重學 Statistics，Cha12 Tests of Goodness of Fit and Independence

Goodness of Fit Test: A Multinomial Population

Multinomial population: each element of a population is assigned to one and only one of several classes or categories
Binomial Distribution: to one and only one of 2 classes

H0: Pa = 0.3 Pb = 0.5 Pc=0.2
Ha: The population proportions are not Pa = 0.3 Pb = 0.5 Pc=0.2

用這個算出 goodness of fit 之後，然後查表找到相應的p-value

1. State the null and alternative hypotheses.
   H0: The population follows a multinomial distribution with specified probabilities for each of the k categories
   Ha: The population does not follow a multinomial distribution with the specified probabilities for each of the k categories
2. Select a random sample and record the observed frequencies fi for each category.
3. Assume the null hypothesis is true and determine the expected frequency ei in each category by multiplying the category probability by the sample size.
4. Compute the value of the test statistic
5. Rejection rule:
   
   1. p-value approach: Reject H0 if p-value <= α
   2. Critical Value Approach: Rejct H0 if X2 >= X2α
   3. where α is the level of significance for the test and there are k-1 degrees of freedom.

12.2 Test of Independence

H0: Beer preference is independent of the gender of the beer drinker
Ha: Beer preference is not independent of the gender of the beer drinker

Summary:

1. State the null and alternative hypotheses.
   
   • H0: The column variable is independent of the row variable
   • Ha: The column variable is not independent of the row variable
2. Select a random sample and record the observed frequencies for each cell of the contingency table.
3. Use equation (12.2) to compute the expected frequency for each cell.
4. Use equation (12.2) to compute the expected frequency for each cell.
5. Use equation (12.3) to compute the value of the test statistic.
6. Rejection rule:
   
   1. p-value approach: Reject H0 if p-value <= α
   2. Critical value approach: Reject H0 if X2 >= X2α
   3. where α is the level of significance, with n rows and m columns providing (n - 1)(m - 1) degrees of freedom.

12.3 Goodness of Fit Test: Poisson and Normal Distributions

Poisson Distribution

H0: The number of customers entering the store during 5-minute intervals has a Poisson probability distribution
Ha: The number of customers entering the store during 5-minute intervals does not have a Poisson distribution

POISSON DISTRIBUTION GOODNESS OF FIT TEST: A SUMMARY

1. State the null and alternative hypotheses.
   
   • H0: The population has a Poisson distribution
   • Ha: The population does not have a Poisson distribution
2. Select a random sample and
   a. Record the observed frequency fi for each value of the Poisson random variable.
   b. Compute the mean number of occurrences μ.
3. Compute the expected frequency of occurrences ei for each value of the Poisson random variable. Multiply the sample size by the Poisson probability of occurrence for each value of the Poisson random variable. If there are fewer than five expected occurrences for some values, combine adjacent values and reduce the number of categories as necessary.
4. Compute the value of the test statistic.
5. Rejection rule:
   
   1. where α is the level of significance and there are k – 2 degrees of freedom.
   2. p-value approach: Reject H0 if p-value <= α
   3. Critical value approach: Reject H0 if X2 >= X2α

關於泊松分佈的好文章：
http://www.ruanyifeng.com/blog/2013/01/poisson_distribution.html
http://www.ruanyifeng.com/blog/2015/06/poisson-distribution.html
http://maider.blog.sohu.com/304621504.html

Normal Distribution