Editing Chi-squared test (section)

==Chi-squared test for variance in a normal population==
If a sample of size {{math|''n''}} is taken from a population having a [[normal distribution]], then there is a result (see [[Variance#Distribution of the sample variance|distribution of the sample variance]]) which allows a test to be made of whether the variance of the population has a pre-determined value. For example, a manufacturing process might have been in stable condition for a long period, allowing a value for the variance to be determined essentially without error. Suppose that a variant of the process is being tested, giving rise to a small sample of {{math|''n''}} product items whose variation is to be tested. The test statistic {{math|''T''}} in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e. the value to be tested as holding). Then {{math|''T''}} has a chi-squared distribution with {{math|''n'' − 1}} [[Degrees of freedom (statistics)|degrees of freedom]]. For example, if the sample size is 21, the acceptance region for {{math|''T''}} with a significance level of 5% is between 9.59 and 34.17.
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==Chi-squared test for contingency table example==
[[Dispute: This example is actually for a goodness-of-fit test, and NOT a test of independence in a contingency table]] [[Dispute claim is valid]]

A chi-squared test may be applied on a [[contingency table]] for testing a null hypothesis of independence of rows and columns.

As an example of the use of the chi-squared test, a fair coin is one where heads and tails are equally likely to turn up after it is flipped. Suppose one is given a coin and asked to test if it is fair. After 200 trials, heads turn up 153 times and tails result 147 times. The following is a chi-squared analysis, where the null hypothesis is that the coin is fair:

{|class="wikitable" align="center"
|+ Chi-squared calculation of coin tosses
| |
| | Heads
| Tails
| Total
|-
| | Observed
| | 53
| | 47
| | 100
|-
| | Expected
| | 50
| | 50
| | 100
|-
| | {{math|(''O'' − ''E'')<sup>2</sup>}}
| | 9
| | 9
| |
|-
| | {{math|1=χ<sup>2</sup> = (''O'' − ''E'')<sup>2</sup>/''E''}}
| | 0.18
| | 0.18
| | 0.36
|}
In this case, the test has one [[Degrees of freedom (statistics)|degree of freedom]] and the chi-squared value is 0.36. In order to see whether this result is [[statistically significant]], the [[p-value]] (the probability that at least as extreme a result is observed when the null hypothesis is true) must be calculated or looked up in a chart. The p-value, {{math|Prob(χ<sup>2</sup> ≥ 0.36)}}, is found to be 0.5485. There is thus a probability of about 55% of seeing data that deviates at least this much from the expected results if indeed the coin is fair. This probability is not considered statistically significant evidence of an unfair coin.-->