Final answer:
The chi-square distribution is a non-symmetrical curve used for hypothesis testing, with different shapes for each degree of freedom. It has applications in goodness-of-fit tests, tests of independence, and tests of a single variance. Calculations usually rely on technological tools and reference tables for certain values.
Step-by-step explanation:
A chi-square distribution is a statistical distribution used for hypothesis testing, particularly suited for use with categorical data. The random variable in a chi-square distribution, often represented as x², is the sum of the squares of k independent standard normal variables: x² = (Z₁)² + (Z₂)² + … + (Zk)², where k is the degrees of freedom (df) of the distribution. This sum of squares results in a curve that is non-symmetrical and skewed to the right. Each degree of freedom corresponds to a different chi-square curve. Three major applications of this distribution include the goodness-of-fit test, the test of independence, and the test of a single variance.
When using the chi-square distribution for hypothesis testing, it is essential to use a chi-square solution sheet and accurately round expected frequencies. Although most calculations rely on technology such as calculators or computer software, it is possible to reference tables for certain values.
As the number of degrees of freedom increases, the chi-square distribution increasingly resembles a normal distribution. For example, with df over 90, the chi-square distribution closely approximates a normal distribution.
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