The samples are independent.
The distributions of the residuals are normal.
The variance of each population from which the samples are taken are the same.
ANOVA involves partitioning of the total sum of squares SST into components (treatment sum of square, SSTr and error sum of squares SSE) related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels
SST=SSTr +SSE with
withwhere I is the number of samples and J is the number in each sample.
Then withwith the distribution of the test statistic we can find if there is a difference in the sample means.
Example. Perform an ANOVA test on the three treatments below.
Treatment 1 
0.56 
1.12 
0.9 
1.07 
0.94 
Treatment 2 
0.72 
0.69 
0.87 
0.78 
0.91 
Treatment 3 
0.62 
1.08 
1.07 
0.99 
0.93 
Comparing withcauses us not to reject the null hypothesis.
]]>From the definition of conditional probability,
Again from the definition of conditional probability, we can express the joint probability by conditioning onto give
Substituting (2) into (1) gives Bayes’ theorem:
If there aremutually exclusive possible outcomes forthen we can write
hence
Bayes theorem gives rise to some surprises. Many people diagnosed with disease are falsely diagnosed. Suppose that one in a thousand adults has a disease. When an individual has a disease, a positive result will be returned 99% of the time, while a positive result will be returned for 2 % of individuals who do not have the disease. Let andthenandsoand
Less that one in twenty positive diagnoses are actually true positives. More than 95% of positives results are false positives.
]]>Clinical trials to test new drugs are designed as randomized, double blind, and placebocontrolled.
Randomized: Each study subject is randomly assigned to receive either the study treatment or a placebo.
Blind: The subjects involved in the study do not know which study treatment they receive. If the study is doubleblind, the researchers also do not know which treatment is being given to any given subject. This 'blinding' is to prevent biases, since if a physician knew which patient was getting the study treatment and which patient was getting the placebo, he/she might be tempted to give the (presumably helpful) study drug to a patient who could more easily benefit from it. In addition, a physician might give extra care to only the patients who receive the placebos to compensate for their ineffectiveness. A form of doubleblind study called a "doubledummy" design allows additional insurance against bias or placebo effect. In this kind of study, all patients are given both placebo and active doses in alternating periods of time during the study.
Placebocontrolled: The use of a placebo (fake treatment) allows the researchers to isolate the effect of the study treatment.
The aim of the trial is to obtain a statistically significant result showing a significant or not difference in outcome between the groups of patients who receive a treatment and those who receive a placebo or a different treatment.
The number of patients enrolled in a study has a large bearing on the ability of the study to reliably detect the size of the effect of the study intervention. This is described as the "power" of the trial. The larger the sample size or number of participants in the trial, the greater the statistical power.
]]>
0 
1 
2 

0 
0.05 
0.05 
0 
0.1 

1 
0.2 
0.05 
0.15 
0.4 

2 
0.1 
0.06 
0.04 
0.2 

3 
0.1 
0.08 
0.12 
0.3 

0.45 
0.24 
0.31 
1The conditional probabilities forgivenare obtained by dividing each entry by the righthandmost entry shown in bold, giving
0 
1 
2 

0 
0.1 

1 
0.4 

2 
0.2 

3 
0.3 

0.45 
0.24 
0.31 
1Note that each row sums to one.
The conditional probabilities forgivenare obtained by dividing each entry by the bottom entry in the column shown in bold, giving
0 
1 
2 

0 
0.1 

1 
0.4 

2 
0.2 

3 
0.3 

0.45 
0.24 
0.31 
1 
Note that each column sums to one.
To generalise, for discrete random variables, the conditional probability mass function ofgiven (the occurrence of) the valueofwithcan be written, using the definition of conditional probability, as:
We can write down also the probability distribution ofgiven
From these we deduce
(1)
Similarly for continuous random variables, the conditional probability density function X given the value y of Y is and the conditional probability density function ofgiven the valueof can be written as
wheregives the joint density ofandwhileforgives the marginal distribution function for
Similarly as for (1) we can write
If for discrete random variablesfor allandor for continuous random variablesoror equivalentlyfor allandthenandare independent.
As a function ofgivenis a probability and so the sum over all(or integral if it is a conditional probability density) is 1. Seen as a function offor givenit is a likelihood function, so that the sum over allneed not be 1.
Because the regression lineis linear in the bi the equations above are linear too. We can solve this system of linear equations to solve for thethese solutions are labelledTheare themselves random variables because they are functions of the random variablesBecause the equations are linear, theare normally distributed with corresponding standard deviationWe can then construct confidence intervals for each
Typically we want to test whether 0 is in the interval. If it is, then at the significance level of the test, there is no evidence of a correlation betweenand
Much of the time theandare found automatically with computer packages.
Example: The table below gives data on the amount of iron, aluminium and phosphate in soil.
Observation 
=iron 
=aluminium 
=phosphate 
1 
61 
13 
4 
2 
175 
21 
18 
3 
111 
24 
14 
4 
124 
23 
18 
5 
130 
64 
26 
6 
173 
38 
26 
7 
169 
33 
21 
8 
169 
61 
30 
9 
160 
39 
28 
10 
244 
71 
36 
11 
257 
112 
65 
12 
333 
88 
62 
13 
199 
54 
40 
A computer package returns the results:
Parameter 
Estimate, 
Estimated standard deviation, 
7.35100 
3.48500 

0.11273 
0.02969 

0.34900 
0.07131 
A 99% confidence interval foris then, with
A 99% confidence interval foris, with
]]>
They consist of points representing a statistic being tracked plotted against time.
The mean of this statistic – the mean, the mean variance typically  using all the samples is calculated.
A centre line is drawn at the value of the mean of the statistic
The standard errorof the statistic in question using all the data is found.
Upper and lower control limits,and andthat indicate the threshold at which the process output is considered statistically 'unlikely' are drawn typically at 3 standard errors from the mean line(above).
The chart may have other optional features, including:
Upper and lower warning lines typically two standard errors above and below the centre line.
Division into zones, with the addition of rules governing frequencies of observations in each zone.
Annotation with events of interest, as determined by the Quality Engineer in charge of the process's quality.
A control chart is shown below.
]]>and similarly
Sinceareare independent,
The formula for the correlation betweenandis
Then the correlation betweenandis
We can follow the same procedure for any linear combination of standard normal distributions.
Ifandthen
]]>