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# Solutions for Chapter 4.11: The Central Limit Theorem ## Full solutions for Statistics for Engineers and Scientists | 4th Edition

ISBN: 9780073401331 Solutions for Chapter 4.11: The Central Limit Theorem

Solutions for Chapter 4.11
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##### ISBN: 9780073401331

This expansive textbook survival guide covers the following chapters and their solutions. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. Chapter 4.11: The Central Limit Theorem includes 20 full step-by-step solutions. Since 20 problems in chapter 4.11: The Central Limit Theorem have been answered, more than 263824 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4.

Key Statistics Terms and definitions covered in this textbook
• `-error (or `-risk)

In hypothesis testing, an error incurred by rejecting a null hypothesis when it is actually true (also called a type I error).

• Additivity property of x 2

If two independent random variables X1 and X2 are distributed as chi-square with v1 and v2 degrees of freedom, respectively, Y = + X X 1 2 is a chi-square random variable with u = + v v 1 2 degrees of freedom. This generalizes to any number of independent chi-square random variables.

• Axioms of probability

A set of rules that probabilities deined on a sample space must follow. See Probability

• Bimodal distribution.

A distribution with two modes

• Causal variable

When y fx = ( ) and y is considered to be caused by x, x is sometimes called a causal variable

• Central limit theorem

The simplest form of the central limit theorem states that the sum of n independently distributed random variables will tend to be normally distributed as n becomes large. It is a necessary and suficient condition that none of the variances of the individual random variables are large in comparison to their sum. There are more general forms of the central theorem that allow ininite variances and correlated random variables, and there is a multivariate version of the theorem.

• Coeficient of determination

See R 2 .

• Components of variance

The individual components of the total variance that are attributable to speciic sources. This usually refers to the individual variance components arising from a random or mixed model analysis of variance.

• Continuity correction.

A correction factor used to improve the approximation to binomial probabilities from a normal distribution.

• Contrast

A linear function of treatment means with coeficients that total zero. A contrast is a summary of treatment means that is of interest in an experiment.

• Counting techniques

Formulas used to determine the number of elements in sample spaces and events.

• Curvilinear regression

An expression sometimes used for nonlinear regression models or polynomial regression models.

• Discrete distribution

A probability distribution for a discrete random variable

• Dispersion

The amount of variability exhibited by data

• Estimator (or point estimator)

A procedure for producing an estimate of a parameter of interest. An estimator is usually a function of only sample data values, and when these data values are available, it results in an estimate of the parameter of interest.

• Event

A subset of a sample space.

• Expected value

The expected value of a random variable X is its long-term average or mean value. In the continuous case, the expected value of X is E X xf x dx ( ) = ?? ( ) ? ? where f ( ) x is the density function of the random variable X.

• Fixed factor (or fixed effect).

In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.

• Frequency distribution

An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on

• Hat matrix.

In multiple regression, the matrix H XXX X = ( ) ? ? -1 . This a projection matrix that maps the vector of observed response values into a vector of itted values by yˆ = = X X X X y Hy ( ) ? ? ?1 .

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