Solutions for Chapter 8: Sampling Variability and Sampling Distributions

The arithmetic mean of a set of numbers x1 , x2 ,…, xn is their sum divided by the number of observations, or ( / )1 1 n xi t n ? = . The arithmetic mean is usually denoted by x , and is often called the average

An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.

Sequences of independent trials with only two outcomes, generally called “success” and “failure,” in which the probability of success remains constant.

The joint distribution of two normal random variables

In experimental design, a group of experimental units or material that is relatively homogeneous. The purpose of dividing experimental units into blocks is to produce an experimental design wherein variability within blocks is smaller than variability between blocks. This allows the factors of interest to be compared in an environment that has less variability than in an unblocked experiment.

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

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

The variance of the conditional probability distribution of a random variable.

If it is possible to write a probability statement of the form PL U ( ) ? ? ? ? = ?1 where L and U are functions of only the sample data and ? is a parameter, then the interval between L and U is called a conidence interval (or a 100 1( )% ? ? conidence interval). The interpretation is that a statement that the parameter ? lies in this interval will be true 100 1( )% ? ? of the times that such a statement is made

A random variable with an interval (either inite or ininite) of real numbers for its range.

A two-dimensional graphic used for a bivariate probability density function that displays curves for which the probability density function is constant.

A square matrix that contains the correlations among a set of random variables, say, XX X 1 2 k , ,…, . The main diagonal elements of the matrix are unity and the off-diagonal elements rij are the correlations between Xi and Xj .

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 .

Another name for factors that are arranged in a factorial experiment.

The cumulative distribution of the standard normal distribution, often denoted as ?( ) x and tabulated in Appendix Table II.

A quality tool that graphically shows the location of defects on a part or in a process.

See U chart

A study in which a sample from a population is used to make inference to the population. See Analytic study

In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a model-itting process and not on replication.

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