Test Statistic in Chi-Square Test

In summary, the conversation discusses the derivation of the test statistic in a chi-square test and the approximation of the statistic by a Chi-Square random variable under the null hypothesis. It is explained that the chi-square distribution with k degrees of freedom arises from the sum of squares of k standard normal distributions and that for each bin, the expected value is equal to the observed value. The conversation also mentions one linear restriction and how it affects the degrees of freedom.
  • #1
ych22
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Can anyone come up with an intuitive explanation or point me to a link that gives a derivation of the test statistic in chi-square test? I am having problems understanding why the particular test statistic is approximated by a Chi-Square random variable under the null hypothesis of the chi-square test. I cannot find any helpful literature online or in my textbooks too.

After all, the chi-square distribution with k degrees of freedom arises from the sum of squares of k standard normal distributions. This implies that for each bin, (observed-expected)^2 /expected ~ chi-square(1). Why?
 
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  • #2
Prepare for the incredibly non-formal discussion:
You have one linear restriction: the sum of the "(observed - expected)" values is zero. You have k statistics, with 1 restriction, hence k - 1 degrees of freedom.
 

Related to Test Statistic in Chi-Square Test

What is a test statistic in a chi-square test?

A test statistic in a chi-square test is a numerical value that is calculated from sample data to determine the likelihood that the observed data is representative of the expected data. It is used to assess whether there is a significant difference between the observed and expected frequencies in a categorical data set.

How is the test statistic calculated in a chi-square test?

The test statistic in a chi-square test is calculated by taking the difference between the observed and expected frequencies for each category, squaring this difference, and then dividing it by the expected frequency. This process is repeated for all categories, and the resulting values are then summed together to get the final test statistic.

What is the purpose of a test statistic in a chi-square test?

The test statistic in a chi-square test is used to determine whether there is a significant difference between the observed and expected frequencies in a categorical data set. It helps to assess whether the observed data is a good fit for the expected data, and whether any differences between the two are due to chance or some other factor.

How is the test statistic interpreted in a chi-square test?

In a chi-square test, the test statistic is compared to a critical value from a chi-square distribution with a certain degree of freedom. If the test statistic is greater than the critical value, it indicates that there is a significant difference between the observed and expected frequencies. However, if the test statistic is lower than the critical value, it suggests that any differences between the observed and expected data may be due to chance.

What are some limitations of using a test statistic in a chi-square test?

One limitation of using a test statistic in a chi-square test is that it can only be used for categorical data. It also assumes that the sample data is representative of the entire population, and that the expected frequencies are known or can be estimated accurately. Additionally, it is important to have a sufficiently large sample size for the test statistic to be reliable.

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