#### Chi-square goodness-of-fit test

Goodness-of-fit Test:

The basic idea behind the chi-square goodness-of-fit test is to see when the sample comes from the population with claimed distribution. The other way of looking at that is to inquire when the frequency distribution fits a particular pattern.

Two values are included, an observed value, that is the frequency of a category from the sample and the predicted frequency that is computed and based on the claimed distribution. The derivation of formula is much similar to that of variance.

The basic idea is that when the observed frequency is actually close to the claimed (or expected) frequency, then the square of deviations will be much small. The square of deviation is divided by the predicted frequency to weight frequencies. The difference of 10 might be much significant if 12 was the predicted frequency, however a difference of 10 is not much significant at all when the expected frequency was 1200.

When the sum of such weighted squared deviations is small, then the observed frequencies are much close to the predicted frequencies and there would be no reason to refuse the claim that it came from the distribution. Whenever the sum is big than it is the reason to question distribution. Thus, the chi-square goodness-of-fit test is for all time a right tail test.

χ2 = Σ [(observed - expected)2/Expected]

The test statistic consists of a chi-square distribution whenever the given assumptions are met:

a) The data are received from the random sample.

b) The expected frequency of each and every category should be at least 5. This goes back to the need that the data be normally distributed. You are simulating a multinomial experiment (by using a discrete distribution) with the goodness-of-fit test and a continuous distribution, and if each and every expected frequency is at least five then you can utilize the normal distribution to approximate (much similar to the binomial).

Properties of the goodness-of-fit test are as shown below:

A) The data are observed frequencies. This signifies that there is just one data value for each and every category.
B) The degree of freedom is one less than the number of classes, not one less than the sample size.
C) This is always a right tail test.
D) This consists of a chi-square distribution.
E) The value of test statistic doesn’t alter if the order of categories is switched.

Interpreting the Claim:

There are four manners you might be provided a claim.

a) The values take place with equivalent frequency. The other words for this are ‘uniform’, ‘no preference’, or ‘no difference’. To determine the expected frequencies, sum the observed frequencies and then divide it by the number of categories. This quotient is expected frequency for each and every category.

b) Specific proportions and probabilities are given. To determine the expected frequencies, multiply the total of observed frequencies through the probability for each and every category.

c) The expected frequencies are provided to you. In this condition, you do not encompass to do anything.

d) The specific distribution is claimed. For illustration, ‘The data is generally distributed’. To work a problem similar to this, you require grouping a data and finding the frequency for each and every class. Then, determine the probability of being in that class by transforming the scores to z-scores and looking up the probabilities. Lastly, multiply the probabilities by the net observed frequency.

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