Statistics Exam 3

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Multiple regression equation

mathematical equation relating expected value or mean value of dependent variable to values of independent variables

Multiple regression model

mathematical equation that describes how dependent variable y is related to independent variables and an error term

Coefficient of determination

measure of goodness of fit of the estimated regression equation. it can be interpreted as the proportion of the variability in the dependent variable y that is explained by the estimated regression equation

Adjusted multiple coefficient of determination

measure of the goodness of fit of the estimated multiple regression equation that adjusts for the number of independent variables in the model and thus avoids overestimating the impact of adding more independent variables

Multiple coefficient of determination

measure of the goodness of fit of the estimated multiple regression equation. can be interpreted as the proportion of variability in the dependent variable that is explained by estimated regression equation

Correlation coefficient

measure of the strength of the linear relationship between two variables

Least squares method

method used to develop estimated regression equation. it minimizes sum of squared residuals (the deviations between the observed values of the dependent variable and the estimated values of the dependent variable)

If Bi=0, this is the relationship between the independent variable xi and the dependent variable

no relationship

Dependent variable

variable that is being predicted or explained (denoted by y)

Independent variable

variable that is doing the predicting or explaining (denoted by x)

Consider the sample regression equation: y hat=50 - 10xi, with an R Squared value of 0.64. What is the value of the correlation coefficient between x and y?

-.8

If r squared = 0, then SSR equals this

0

The sum of error terms should approximate this number

0

This method minimizes the Sum of Squared Errors, ie. Minimizes SSE

Least Squares Method

The formula for Mean Squares Within is:

MSE=SSE/(nt-k)

Assumptions About Error Term in Regression Model

1. The error term is a random variable with a mean or expected value of zero 2. The variance of epsilon, denoted by sigma squared, is the same for all values of x 3. The values of epsilon are independent 4. The error term is a normally distributed random variable for all values of x

A regression analysis involved 6 independent variables and 25 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have

18

At least this many populations are under consideration for ANOVA

2

An ANOVA procedure is applied to data obtained from 5 samples where each sample contains 10 observations. The degrees of freedom for the critical value of F are

4 and 45

If the coefficient of correlation is -0.7, the percentage of variation in the dependent variable explained by the estimated regression equation is

49%

In a 1-way ANOVA, given R Squared = .85 and SSR = 700, this is the SST

700/.85

The mean square of the treatments given the SSTR=25.08 and df=3 is

8.36

Given the above and an alpha of 5%, if the parameter of X1 was statistically significant, its p-value would need to be this

< .05

Another term for R Squared

Coefficient of Determination

The null hypothesis to determine if the slope of Bi differs from 2

Ho: Bi=2

Null hypothesis to test whether or not there is a difference between treatments A, B, and C, a sample of 8 observations has been randomly assigned to the 4 treatments

Mu1=Mu2=Mu3=Mu4

SSR = SST -

SSE

What is the implication of the assumption that the error term, epsilon, is a normally distributed random variable?

The dependent variable, y, will also be a normally distributed variable

Which of the following is not true with respect to the error in a least squares linear regression?

The sum of squares due to error is minimized by the least squares method

In a simple linear regression, there are two ways to explain R Squared: the variation of the response variable can be explained by 1) the regression model or explained by this

Variation of the explanatory (independent) variable

ANOVA table

a table used to summarize the analysis of variance computations and results. It contains columns showing the source of variation, the sum of squares, the degrees of freedom, the mean square, the F value, and the p-value

Single-factor experiment

an experiment involving only one factor with k populations or treatments

Randomized block design

an experimental design employing blocking

Completely randomized design

an experimental design in which the treatments are randomly assigned to the experimental units

Residual analysis

analysis of residuals used to determine whether the assumptions made about the regression model appear to be valid. also used to identify outliers and influential observations

Response variable

another word for the dependent variable of interest

Factor

another word for the independent variable of interest

A regression analysis between supply (y in 10 widgets) and price (x in dollars) resulted in the following equation: =5 - 3x The above equation implies that if the price is increased by $1, the supply is expected to:

decrease by 30 widgets

ith residual

difference between observed value of dependent variable and value predicted using estimated regression equation

Treatments

different levels of a factor

Regression equation

equation that describes how mean or expected value of dependent variable is related to independent variable

Regression model

equation that describes how y is related to x and an error term

Estimated multiple regression equation

estimate of multiple regression equation based on sample data and least squares method

Estimated regression equation

estimate of the regression equation developed from sample data by using the least squares method

Scatter diagram

graph of bivariate data in which independent variable is on horizontal axis and dependent variable is on vertical axis

In regression analysis, the variable that helps to explain:

independent (or explanatory) variable

The adjusted multiple coefficient of determination is adjusted for the number of these type of variables

independent variables

Prediction interval

interval estimate of an individual value of y for a given value of x

Confidence interval

interval estimate of the mean value of y for a given value of x

P-value where value of F is 1.5 with df1, df2 equal to 5 and 10, respectively

look up

least squares method

procedure used to develop estimated regression equation

The error terms being "this" ensures they are not correlated with any of the explanatory variables

random, no pattern

Simple linear regression

regression analysis involving one independent variable and one dependent variable in which the relationship between the variables is approximated by a straight line

Multiple regression analysis

regression analysis involving two or more independent variables

Standard error of the estimate

square root of the mean square error, denoted by s. is is estimate of sigma, the standard deviation of the error term

Multiple comparison procedures

statistical procedures that can be used to conduct statistical comparisons between pairs of population means

Multicollinearity

term used to describe correlation among independent variables

Experimental units

the objects of interest in the experiment

Comparisonwise Type I error rate

the probability of a Type I error on at least one of several pairwise comparisons

Experimentwise Type I error rate

the probability of making a Type I error on at least one of several pairwise comparisons

Partitioning

the process of allocating the total sum of squares and degrees of freedom to the various components

Blocking

the process of using the same or similar experimental units for all treatments. the purpose of blocking is to remove a source of variation from the error term and hence provide a more powerful test for a difference in population or treatment means

Mean square error

the unbiased estimate of the variance of the error term sigma squared (denoted by s squared)


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