From a Chi-Square Curve to a Probability
In “Shape of Chi-Square Distributions,” you learned that the degrees of freedom determine the chi-square reference curve. In “The Chi-Square Statistic Formula,” you learned how a table’s cell counts produce a statistic. The next task is to find how much area lies at or to the right of that statistic on the curve with the correct degrees of freedom.
Chi-square tests use the right tail because larger values of \(X^2\) indicate greater differences between observed and expected counts. The probability in that tail is the p-value for the test, provided the chi-square model is appropriate. A table can usually give a range for the p-value; a calculator can give a more precise value.
Because a chi-square distribution is continuous, \(P(X\ge x)\) and \(P(X>x)\) are equal. The chance of getting exactly one particular value is zero, so including or excluding the endpoint does not change the area.
Reading an Upper-Tail Chi-Square Table
A standard chi-square table is organized by degrees of freedom and upper-tail probabilities. The entry in a row and column is a critical value: the value of \(X^2\) with the stated area to its right. For example, if the critical value in the \(df=2\) row and the 0.05 column is 5.991, then \(P(X\ge5.991)=0.05\) for a chi-square distribution with 2 degrees of freedom.
To use the table with an observed statistic, find its row using \(df\), then compare the statistic with the critical values in that row. Since larger statistics have smaller right-tail areas, the two neighboring critical values tell you between which two probabilities the p-value lies. Check the column headings carefully: some references arrange tables differently, but the examples here use upper-tail probabilities.
A table’s range is useful, but it is not an exact p-value. Do not report the probability for the nearest critical value as if it were the probability for your statistic. Unless the statistic matches a listed critical value, report a bound from the table or use a calculator for a more precise area. Linear interpolation between two critical values is not a dependable way to find the probability; the curve does not change at a constant rate.
Finding the Area With a Calculator
A cumulative distribution function, or cdf, gives the area to the left of a value when its upper bound is that value. On a TI-84, the chi-square cdf command accepts a lower bound, an upper bound, and degrees of freedom. To obtain the upper-tail area, enter the observed statistic as the lower bound and a very large number as the upper bound. The value \(1\mathrm{E}99\) acts as infinity for this calculation.
For example, to find the area to the right of 6 for \(df=2\), enter \(\operatorname{chi2cdf}(6,1\mathrm{E}99,2)\). The result is approximately 0.0498. A second way to express the same area is one minus the area to the left: \(1-\operatorname{chi2cdf}(0,x,df)\). The direct upper-tail entry is generally convenient and makes the direction of the area clear.
Worked Example: Bracketing a Probability With a Table
Worked Example: Bracketing a Probability With a Table
A chi-square test has statistic \(X^2=6.00\) and \(df=2\). Use a standard upper-tail table to bracket its p-value, then check the result with a calculator.
In the \(df=2\) row, the critical value for an upper-tail area of 0.05 is 5.991, and the critical value for an upper-tail area of 0.025 is 7.378. The observed statistic falls between them:
The right-tail area decreases as the statistic increases. Therefore, the area to the right of 6.00 is less than 0.05 but greater than 0.025:
On a TI-84, enter \(\operatorname{chi2cdf}(6,1\mathrm{E}99,2)\). The calculator returns approximately 0.0498, rounded to four decimal places. This is consistent with the table’s range. As a numerical check, the \(df=2\) right-tail area at 6 is \(e^{-6/2}=e^{-3}\approx0.0498\).
Answer: The table shows that the p-value is between 0.025 and 0.05. The calculator gives the more precise result \(p\approx0.0498\). The value is close to, but below, 0.05; it is not exactly 0.05.
Worked Example: Using the Calculator for a More Precise Area
Worked Example: Using the Calculator for a More Precise Area
A chi-square statistic is \(X^2=8.00\) with \(df=4\). Find its upper-tail probability with a calculator and check that the table gives a compatible range.
On the calculator, enter \(\operatorname{chi2cdf}(8,1\mathrm{E}99,4)\). The result is approximately 0.0916. This is the area under the \(df=4\) curve at or to the right of 8.
Check the result against the \(df=4\) row of an upper-tail table. The critical value for an upper-tail probability of 0.10 is 7.779, and the value for 0.05 is 9.488. Since \(7.779<8.00<9.488\), the statistic’s right-tail area must be between 0.05 and 0.10. The calculator result, 0.0916, is within that interval.
As an arithmetic check, for \(df=4\) the upper-tail area at 8 can be written \(e^{-8/2}(1+8/2)=e^{-4}(5)\). This is approximately \(0.0183156\times5=0.091578\), which rounds to 0.0916. The check agrees with the calculator.
Answer: \(P(X\ge8.00)\approx0.0916\) for \(df=4\). The table alone gives the range \(0.05<p<0.10\); the calculator supplies a more precise probability.
Worked Example: Interpreting a Chi-Square P-Value
Worked Example: Interpreting a Chi-Square P-Value
Suppose a fictional community survey uses a chi-square test to examine whether residents’ preferred way to receive local alerts is associated with age group. The test produces \(X^2=14.00\) with \(df=6\). Find the upper-tail probability and explain what it means if the significance level is \(\alpha=0.05\).
State. Let \(X\) be a chi-square random variable with 6 degrees of freedom. The requested probability is the area to the right of the observed statistic, \(P(X\ge14.00)\).
Plan. Use the chi-square distribution with \(df=6\), as determined by the test’s table structure. Use the upper-tail cdf because chi-square test statistics at least as large as the observed value count as evidence against the null hypothesis. This calculation assumes the chi-square reference model is appropriate for the test.
Do. Enter \(\operatorname{chi2cdf}(14,1\mathrm{E}99,6)\). The calculator gives approximately 0.0296. A table check agrees: for \(df=6\), the critical values for upper-tail probabilities 0.05 and 0.025 are 12.592 and 14.449. Since \(12.592<14.00<14.449\), the probability is between 0.025 and 0.05. More precisely, the \(df=6\) calculation gives \(e^{-7}(1+7+7^2/2)=e^{-7}(32.5)\approx0.0296\).
Conclude. If the null hypothesis of no association is true and the chi-square model is appropriate, the probability of obtaining a statistic of 14.00 or larger is about 0.0296. Because this is less than \(\alpha=0.05\), the result would lead to rejecting \(H_0\). In context, the survey provides convincing evidence of an association between residents’ age group and preferred way to receive local alerts. The p-value does not measure the probability that the null hypothesis is true.
Common Mistakes and AP Exam Tip
- Using the left-tail area: A chi-square cdf with bounds from zero to the statistic gives area to the left, not the p-value for a chi-square test. Use the statistic as the lower bound and a very large upper bound, or subtract the left-tail area from 1.
- Forgetting degrees of freedom: The same statistic can have different upper-tail probabilities for different \(df\). Use the degrees of freedom from the test, not a nearby row that looks convenient.
- Reporting a table bound as an exact p-value: If the statistic falls between two critical values, report the probability range. For a more precise value, use the calculator. Do not claim the p-value equals a column heading unless the statistic matches that column’s critical value.
- Reversing the table probabilities: A larger chi-square statistic has a smaller upper-tail area. If the statistic lies between the 0.05 and 0.025 critical values, its p-value is between 0.025 and 0.05—not the other way around.
- Giving an incomplete interpretation: A p-value is not the probability that the null hypothesis is true. State it as a probability of a statistic at least as large as the observed one, assuming the null hypothesis and the model are appropriate.
Check Your Understanding
Use the table direction, statistic, and degrees of freedom together to find or interpret each upper-tail probability.
- A statistic is 10.2 with \(df=4\). Which calculator bounds would you enter to find its upper-tail probability?
- For \(df=2\), a statistic falls between the critical values 4.605 for an upper-tail area of 0.10 and 5.991 for an area of 0.05. What range does the table give for its upper-tail probability?
- Why is the calculator entry \(\operatorname{chi2cdf}(0,x,df)\) not itself the upper-tail p-value?
- A \(df=6\) statistic lies between the critical values for upper-tail areas 0.05 and 0.025. Which probability is larger: the p-value or 0.025? Explain.
- In context, what does an upper-tail p-value of 0.03 mean under the null hypothesis?