statistical formulas - sonoma state universitysonoma.edu/users/s/seward/211file/computer...

BUS 211 NotesChapter 1 Introduction and Data Collection

Categorical Variables – responses are a selection i.e. Gender (male or female), Class (freshman, sophomore, junior, senior), Smoke (yes or no), etc.

Numerical Variables – responses are numbers i.e. Income ($30,000), Age (25), etc.Can be Discrete (Integer) or Continuous (fractional parts),

Chapter 2 Presenting Data in Tables and ChartsSort Data – Data | Sort

Stem-and-Leaf Graph – PHStat | Descriptive Statistics | Stem-and-Leaf Display

Frequency Distribution - PHStat | Descriptive Statistics | Frequency DistributionSet up classes then array (bin) the upper limit of the desired frequency distributionBe sure to include a label for the array (use Upper Limit)

Relative Frequency distribution – Divide the frequency distribution by the total

Percentage Distribution - Divide the frequency distribution by the total and multiply by 100 Or use Format | Cells… | Percentage

Cumulative Distribution – Sum the frequencies from top to bottom listing each total as you go.

Graphs - PHStat does not work well for most graphs use the chart wizard in Excel

Histogram also known as a Vertical Bar Chart or Column Chart - Set up the frequency distribution then use the midpoints for labelsDouble click the chart icon and select a column graph typeSelect the frequency without labels as the dataSelect the Series tab, mouse into the X-axis label box then select the midpointsSelect Next to insert the title and axis labels and make any other changesSelect Next to pick a location for the chart then FinishDouble click a bar and select Options, set gap width to 0

Polygon also known as a line graph - Set up the frequency distribution then use the midpoints for labels. Insert a class with O frequency and an appropriate label at the top and the bottom.Double click the chart icon and select a line graph typeSelect the frequency without labels as the data Select the Series tab, mouse into the X-axis label box then select the midpointsSelect Next to insert the title and axis labels and make any other changesSelect Next to pick a location for the chart then Finish

Ogive also known as a cumulative line graph or cumulative polygonSet up the cumulative frequency distribution use the upper class limit for labels. Insert a class with O frequency and an appropriate label at the top but not the bottom.Double click the chart icon and select a line graph type and complete the steps

XY Scatter Set up the data in columns with the X values first and the Y in the second columnDouble click the chart icon and select XY Scatter graphSelect both columns as the data, do not select the labels, and complete the steps

Bar Chart Same as Histogram but for categorical data. Use the category labels: if not numerical values they can be selected with the data.

Pie Chart Same as above. Be sure to remove legend, select Data Labels, check Category name

Pareto Chart Raw Data: use line chart on 2 axis or Select Descriptive Statistics | One-Way Tables & Charts…Be sure to select labels as the model will not work otherwiseCheck table of frequencies and Pareto Diagram

Bivariate Categorical Tables and Charts Use PHStat (also available in Excel - Data | Pivot Wizard)In PHStat select Descriptive Statistics | Two-Way Tables & Charts

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Chapter 3 Numerical Descriptive Measures

Use Tools | Data Analysis | Descriptive Statistics, check the Summary statistics box to get the following:

sample mean, median, mode, standard deviation, variance, rangepopulation mean, median, mode, range

Use fx the individual functions for the following measures

geometric mean (GEOMEAN), population variance (VARP) and standard deviation (STDEVP)approximate quartiles (QUARTILE), approximate percentiles (PERCENTILE)

Coefficient of variation: Divide the standard deviation by the mean and multiply by 100%

Box-and-Whisker Plot and Five-Number Summary

PHStat | Descriptive Statistics | Box-and-Whisker Plot then check Five-Number SummaryGives the exact quartiles not approximations

Coefficient of Correlation: fx (CORREL), or Tools | Data Analysis | Correlation

Chapter 4 Basic Probability

Probability of A or B: If A and B are Mutually Exclusive:

Conditional probability of A given B: If A and B are Independent:

Joint Probability of A and B: If A and B are Independent:

Bayes' Theorem

Chapter 5 Some Important Discrete Probability Distributions

Combinations:

Binomial distribution: (for an infinite population) PHStat | Probability & Prob. Distributions | Binomial then check Cumulative Probabilities

Hypergeometric distribution: (for a finite population) PHStat | Probability & Prob. Distributions | Hypergeometric no cumulative probabilities available

Poisson distribution: PHStat | Probability & Prob. Distributions | Poisson then check Cumulative Probabilities-

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Chapter 6 The Normal Distribution and Other Continuous DistributionsNormal Distribution

PHStat | Probability & Prob. Distributions | Normal then check the desired calculationTo check the normality assumption construct a stem-and-leaf, box-and-whisker, histogram or aNormal probability plot PHStat | Probability & Prob. Distributions | Normal Probability Plot

Uniform Distribution

where a and b are the endpoints of the uniform distribution.

Exponential distribution PHStat | Probability & Prob. Distributions | Exponential Only returns results for X, for > x use 1-probability, for results between two values find the probability for each and subtract the smaller from the larger

Sampling distribution of the meanCalculate the standard deviation of the sampling distribution also called the Standard error of the mean then use the Normal Distribution calculator if the population is normally distributed or the sample size is > 30 or the population distribution is symmetrical and the sample size is > 15

Infinite population Finite population

Sampling distribution of the proportion: Calculate the standard deviation of the sampling distribution (Standard Error of the Mean) then If np > 5 and n(1-p) > 5 use the Normal Distribution calculator PHStat | Probability & Prob. Distributions | Normal

ps = sample proportion p = population proportion

Infinite population Finite population

Chapter 7 Confidence Interval EstimationInterval estimate of the population mean (mx) with sx unknown:

PHStat | Confidence Intervals | Estimate for the Mean, sigma unknownbe sure to check the finite box for finite populations

Interval estimate of the population proportion: PHStat | Confidence Intervals | Estimate for the Proportionbe sure to check the finite box for finite populations

Interval estimate of the population total: PHStat | Confidence Intervals | Estimate for the Population Total

Sample size (n) for estimating a mean: PHStat | Sample Size | Determination for the Meanbe sure to check the finite box for finite populations Estimate of parameters would be from a preliminary sample

Sample size for estimating a proportion: PHStat | Sample Size | Determination for the Proportionbe sure to check the finite box for finite populationsEstimate of True Proportion would be the proportion from a preliminary sampleIf a preliminary sample is not available use .5

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Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests

One Sample numerical data s unknown Hypothesis

Ho: mx = value a two tail test Ho: mx value Ha: mx value upper tail testHa: mx value Ho: mx ³ value Ha: mx value lower tail test

Test Statistic t

Procedure Summary Data: PHStat | One-Sample Tests | t Test for the Mean, sigma unknown

Decision Rule If the p-value is less than alpha Reject the Hypothesis If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis

Conclusion If rejected – There is sufficient evidence that (Question asked)If not rejected – There is not sufficient evidence that (Question asked).

Parentheses indicate information to be taken from the problemOne Sample Categorical Data

HypothesisHo: p = value a two tail test Ho: p value Ha: p value upper tail testHa: p value Ho: p ³ value Ha: p value lower tail test

Test Statistic Z

Procedure Summary Data: PHStat | One-Sample Tests | Z Test for the ProportionRaw Data: No Tests available, calculate p and use PHStat


Conclusion If rejected – There is sufficient evidence that (Question asked)If not rejected – There is not sufficient evidence that (Question asked)

Parentheses indicate information to be taken from the problem

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Chapter 9 Two-Sample Tests Procedure to determine the proper two sample mean test for numerical data:

Two Sample test of Means with Paired numerical data

Hypothesis Ho: m1 = m2 a two tail test Ho: m1 m2 Ha: m1 m2 upper tail testHa: m1 m2 Ho: m1 ³ m2 Ha: m1 m2 lower tail test

Procedure Summary Data: no PHStat calculation available Raw Data: Data Analysis | t Test: Paired Two Sample for Means

Test Statistic t



Interval estimate of the difference To get t use function TINV(1-Confidence, df)Use Descriptive Statistics to get D and sd

Or PhStat | Confidence Intervals | Estimate for the Mean, sigma unknown - Select the differences as the data

Two Sample test of Variances with numerical data Hypothesis Ho: s2

1 = s22 a two tail test Ho: s2

1 s22 Ha: s2

1 s22 upper tail test

Ha: s21 s2

2 Ho: s21 ³ s2

2 Ha: s21 s2

2 lower tail test

Procedure Summary Data: PHStat | Two-Sample Tests | F Test for the Difference in Two VariancesRaw data: Data Analysis | F Test Two Sample for Variances Do not use only gives lower tail value

Test Statistic F


Conclusion If rejected – There is sufficient evidence that (Question asked)If not rejected–There is not sufficient evidence that (Question asked)

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Are Data PairedYes Use Paired Data Model

No

F TestAre s2's Equal Use s2 Equal Model Use s2 Unequal Model

No Yes

Two Sample test of Means with numerical data s 2 ’s not proven unequal with the F test


Procedure Summary Data: PHStat | Two-Sample Tests | t Test for Differences in Two MeansRaw Data: Data Analysis | t Test: Two Sample Assuming Equal Variances

Test Statistic t



Interval estimate of the difference

To get t use function TINV(1-Confidence, df)

Two Sample test of Means with numerical data s 2 ‘s proven unequal with the F test


Procedure Summary Data: Use spreadsheet downloaded from the Homework web page

Raw Data: Data Analysis | t Test: Two Sample Assuming Unequal Variances

Test Statistic t




To get t use function TINV(1-Confidence, df)

Two Sample test of a Proportion with categorical data

Hypothesis Ho: p1 = p2 a two tail test Ho: p1 p2 Ha: p1 p2 upper tail testHa: p1 p2 Ho: p1 ³ p2 Ha: p1 p2 lower tail test

Procedure PHStat | Two-Sample Tests | Z Test for the Differences in Two Proportions

Test Statistic Z




To get Z use function NORMSINV(two tail)where two tail=Confidence+(1-Confidence)/2

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Chapter 10 Analysis of Variance ( Multi (c) Sample tests with numerical data) Equality of Variances

Hypothesis Ho: s21 = s2

2= s23 a two tail test

Ha: not all s’s are equal

Procedure Raw data: PHStat | Multiple-Sample Tests | Levene’s Test

Test Statistic F


Conclusion If rejected – There is sufficient evidence that (Question asked)If not rejected–There is not sufficient evidence that (Question asked)

One Factor ANOVAHypothesis Ho: m1 = m2 = m3 … = mc c = the number of populations

Ha: not all m’s are equal

Procedure Tools | Data Analysis |Anova: Single Factor

Test Statistic F from the computer printout P-value = The Probability of



Tukey's multiple comparison method: (determines which of the c means are different from each other).

Procedure PHStat | Multiple-Sample Tests | Tukey-Kramer Procedure

Test Statistic Critical Range

Input Q found in the Studentized Range Table where column = c and row = n-c c = number of groups n = total number of data points in all groups

Decision Rule If the absolute difference between any two pairs of means is greater than the critical range the pair is different.

Two Factor With Replication Hypothesis Ho1: mA1 = mA2 = mA3 … = mr r = the number of levels in Factor A

Ha1: not all m’s are equal

Ho2: mB1 = mB2 = mB3 … = mc c = the number of levels in Factor BHa2: not all m’s are equal

Ho3: No InteractionHa3: Interaction

Procedure Tools | Data Analysis |Anova: Two Factor With Replication

Test Statistic F from the computer printout. p-value = The Probability of For differences in rows see p-value for the Sample row of the ANOVAFor differences in columns see p-value for the Columns row of the ANOVAFor interaction between factors see p-value for the Interaction row of the ANOVA


Conclusion H1 If rejected – There is sufficient evidence of a difference in (factor A) H2 If rejected – There is sufficient evidence of a difference in (factor B)H3 If rejected – There is sufficient evidence of an interaction termIf not rejected – There is not sufficient evidence to make a conclusion about …

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Tukey's multiple comparison method for Two Factor ANOVA with replication: No spreadsheet, hand calculate with the following formulas:

MSW from ANOVA MS Within

Q table column is r the number of levels in Factor A

Q table row is rc(n’-1) where c is the levels in Factor B, and n’ is the number of replications

MSW from ANOVA MS Within

Q table column is c the number of levels in Factor B

Q table row is rc(n’-1) where r is the levels in Factor A, and n’ is the number of replications

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Chapter 11 Chi-Square Tests and Nonparametric TestsTwo Sample test of a Proportion with categorical data (Alternate Procedure)

Hypothesis Ho: p1 = p2 Ha: p1 p2 (No <, or > Hypothesis)

Procedure PHStat | Two-Sample Tests | Chi-Square Test for the Differences in Two Proportions

Test Statistic 2



Multi (c) Sample test of Proportions with categorical data Hypothesis Ho: p1 = p2 = p3 … pc c = the number of samples

Ha: not all p’s are equal

Procedure PHStat | Multiple-Sample Tests | Chi-Square Test

Test Statistic 2



Be sure to check the box for the Marascuilo Procedure to determine which proportions are different.

2 Test of Independence

Hypothesis Ho: Two categorical variables are independentHa: Two categorical variables are related

Procedure PHStat | Multiple-Sample Tests | Chi-Square Test

Test Statistic 2


Conclusion If rejected – There is sufficient evidence that the variables are relatedIf not rejected – There is not sufficient evidence that the variables are related.

Two Sample test of Medians with numerical dataHypothesis Ho: M1 = M2 a two tail test Ho: M1 M2 Ha: M1 M2 upper tail test

Ha: M1 M2 Ho: M1 ³ M2 Ha: M1 M2 lower tail test

Procedure Raw Data PHStat | Two-Sample Tests | Wilcoxon Rank Sum TestSummary Data No Tests available.

Test Statistic Z



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Kruskal-Wallis Rank Test for Differences Between c MediansHypothesis Ho: M1 = M2 = M3 = MC

Ha: Not all Mj are equal ( j=1,2,…C)

Procedure Raw Data PHStat | Multiple-Sample Tests | Kruskal-Wallis Rank TestSummary Data No PHStat or Excel calculation available

Test Statistic H



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Chapter 12 Simple Linear Regression

Linear Regression Model: relationship represented as

Determining if the linear model is significant

Hypothesis Ho: 1 = 0Ha: 1 0

Procedure PHStat | Regression | Simple Linear Regression orTools | Data Analysis | Regression

Test Statistic F

Decision Rule If the significant F (a p-value) is less than alpha Reject the Hypothesis If the significant F is greater than or equal to alpha Fail to Reject the Hypothesis

Conclusion If rejected – There is sufficient evidence to accept the linear regression modelIf not rejected – There is not sufficient evidence of a linear model end the analysis

Confidence Interval estimate of 1 found on the ANOVA output. See the independent variable line under Lower 95% and Upper 95%.

Confidence interval estimates for the dependent variable be sure to check the input box and insert a value.

Durbin Watson statistic for autocorrelation be sure to check the input box.

Additional measures from the regressionStandard error of the estimate: a measure of variability of the data around the regression line

Coefficient of determination (r 2 ) : measures the percent of the variation in the dependent variable Y that is explained by the independent variable X in the regression model. Shows the strength of the relationship.

Adjusted r 2 : modifies the r2 for the number of explanatory variables in the model and the sample size

Sample coefficient of correlation (r): estimator of

Checking the Assumptions of regression: 1. Normality - to check normality analyze the normal probability plot of the sample values.

2. Homoscedasticity - variation around the regression line must be constant for all values of Xto check analyze the residual plot for horn shape.

3. Independent residuals - to check analyze residual plot for randomness.

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Chapter 13 Introduction to Multiple Regression

Multiple Regression Model: represented as

Determining if the multiple linear model is significant Hypothesis Ho: 1 = 2 =…= k = 0 where k equals the number of variables

Ha: Not all ’s = 0

Procedure PHStat | Regression | Multiple Regression orTools | Data Analysis | Regression

Test Statistic F

Decision Rule If the significant F (a p-value) is less than alpha Reject the Hypothesis If the significant F is greater than or equal to alpha Fail to Reject the Hypothesis

Conclusion If rejected – There is sufficient evidence that all or part of the model is significant, (proceed with the analysis)

If not rejected – There is not sufficient evidence of a linear model (end the analysis)

Determining which variables are significantHypothesis Ho: 1 = 0 Ho: 2 = 0 … Ho: k = 0

Ha: 1 0 Ha: 2 0 … Ha: k 0

Procedure PHStat | Regression | Multiple Regression orTools | Data Analysis | Regression

Test Statistic t

Decision Rule If the p-value of the t statistic is less than alpha Reject the Hypothesis If the p-value is greater than or equal to alpha Fail to Reject the Hypothesis

Conclusion If rejected – There is sufficient evidence that (variable) is significantIf not rejected – There is not sufficient evidence to prove (variable) is significant

Confidence Interval estimates of ’s found on the ANOVA output. See the variable lines under Lower 95% and Upper 95%.

Confidence interval estimates for the dependent variable be sure to check the input box and insert a value.

Durbin Watson statistic for autocorrelation be sure to check the input box.

Additional measures from the regressionStandard error of the estimate: a measure of variability of the data around the regression line

Coefficient of multiple determination (r 2 ) : measures the percent of the variation in the dependent variable Y that is explained by the independent variables in the regression model. Shows the strength of the relationship.

Adjusted r 2 : modifies the r2 for the number of explanatory variables in the model and the sample size

Sample coefficient of correlation (r): estimator of

Coefficient of partial determination (r 2 Y) contribution of each variable holding the others constant

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Dummy Variables Model used to include categorical variables.

Prepare a data matrix with Y, X1..Xn and dummy variables with 1 representing the characteristic and 0 its absence

Y X1 … Xn XD1 … XDn

3.8 3 14.2 2 0. . 0. . 1.

Follow the usual multiple regression procedures. Then test for an interaction term between numerical and categorical variables. If the interaction term is significant you can not use the dummy variable.

Dummy Variables Interactions Model Used to check on the interaction between the numerical and categorical variables.

To test for an interaction term prepare a data matrix with Y, X1..n , the dummy variables and the product of the dummy variable and the numerical variables

Y X1..n XD1..n X1 * XD

3.8 3 1 34.2 2 0 0. . 0. . 1

Include all possible combinations Follow the usual multiple regression procedures.

Independent Variables Interactions Model used to check on interaction between numerical variables.

To test for an interaction term prepare a data matrix with Y, X1..Xn and the product of all pairs of numerical variables

Prepare a data matrix with Y, X1..Xn and Xa*Xb

Y X1 X2 Xa * Xb

3.8 3 11 334.2 2 15 30. . 12. . 18

Include all possible combinations Follow the usual multiple regression procedures.

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Chapter 14 Multiple Regression Model BuildingThe Quadratic model

b0 = estimated Y interceptb1 = estimated linear effect on Yb11 = estimated curvilinear effect on Y

Prepare a data matrix with the dependent variable Y and the independent variables X and X2

Y X X2

3.8 3 94.2 2 4…

Do a multiple regression with X and X2 as the independent variables.

The Square-Root Transformation Model

Prepare a data matrix with the dependent variable Y and the independent variable square root of X

Y X3.8 9 34.2 4 2…

Do a simple linear regression with as the independent variables.

The Log Multiplicative Model

Prepare a data matrix with Y , X1 , X2 and their logs: use =Log(..)

Y X1 X2 Log Y Log X1 Log X2

3.8 3 9 .579784 .477121 .9542434.2 5 8 .623249 .69897 .90309…

Do a multiple regression with Log Y, Log X1 and Log X2 as the independent variables. To convert your predictions to the original data range take 10 to the power Log Y

The Natural Log Exponential Model

Prepare a data matrix with Y , X1 , X2 and their logs: use =ln(..)

Y X1 X2 Ln Y Ln X1 Ln X2

3.8 3 9 1.335001 1.098612 2.1972254.2 5 8 1.435085 1.609438 2.079442…

Do a multiple regression with X and X2 as the independent variables. To convert your predictions to the original data range take e to the power Ln Y or use =Exp(Ln Y)

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Model BuildingStepwise Regression – limited evaluation of alternative models

Procedure: PHStat | Regression | Stepwise Regression

Best-Subsets – all possible subsets of the independent variables.

Procedure: PHStat | Regression | Best Subsets

1. Fit a model with all the independent variables and check the VIF box.

2. If all VIF’s are 10 proceed to the next step, else eliminate the variable with the highest VIF and go to back to step 1

3. Sort the results by the adjusted r2 select the model with the least variables if the r2’s are close. Or Sort the results by Cp and pick models with Cp to k+1 (k=total number of variables)and pick the best.

15

Chapter 15 Time-Series Forecasting and Index Numbers

Time-Series models use the same least squares technique as regression models. Only the data is different.

The Linear model

b0 = estimated Y interceptb1 = estimated linear effect on Y

Prepare a data matrix with the dependent variable Y and the independent variable X

Y X3.8 14.2 2…

Do a simple linear regression with X as the independent variable.

Forecast by plugging the next X value into the linear equation

The Quadratic model

b0 = estimated Y interceptb1 = estimated linear effect on Yb11 = estimated curvilinear effect on Y

Prepare a data matrix with the dependent variable Y and the independent variables X and X2

Y X X2

3.8 1 14.2 2 4…

Do a multiple regression with X and X2 as the independent variables.

Forecast by plugging the next X value into the quadratic equation

The Exponential model

b0 = estimated Y interceptb1 = is the compound growth factor where (b1 -1)*100% is the compound growth rate

Prepare a data matrix with the independent variable X and the common log of the dependent variable Y

Y log Y X3.8 .5798 14.2 .6232 2…

The data for the independent variable is often time series data where X is the year or month.

Do a linear regression with X as the independent variable, and the log of Y as the dependent variable.

This provides the following transformation

Forecast by plugging the X value into this linear equation yielding the log of Y.Take 10 to the power (log of Y) to get the antilog which is the actual Y forecast. or

Take 10 to the power (log of b0) to get the antilog which is the actual b0 then

Take 10 to the power (log of b1) to get the antilog which is the actual b1 and use the exponential equation

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Autoregressive Models

First-Order autoregressive modelSecond-Order autoregressive modelThird-Order autoregressive modelpth-Order autoregressive model

Autoregressive models lag the dependent variable data by one or more periods to provide a weighted moving average of the previous values of the variable Y.

Third-Order autoregressive model

Prepare a data matrix with the dependent variable Y and lagged versions of Y as the independent variables Y

X Y Y lag 1 Y lag 2 Y lag 31 3.82 4.2 3.83 3.0 4.2 3.84 4.6 3.0 4.2 3.8

5.0 4.6 3.0 4.2. . .. . .. . .

For this type of autoregressive analysis the X variable is not needed. The dependent variable is Y and the independent variables are the lagged versions of Y

For first-order autoregressive, do a multiple regression with Y lag 1 as the independent variable and Y as the dependent. Forecast by plugging the last Y value into the equation. Forecast additional periods into the future by using the most recently forecast value as the independent variable.

For second-order autoregressive, do a multiple regression with Y lag 1, and Y lag 2 as the independent variables. Forecast by plugging the last two Y values into the equation. Forecast additional periods into the future by using the most recently forecast values and previous values of Y as needed for the independent variables.

For third-order autoregressive, do a multiple regression with Y lag 1,Y lag 2, and Y lag 3 as the independent variables. Forecast by plugging the last three Y values into the equation. Forecast additional periods into the future by using the most recently forecast values and previous values of Y as needed for the independent variables.

Choosing the Best Model

Choose the model with the best adjusted r2, where r2’s are close choose the simplest model.

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statistical formulas - sonoma state universitysonoma.edu/users/s/seward/211file/computer...

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