the formula for conditional probability:

Let there be successes out of Bernoulli trials. The sample proportion is the fraction of samples which were successes, so

(3)

"since er are using normal approximation to the binomial, both np and n(1-p) shouldbe at least 10. Furthermore, in making calculations and drawing conclusions from a specific sample, it is important that the sample be a simple random sample"
"the sample size should be no larger than 10% of the population."

The sampling distribution of the mean is a very important distribution. In later chapters you will see that it is used to construct confidence intervals for the mean and for significance testing.
Given a population with a mean of μ and a standard deviation of σ, the sampling distribution of the mean has a mean of μ and a standard deviation of
,
where n is the sample size. The standard deviation of the sampling distribution of the mean is called the standard error of the mean. It is designated by the symbol: σM . Note that the spread of the sampling distribution of the mean decreases as the sample size increases.
An example of the effect of sample size is shown above. Notice that the mean of the distribution is not affected by sample size.
Click here for an interactive demonstration of sampling distributions.

In probability theory, the central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed.[1] That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the computed average will not always be the same each time; the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a "bell curve").
http://en.wikipedia.org/wiki/Central_limit_theorem

dont be confused by the several different distributions being discussed:
1. the distribution of the original population, which may be uniform, bell-shaped, strongly skewed--anything at all.
2. there's the distribution of the data in the sample, and the larger the sample size, the more will look like the population distribution.
3. there's distribution of the means of many samples of a size, the shape can be discribed by normal model
Sampling distribution of a different between two independent sample proportions：
The mean of the sampling distribution of the difference between two independentproportions (p1 - p2) is:
.
.

Distribution of the difference between two means
It often becomes important to compare two population means. Knowledge of the sampling distribution of the difference between two means is useful in studies of this type. It is generally assumed that the two populations are normally distributed.
Sampling distribution of
Plotting sample differences against frequency gives a normal distribution with mean equal to
which is the difference between the two population means.
Variance
The variance of the distribution of the sample differences is equal to ( / ) + ( / ).
Therefore, the standard error of the differences between two means would be equal to .
Converting to a z score
To convert to the standard normal distribution, we use the formula, . We find the z score by assuming that there is no difference between the population means.

Confidence Intervals:
Typically, we consider 90%, 95%, and 99% confidence interval estimates, but any percentage is possible.
First there is the confidence interval, usually expressed in the form: estimate+margin of error.
(Note that we canoot say there is a .99 probability that the population proportion is within 2.576 standard deviation of a given proportion. For a given proportion, the population proportion either is or isn't within the specific interval, and so probability is either 1 or 0)
We know that 99%of the sample proportion should be within 2.576 standard deviationsof the population proportion.(the normal distribution)
Margin of Error= standard deviation * calculated by percentage(2.576)

In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values. The least-squares estimates b0 and b1 are usually computed by statistical software. They are expressed by the following equations:

【more on power and type II errors】
"the power is the probability of rejecting that false hypothesis"
"power gives the probability of avoiding a Type II error"
"It is actually a function where the indepenednt variable ranges over spefific alternative hypothesis"
"Choosing a smaller a(that is tougher standard to reject H0) results in a high riskof Type II error and a lower power"
"the greater the different between the null hypothesis p0 and the true value p, thesmaller the risk of a Type II error and the greater the power."

【test of significance--Chi-square and slope of least square line】
If Z1, ..., Zk are independent, standard normal random variables, then the sum of their squares,

X^2= the sum of the square of the difference of observation and experience over experience
(obs-exp)^2/exp

In a test of independence, we ask whether the two or more samples might reasonably have come from some larger set.
The smaller the resulting X^2 value, the more reasonable the null hypothesis of independence. If the X^2 value is large is enough, that is, if the P-value is small enough, we can say that the evidence is sufficient to reject the null hypothesis and to claim that there is some relationship between the two variables or methods of classification.
In this type of problem:
df=(r-1)(c-1)
where df is the degree of freedom, r is the number of rows and c is the number of columns