- 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 p-value 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 two-sided 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
a-error (or a-risk)
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.
The probability 1?a associated with a conidence interval expressing the probability that the stated interval will contain the true parameter value.
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the in-control 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 in-control, or free from assignable causes. Points beyond the control limits indicate an out-of-control process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.
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.
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.
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
Defects-per-unit 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.
Another name for a cumulative distribution function.
The variance of an error term or component in a model.
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.
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.
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.
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
A function that is used to determine properties of the probability distribution of a random variable. See Moment-generating function
Effects in a fractional factorial experiment that are used to construct the experimental tests used in the experiment. The generators also deine the aliases.
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 .