 9.7.1: Consider again the situation described in Exercise 11 of Sec. 9.6. ...
 9.7.2: Suppose that a random variable X has the F distribution with three ...
 9.7.3: Suppose that a random variable X has the F distribution with one an...
 9.7.4: Suppose that a random variable X has the F distribution with m and ...
 9.7.5: What is the value of the median of the F distribution with m and n ...
 9.7.6: Suppose that a random variable X has the F distribution with m and ...
 9.7.7: Consider two different normal distributions for which both the mean...
 9.7.8: Consider again the conditions of Exercise 7, but suppose now that i...
 9.7.9: Consider again the conditions of Exercise 7, but suppose now that i...
 9.7.10: Suppose that a random sample consisting of 16 observations is avail...
 9.7.11: Consider again the conditions of Exercise 10. Use the results of th...
 9.7.12: Suppose that a random variable Y has the 2 distribution with m0 deg...
 9.7.13: The final column in the table of the 0.95 quantile of the F distrib...
 9.7.14: Consider again the conditions of Exercise 7. Find the power functio...
 9.7.15: Prove Theorem 9.7.5. Also, compute the pvalue for Example 9.7.4 us...
 9.7.16: Let V be as defined in Eq. (9.7.4). We wish to determine the size 0...
 9.7.17: Prove that the test found in Exercise 9 is not a likelihood ratio t...
 9.7.18: Let be the twosided F test that rejects H0 in (9.7.3) when either ...
 9.7.19: Suppose that X1,...,X11 form a random sample from the normal distri...
Solutions for Chapter 9.7: Testing Hypotheses
Full solutions for Probability and Statistics  4th Edition
ISBN: 9780321500465
Solutions for Chapter 9.7: Testing Hypotheses
Get Full SolutionsSince 19 problems in chapter 9.7: Testing Hypotheses have been answered, more than 15155 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Probability and Statistics, edition: 4. Probability and Statistics was written by and is associated to the ISBN: 9780321500465. Chapter 9.7: Testing Hypotheses includes 19 full stepbystep solutions.

aerror (or arisk)
In hypothesis testing, an error incurred by failing to reject a null hypothesis when it is actually false (also called a type II error).

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.

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

Control chart
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the incontrol value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be incontrol, or free from assignable causes. Points beyond the control limits indicate an outofcontrol process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.

Convolution
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.

Covariance
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 .

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

Cumulative sum control chart (CUSUM)
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

Defectsperunit control chart
See U chart

Degrees of freedom.
The number of independent comparisons that can be made among the elements of a sample. The term is analogous to the number of degrees of freedom for an object in a dynamic system, which is the number of independent coordinates required to determine the motion of the object.

Distribution function
Another name for a cumulative distribution function.

Error variance
The variance of an error term or component in a model.

Expected value
The expected value of a random variable X is its longterm 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.

Experiment
A series of tests in which changes are made to the system under study

Extra sum of squares method
A method used in regression analysis to conduct a hypothesis test for the additional contribution of one or more variables to a model.

Ftest
Any test of signiicance involving the F distribution. The most common Ftests 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.

Gaussian distribution
Another name for the normal distribution, based on the strong connection of Karl F. Gauss to the normal distribution; often used in physics and electrical engineering applications

Generating function
A function that is used to determine properties of the probability distribution of a random variable. See Momentgenerating function

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

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 .