can be one sided)

HA: p > p 0

HA: p < p 0

Test Statistic

Z =

𝑝̂ −𝑝 0 √𝑝^0 𝑛𝑞 0 = NORM.S.INV(prob)

P-Value

HA: p > p 0 p = Pr(z > z-stat) (^) = 1 – NORM.S.DIST(z-

stat, cumulative) HA: p < p 0 p = Pr(z < z-stat) (^) = NORM.S.DIST(z-stat,

cumulative) HA: p ≠ p 0 p = 2 x Pr(z < z- stat)

= 2 x NORM.S.DIST(z- stat, cumulative)

Critical Values (z) (if you cannot find p-value)*

Sig Level (%)

One sided

Two sided

5 1.645 1.96 = NORM.S.INV(prob) 1 2.33 2. 0.1 3.09 3. Confidence Interval (Always 2-sided) 𝑝̂ ± z*√

𝑝̂ ( 1 −𝑝̂ ) 𝑛 Sample Size for a certain confidence interval n =^

(𝑧∗)^2 𝑝̂ ( 1 −𝑝̂ ) (𝑀𝐸)^2 (approximation with 95% CI) n =

1 (𝑀𝐸)^2

Two Samples z test

When to use: When there are 2 binary predictors (2 categories) and 2 binary outcomes ( categories).

Excel

Parameter p 1 – p2, Mean of (𝑝̂ 1 – 𝑝̂ 2 ) Estimate 𝑝̂ 1 – 𝑝̂ 2 , where (𝑝̂ 1 = X 1 / n 1 and 𝑝̂ 2 = X 2 / n 2 ) Model Two sample z test

SE (estimate of standard dev) (^) √𝑝̂^1 𝑞̂^1 𝑛 1

+

𝑝̂ 2 𝑞̂ 2

𝑛 2

Null Hypothesis Ho: p 1 – p 2 = Δ 0

Alternative Hypothesis Ha: p 1 – p 2 ≠ Δ 0

Test Statistic Z^ =^

(𝑝̂ 1 – 𝑝̂ 2 )−∆ 0

√𝑝

̂ 1 𝑞̂ 1 𝑛 1 +

𝑝̂ 2 𝑞̂ (^2) 𝑛 2 (If Δ 0 =0, we use a pooled p) 𝑝̂ 1 =

𝑋 1 𝑛 1 and^ 𝑝̂^2 =^

𝑋 2 𝑛 2 so 𝑝̂ = 𝑋 𝑛^1 +𝑋^2 1 +𝑛 2 Z =

(𝑝̂ 1 − 𝑝̂ 2 ) √𝑝̂ 𝑞̂ ( (^) 𝑛^11 + (^) 𝑛^12 )

where 𝑞̂ = 1–𝑝̂

P-Value

Ha: p 1 – p 2 > Δ 0 p = Pr(z > z- stat)

= 1 – NORM.S.DIST(z- stat, cumulative) Ha: p 1 – p 2 < Δ 0 p = Pr(z < z- stat)

= NORM.S.DIST(z-stat, cumulative) Ha: p 1 – p 2 ≠ Δ 0 p = 2 x Pr(z < z-stat)

= 2 x NORM.S.DIST(z- stat, cumulative)

Critical Values (z) (if you cannot find p-value)*

Sig Level (%)

One sided

Two sided

5 1.645 1.96 = NORM.S.INV(prob) 1 2.33 2. 0.1 3.09 3. Confidence Interval (Always 2-sided) (𝑝̂^1 –^ 𝑝̂^2 )^ ±^ z

* √𝑝̂ 1 𝑞̂ 1

𝑛 1 +^

𝑝̂ 2 𝑞̂ 2 𝑛 2

Inference and Hypothesis Tests for Means

One sample t test of means

Excel

Parameter μ

Estimate (^) 𝑥̅

Model One sample t test

SE (estimate of standard dev)

𝑠

√𝑛

Null Hypothesis H 0 : μ = 0

Alternative Hypothesis

HA: μ ≠ 0 HA: μ > 0 HA: μ < 0

Test Statistic t =^

𝑥̅ −𝜇 0 𝑠 √𝑛

,

with degrees of freedom n-

P-Value

HA: μ > 0 Pr(t > t-stat) (^) =T.DIST.RT(t-stat,df)

HA: μ < 0 Pr(t < t-stat) (^) =T.DIST.RT(t-stat,df)

HA: μ ≠ 0 2 x Pr(t > t-stat) (^) =T.DIST.2T((t-stat,df)

Critical Values (t) (for CI)*

=T.INV. 2T(sig level,df)

Confidence Interval (Always 2-sided)

𝑥̅ ± t*n- 1

𝑠 √𝑛 Sample Size n =

(𝑡∗)^2 𝑠^2 (𝑀𝐸)^2 s ≈

𝑟𝑎𝑛𝑔𝑒 6

Two-sample t-test of Independent Means

Characteristic Unequal (or separate) Variance Version

Equal (or pooled) Variance Version

Excel

Parameter μ 1 – μ 2 Estimate 𝑥̅ 1 – 𝑥̅ 2 SE

√

𝑠 12

𝑛 1

+

𝑠 22

𝑛 2

You cannot add

standard deviations

(add the variances)

Null Hypothesis Ho: μ 1 – μ 2 = Δ 0

Alternative Hypothesis

Ha: μ 1 – μ 2 ≠ 0

When to use pooled version

𝐿𝑎𝑟𝑔𝑒𝑟 𝑠

𝑆𝑚𝑎𝑙𝑙𝑒𝑟 𝑠

< 2

SE

√

𝑠 12

𝑛 1

+

𝑠 22

𝑛 2

𝑠𝑝^2 =

(𝑛 1 − 1 )𝑠 12 +(𝑛 2 − 1 )𝑠 22

𝑛 1 +𝑛 2 − 2

= pooled variance

Regression

t-statistic

t =

(𝑥̅ 1 – 𝑥̅ 2 )−∆ 0

√𝑠^1

2

𝑛 1 +

𝑠 22 𝑛 2

t =

(𝑥̅ 1 – 𝑥̅ 2 )−∆ 0

𝑠𝑝√

1

𝑛 1 +^

1 𝑛 2

Regression

Distribution (^) Approximate Exact Degrees of freedom

• Simplest choice: df = minimum of (n 1 -1, n 2 -1).
• Use the sum of n 1 - and n 2 -1.
• A lengthy formula that we will only use if we do computations via excel software.

n 1 + n 2 – 2

Both If n1=n2, the two test statistics are the same. The P-value, however, is not.

• In the final, they should be the same.

=T.DIST.2T(t-stat, degrees of freedom) =T.DIST.RT(t-stat, degrees of freedom) Critical Value =T.INV.2T(sig level,df), Regression

Confidence Interval

(x̅ 1 – x̅ 2 ) ± t*min(n1-

1,n2-1) √

s 12 n 1

+

s 22 n 2

(𝑥̅ 1 – 𝑥̅ 2 ) ± t*(n1+n2-2)

𝑠𝑝√

1

𝑛 1 +^

1 𝑛 2 Regression

Two-sample t-test of dependent means (paired t-test)

When to use: same sample, different situations. EX: same person’s midterm and final results.

Excel

Parameter di = x1i – x2i

Estimate Sample mean of the differences, 𝑑̅

Model Paired t test

sd (estimate of standard dev)

Compute the differences and then find their standard deviation on excel. SE 𝑠𝑑

√𝑛

Null Hypothesis H 0 : μd = 0

Alternative Hypothesis

HA: μd ≠ 0 HA: μd > 0 HA: μd < 0

Test Statistic t =^

𝑑̅ − 0

𝑠𝑑 √𝑛

,

with degrees of freedom n-

P-Value

HA: μ > 0 Pr (t > t-stat) (^) =T.DIST.RT(t-stat,df)

HA: μ < 0 Pr(t < t-stat) (^) =T.DIST.RT(t-stat,df)

HA: μ ≠ 0 2 x Pr(t > t-stat) (^) =T.DIST.2T((t-stat,df)

Critical Values (z) (for CI)*

=T.INV. 2T(prob,df)

Confidence Interval

(Always 2-sided) 𝑑

̅ ± t*n- 1

𝑠𝑑

√𝑛

Types of Errors in Hypothesis Testing

Decision Accept Ho Reject Ho True State

Ho True Correct Type I error (False positive) Ho False Type II error (False negative)

Correct

Chapter 13: Inference for Counts—Chi Square Tests

‘Goodness of Fit’ Test

When to use: by assessing the discrepancy between observed frequencies for a categorical variable and the frequencies that would be expected under a certain theory.

Excel

Parameter NO PARAMETERS

Model Chi Square test

Null Hypothesis H 0 : p 1 = p 2 = p 3 = (observed proportions)

Alternative Hypothesis

HA: at least one p 1 is not observed value

Test Statistic (always one sided)

𝜒^2 =

𝑑̅ − 0

𝑠𝑑 √𝑛

,

with degrees of freedom n-

P-Value (Always one sided) P-value = Pr (𝜒^2 > 𝜒^2 -stat)

=CHISQ.DIST.RT(test- statistic, degrees of freedom)

Critical Values (X) (for CI)*

= CHISQ.INV.RT

Chi-square test of two way tables (See summary for steps)

When to use:

• Test of independence
• Test of homogeneity

A 2x2 table can be tested using either a two-sample z-test for two proportions or a 𝜒^2 table.

• Chi-square does not have a confidence interval.
• The only time the z test HAS to be the one chosen is when you need to later perform a confidence interval.

Excel

Parameter NO PARAMETERS

Model (^) Chi Square test

Null Hypothesis Ho: Row and column classifications are independent (or unassociated, or unrelated)

Alternative Hypothesis

Ha: Row and column classifications are dependent (or associated, or related)

Expected Counts

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝐶𝑜𝑢𝑛𝑡𝑠

=

𝑟𝑜𝑤 𝑡𝑜𝑡𝑎𝑙 × 𝑐𝑜𝑙𝑢𝑚𝑛 𝑡𝑜𝑡𝑎𝑙

𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑡𝑜𝑡𝑎𝑙

Test Statistic 𝜒^2 = ∑

(𝑂𝑏𝑠−𝐸𝑥𝑝)^2 𝐸𝑥𝑝

, where the

sum is taken over all the cells in the table.

P-Value (Always one sided) (^) P-value = Pr (𝜒^2 > 𝜒^2 - stat); Degrees of freedom = (r–1)(c–1), (r = number of rows, c = number of columns)

=CHISQ.DIST.RT(test- statistic, degrees of freedom)

Critical Values (X) (for CI)*

= CHISQ.INV.RT

Chapter 16: Introduction to Statistical Models

𝑌 = 𝛽 0 + 𝛽 1 𝑋 + 𝜀 Y is the unknown dependent variable (outcome) X is the known independent variable (predictor) ε is a random variable representing random error β 0 and β 1 are parameters

PARAMETER

β 1

PARAMETER

β 0

Estimate b 1 = 𝑟 𝑆𝑦

𝑆𝑥

b 0 = 𝑏 0 = 𝑦̅ − 𝑏 1 𝑥̅

Mean β 1 β 0

SE 𝑠𝑒

𝑠𝑥√𝑛− 1

, or

𝑠𝑒

√∑(𝑥𝑖−𝑥̅ )^2 𝑠𝑒√

1 𝑛

𝑥̅ 2 (𝑛− 1 )𝑠𝑥^2

, or

𝑠𝑒√

1 𝑛

𝑥̅ 2 ∑(𝑥𝑖−𝑥̅ )^2 S^2 e

𝑠𝑒^2 =

∑ 𝑒𝑖^2

𝑛 − 2

Sampling distribution Normal Null Hypothesis HO: β 1 = 0 i.e.:

1. No linear association/trend between X and Y.
2. X is not a useful predictor of Y.
3. Knowing X does not tell you anything about Y.
4. Model is not worthwhile.

HO: β 1 = 0 (Not often tested)

Alternative Hypothesis (^) Ha: β 1 ≠ 0 (Always two sided for this case)

Ha: β 0 ≠ 0

Test statistics 𝑡 =

𝑏 1 – 𝛽 1 ∗

SE(𝑏 1 )

𝑏 0 – 𝛽 0 ∗

SE(𝑏 0 )

P value P-value = 2 x Pr (t > |t-stat|) P-value = 2 x Pr (t > |t-stat|)

Confidence Intervals β 1 : 𝑏

1 ±^ t*n-^2

𝑠𝑒 𝑠𝑥√𝑛− 1

β 0 : 𝑏 0 ± t*n- 2 𝑠𝑒√

1 𝑛

𝑥̅ 2 (𝑛− 1 )𝑠𝑥^2

Confidence Interval for a Mean Response

When to use:

• Predict the mean Y for all units having a particular X-value, called x*
• Used for a group of people who have something in c ommon

SE SE(𝜇̂^ 𝑥∗^ ) =^ √𝑆𝐸^2 (𝑏 1 )^ ×^ (𝑥∗^ −^ 𝑥̅^ )^2 +^

𝑠𝑒^2 𝑛

100(1 – α)% Confidence Interval (𝑏 0 +^ 𝑏 1 𝑥∗)^ ±^ 𝑡𝑛^ ∗^ − 2 √𝑆𝐸^2 (𝑏 1 )^ ×^ (𝑥∗^ −^ 𝑥̅^ )^2 +^

𝑠𝑒^2 𝑛

Confidence interval depends on:

• Standard error of the slope
• X value where the prediction is being made
• Standard deviation of the residuals
• Sample size

Prediction Interval for a Future Observation

• Predict a single value of Y for a single unit having a particular X-value, called x*
• P articular to you

SE SE(𝑦̂𝑥∗^ ) =^ √𝑆𝐸^2 (𝑏 1 )^ ×^ (𝑥∗^ −^ 𝑥̅^ )^2 +^

𝑠𝑒^2

𝑛 +^ 𝑠𝑒

2

100(1-α)% Prediction Interval

(𝑏 0 + 𝑏 1 𝑥∗) ± 𝑡𝑛^ ∗−^2 √𝑆𝐸^2 (𝑏 1 ) × (𝑥∗^ − 𝑥̅ )^2 +

𝑠𝑒^2

𝑛 +^ 𝑠𝑒

2

EXCEPTION: When n is large, this simply becomes:

𝑦̂𝑥∗^ ± 𝑡𝑛−2∗^ 𝑠𝑒 where 𝑦̂𝑥∗^ = 𝑏 0 + 𝑏 1 𝑥∗

For a 95% prediction interval, this becomes simply:

𝑦̂𝑥∗^ ± 2𝑠𝑒

Extrapolation Penalty = 𝑆𝐸^2 (𝑏 1 ) × (𝑥∗^ − 𝑥̅)^2

First Hypothesis Test for Correlation

When to use:

• Testing whether β 1 = 0 is equivalent to testing whether ρ = 0.
• To see if the model has predictive value Excel

Parameter Correlation coefficient of the population (ρ). Estimate r

Null Hypothesis Ho: ρ = 0

Alternative Hypothesis Ha: ρ ≠ 0

Test Statistic t =

𝑟√𝑛− 2

√ 1 −𝑟^2

, df = n- 2

P-Value (Two sided)

P-value = Pr (𝜒^2 > 𝜒^2 -stat) =T.DIST(t-stat,n-2)

Another Test for β1: Analysis of Variance for Regression (ANOVA TABLE)

When to use: to answer whether the X variable is a useful predictor of Y; if the model is worthwhile.

∑(𝑦𝑖 − 𝑦̅ )^2 = ∑(𝑦̂𝑖 − 𝑦̅ )^2 + ∑(𝑦𝑖 − 𝑦̂𝑖)^2

Sum of Squares Total (SST) = Sum of Squares Model (SSM) + Sum of Squares Error (SSE) (Regression) (Residual)

Source of Variation

Sum of Squares

Degrees of Freedom

Mean Square F-stat

Model SSM 1 MSM=SSM/df MSM/MSE Error SSE n-2 MSE=SSE/(n-2) Total SST n-

SSE = ∑(𝑦𝑖 − 𝑦̂𝑖)^2 = ∑ 𝑒𝑖^2

MSE = SSE/(n-2) = 𝑠𝑒^2 , so se = √𝑀𝑆𝐸

Analysis of Variance to Test the Worthiness of the Model

Excel

Parameter β 1

Se^2 𝑠𝑒^2 =

∑ 𝑒𝑖^2

𝑛 − 2

“Standard Error” in the Regression analysis of excel, (standard error of the residuals) Null Hypothesis Ho: β 1 = 0 (The model is not worthwhile; x has no predictive info about y)

Alternative Hypothesis

Ha: β 1 ≠ 0 (The model is worthwhile; x has some predictive information over y) Test Statistic (Overall F test)

Test statistic: F = MSM / MSE = F.DIST.RT(F- stat,df1(k),df2(n-k-1) = T.DIST.2T(SQRT(F- stat), n-k-1) F-stat and t-stat (for simple regression only)

t^2 n- 2 = F1, n- 2 t = (^) √𝐹

P-Value (Always one sided) P-value = Pr (F > F-stat) = F.DIST.RT(f-stat,k,n-

R^2 =

𝑆𝑆𝑀 𝑆𝑆𝑇

= proportion of total variation explained by the model

r = √

𝑆𝑆𝑀 𝑆𝑆𝑇 , (excel says R-multiple, but you have to find the sign from the slope sign)

Chapter 15: Multiple Regression

When to use: to answer the questions

• What set of independent variables provides a ‘good’ explanation of the variation in Y?
• How much further reduction in the remaining variation in Y is there as a result of including a particular variable in the model?

𝑌 = 𝛽 0 + 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 + ⋯ + 𝛽𝑘𝑋𝑘 + 𝜀 k = # of X-variables

Excel

Parameter 𝛽 0 , 𝛽 1 , 𝛽 2 , … , 𝛽𝑘

Estimates (^) b 0 , b 1 , b 2 , …, bk

𝒔𝒆^ 𝟐 𝑠𝑒^2 =

∑ (^) 𝑒𝑖^2

𝑛 − 𝑘 − 1

“Standard error” in the ANOVA table (the standard error of the residuals) Model ANOVA

Null Hypothesis Ho: β 1 = β 2 = … = βk = 0

(The model is not worthwhile; none

of the X-variables has any explanatory power or value)

Alternative Hypothesis

Ha: at least one βj is not 0 (The model is worthwhile; there are some valuable X-variables somewhere in the model) Test Statistic (Overall F test)

Test statistic: F = MSM / MSE = F.DIST.RT(F- stat,df1(k),df2(n-k-1) = T.DIST.2T(SQRT(F- stat), n-k-1)

P-Value (Always one sided) P-value = Pr (F > F-stat) = F.DIST.RT(f-stat,k,n- k- 1 )

Note: If H0 is rejected, the next steps are to find the smallest set of predictors that adequately fit the data.

• This means that if an exam has follow up questions to the hypotheses test, the null hypothesis was probably rejected.

ANOVA Table (The same, with minor changes in the Degrees of Freedom column).

Source of Variation

Sum of Squares

Degrees of Freedom

Mean Square F-stat

Model/Regression SSM k MSM=SSM/k MSM/MSE Error/Residual SSE n-k-1 MSE=SSE/(n-k-1) Total SST n-

R-squared is now formally called the Coefficient of Multiple Determination, but is defined the same way:

R^2 =

𝑆𝑆𝑀

𝑆𝑆𝑇

= proportion of total variation explained by the model

F-test to Compare Full and Reduced Models

When to use: to see if all present variables are relevant, and to avoid multicollinearity. Full Model: Use all available predictor variables Reduced Model: Subset of Full Model: drop q of the predictor variables

Null Hypothesis H 0 : The reduced model is adequate (The extra variables in the full model that are not in the reduced model do not provide a significant improvement in the model's predictive ability)

Alternative Hypothesis

Ha: The reduced model is not adequate (At least one of the extra predictors is worthwhile, so the reduced model discards some worthwhile predictors; therefore, it is better to keep the full model rather than this reduced model.) Test Statistic (F test to compare full and reduced models)

(

𝑛 − 𝑘 − 1

𝑞

) (

𝑅^2 (𝑓𝑢𝑙𝑙)^ − 𝑅^2 (𝑟𝑒𝑑𝑢𝑐𝑒𝑑)

1 − 𝑅^2 (𝑓𝑢𝑙𝑙)

)

k = number of X-variables in the Full Model q = number of variables dropped from the Full Model to get the Reduced Model

EXCEL WILL NOT

DO IT FOR YOU

P-Value (Always one sided) P-value = Pr (F > F-stat) = F.DIST.RT(f- stat,k,n-k-1)

Regression Output

Other formulas:

(3): R^2 : 1 – (1–R^2 )[(n–1)/(n–2)]

(15): Pr[F1,n- 2 > (14)]

(18): 𝑠𝑒√

1 𝑛

𝑥̅ 2 ∑(𝑥𝑖−𝑥̅ )^2

(19):

𝑠𝑒 𝑠𝑥√𝑛−

(22): 2×Pr[ t n- 2 > |(20)|]

(23): 2×Pr[ t n- 2 > |( 21 )|]

(24): (16) – t *n- 2 ×(18)

(25): (17) – t *n- 2 ×(19)

(26): (16) + t *n- 2 ×(18)

(27): (17) + t *n- 2 ×(19)