Critical Value

A critical value (C.V.) is a number that is used to make estimates and test hypotheses. Critical values always correspond to a probability.

This number represents the distance from itself to the center of a bell-shaped graph, either the z or t distribution. The area in this section represents the probability of the C.V.

For example, using the z distribution, the number 1.96 is 47.5% likely. When you also include -1.96, then the likelihood is doubled.

Alpha and Confidence Level are probabilities that correspond to critical values.

Degrees of Freedom

Degrees of freedom (df) equals the sample size minus the number of estimated parameters. Commonly there is only one estimator - the sample standard deviation.

df=n-k, where k is the # of parameters being estimated.

df=n-1, k = 1 for a single population using an estimated standard deviation

t-table

The t-table is used when the population standard deviation ("sigma") is unknown. Sigma is estimated using the sample standard deviation ("s"). The graph is bell-shaped.

It is related to the z-table, but with greater dispersion. Each sample size has a unique set of t-values, determined by the "degrees of freedom (d.f.)"

Central Limit Theorem

Central Limit Theorem (CLT) is the fundamental concept that allows researchers to make conclusions about a population, having drawn only a sample (inferential statistics).

When sufficient samples are drawn, the means of each sample form a normal, that is, a bell-shaped, distribution. This is called the sampling distribution.

The center of the sampling distribution is equal to the mean of the population (mu) from which the samples were drawn.

It's standard deviation is proportionate to the population's standard deviation, relative to the size of the samples (n). The smaller the sample size, the greater the variation, so, the wider the graph. Larger samples will produce a more narrow graph, that is, have smaller variation.

Samples are sufficent either when drawn from a normal population or are "sufficiently large." The common rule of thumb is for n >= 30.