Section 8.3

# Section 8.3 - Section 8.3 Test for a Categorical Population...

This preview shows pages 1–6. Sign up to view the full content.

Section 8.3 Test for a Categorical Population Assume each unit in the population can be placed in to one of k non-overlapping categories. For example, we may want to study a characteristic of the people belonging to different age groups say 15-30, 40-50 and 50-70. Sometimes a unit may be classified with respect to 2 variables 1 x and 2 x are independent. To test a hypothesis here, we use a new probability distribution known as 2 χ - distribution . Definition. The random variables X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chi-square distribution. This is usually written The chi-square distribution has one parameter: k - a positive integer that specifies the number of degrees of freedom (i.e. the number of X i ) 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The following graphs are some of the of the 2 χ - densities. Probability density function A probability density of the chi-square distribution is where Γ denotes the Gamma function which has closed-form values at the half-integers. 2
Test for Univariate data : Assume each unit in the population can be classified into one of k non-overlapping categories. That is the characteristic X is a discrete variable and } ,..., 2 , 1 { k X with ) ( i X P i = = π . 1 = proportion of units belonging to 1 st category; . . . k = proportion of units belonging to k th category. Also, i = probability that a randomly chosen individual falls into i-th category. Then, 1 ... 1 = + + k . Consider 0 10 1 0 ,..., : k k H = = (specified) against 0 : H H a is not true. Take a sample of size n from the population and let 1 n = no. of individuals belonging to 1 st category . . . k n = no. of individuals belonging to k th category. so that = k i n n 1 . 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The value k n n ,..., 1 are called observed counts. The basic idea is to compare observed counts with expected (under 0 H ) counts 0 10 ,..., k n n π . How do we do that? A simple way is - k i i n n 1 2 0 ) ( But, for example, if 1 n = 95, 100 10 = n 2 n = 15, 20 20 = n Then both have same difference = 5. But the first frequency is less than 5%, while 0 H second one is 25% less than their expected frequencies. The 2 χ - test takes into account the percentage derivatives. 2 - test for Univariate Data Hypothesis 0 10 0 ,..., : k k H = = ( 0 i are specified) against 1 H : 0 H is not true. Test Statistic: = - = k i i i i n n n X 1 0 2 0 2 ) ( 4
, ) ( 2 - = k i i i i E E O where i O = observed frequency, i E = expected frequency of the i th category. When 0 H is true, 2 X - follow chi-square distribution with (k-1) df, when all 5 0 i n π for all i. Also, P – value = ) ( 2 2 ) 1 ( x X P k - (Note 2 x is the observed value) Note : If for some 5 0 < i n , categorizes should be combined so that the conditions are satisfied. Example

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 18

Section 8.3 - Section 8.3 Test for a Categorical Population...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online