Solutions for Chapter 9.5: 2p Factorial Experiments

A full factorial experiment with k factors and all factors tested at only two levels (settings) each.

The portion of the variability in a set of observations that can be traced to speciic causes, such as operators, materials, or equipment. Also called a special cause.

See Arithmetic mean.

Data consisting of counts or observations that can be classiied into categories. The categories may be descriptive.

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

A second-order response surface design in k variables consisting of a two-level factorial, 2k axial runs, and one or more center points. The two-level factorial portion of a CCD can be a fractional factorial design when k is large. The CCD is the most widely used design for itting a second-order model.

The probability mass function of the conditional probability distribution of a discrete random variable.

When a factorial experiment is run in blocks and the blocks are too small to contain a complete replicate of the experiment, one can run a fraction of the replicate in each block, but this results in losing information on some effects. These effects are linked with or confounded with the blocks. In general, when two factors are varied such that their individual effects cannot be determined separately, their effects are said to be confounded.

The probability 1?a associated with a conidence interval expressing the probability that the stated interval will contain the true parameter value.

Another term for the conidence coeficient.

A continuous random variable with range of a inite interval and a constant probability density function.

A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).

A measure of association between two random variables obtained as the expected value of the product of the two random variables around their means; that is, Cov(X Y, ) [( )( )] =? ? E X Y ? ? X Y .

A control chart in which the point plotted at time t is the sum of the measured deviations from target for all statistics up to time t

A subset of effects in a fractional factorial design that deine the aliases in the design.

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.

Any test of signiicance involving the F distribution. The most common F-tests are (1) testing hypotheses about the variances or standard deviations of two independent normal distributions, (2) testing hypotheses about treatment means or variance components in the analysis of variance, and (3) testing signiicance of regression or tests on subsets of parameters in a regression model.

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.

A function used in the probability density function of a gamma random variable that can be considered to extend factorials

Effects in a fractional factorial experiment that are used to construct the experimental tests used in the experiment. The generators also deine the aliases.