Normal Probabilities
The normal curve results when a large number of observations are piled together. The area under any normal curve is 1. The probability that a variable lies within an interval is given by the area under the curve.  Below is an unlabeled example:

Each curve is distinguished from other normal curves by two features: a mean (m), which tells where the center of the distribution is, and standard deviation (s), which tells how wide the distribution is. The examples below assume a mean of 40 and a standard deviation of 1.

 P(X > 43) P(39 < X < 42)

The standard normal curve has mean=0 and standard deviation=1. The problems above would be normalized with the formula: z = (x-
m)/s.

Hypothesis Tests t and z
hypothesis test
A hypothesis is a claim about a parameter (e.g., mean, variance, correlation, proportion). The hypothesis comes in two statements: a null hypothesis - which states that there is no difference between the parameter and say 10, and an alternate hypothesis - which states that there is a difference.
t-test
The t-test compares the mean of one set of data against the mean of another set.

A one-sample t-test compares the mean of a data set against an already known population mean. For instance, a set of IQ tests is measured against the national average IQ of 100.

A two-sample t-test compares the mean of one data set against the mean of another data set. For instance, a set of men's IQ scores might be compared with a set of women's IQ scores.

The difference between the z-test and the t-test is the formula used to compute the statistic.

z-test
The z-test compares the mean of one set of data against the mean of another set.

A one-sample z-test compares the mean of a data set against an already known population mean. For instance, a set of IQ tests is measured against the national average IQ of 100.

A two-sample z-test compares the mean of one data set against the mean of another data set. For instance, a set of men's IQ scores might be compared with a set of women's IQ scores.

The difference between the z-test and the t-test is the formula used to compute the statistic.

Confidence Intervals
A 95% confidence interval is an interval that you are 95% confident contains the mean. This does not mean that when you grab a sample from the population, 95% of the time, the mean will fall into that interval. The mean is the mean of the population. It doesn’t change – ever. What it means is this: Each sample has a related confidence interval. 95% of those intervals will contain the population mean.