Khan Open Class-Statistics Study Notes: (10) chi-square distribution

X-2 Distribution

Random Variable X is an independent standard normal distribution variable ~ N (0, 1), that is, E (x) = 0, VAR (x) = 1.

Q1 = X12, Q1 is a chi-square distribution, recorded as, degree of freedom is 1

Q2 = X12 + x22, Q2 is a chi-square distribution, recorded as, degree of freedom is 2

And so on. The figure shows the distribution chart of Chi-square. There are tables available for query, such as P (Q2> 2.41) = 0.3

Pearson's chi-squared test

The square of the difference between the actual observed number o and a certain theoretical number (E, also known as the expected number of times) is then divided by the theoretical number of times, which is a frequency distribution that is very close to one of the sample distributions.

The degree of freedom is the number of irrelevant events.

Just as N is large enough, the two distributions are very consistent with the normal distribution. Here we do not understand that, given by French mathematician Pearson, we believe that they are effective when a tool is provided, to use tools,Used to check the frequency of occurrence.

Example 1: One-dimensional X2 test

It is known that from Monday to Saturday, the customer distribution ratio is 10%, 10%, 15%, 20%, 30%, and 15%, while the observed number of customers is 30, 14, 34, 45, 57, and 20, ask if the ratio is correct under significance level α = 5%.

H0: the ratio is correct; H1: the ratio is incorrect.

The number of degrees of freedom is six, but the number of degrees of freedom is n-1 = 5. As long as you know five, 6th parameters can be calculated. Query the X-table. For rows with degrees of freedom 5, if α = 5%, X-C2 = 11.07. The calculated value is more extreme than the queried value. All rows reject the distribution ratio H0.

Example 2: Two-Dimensional join table X2 test

The test may be performed in a two-dimensional table. For example, if the customer has divided the two men and women, since the last row and the last column can be obtained from other information, it is not an independent variable and the degree of freedom is m-1) (n-1 ). The join table also becomes the contingency table. The following example shows whether herbs are valid at significance level α = 10% during a season of onset.

Based on the above information, we will add:

H0: Herb do nothing; H1: herbs do something

2 rows and 3 columns, degrees of freedom (2-1) (3-1) = 2, query chi table, α = 5% when x c2 = 4.5, we cannot reject H0, that is, herbs cannot be considered as useful.

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