CHISQINV

Calculates the inverse of the CHISQDIST function.

Syntax

CHISQINV(p, k)

k is the degrees of freedom for the $χ^{2} - d i s t r ib u t i o n$ .

Constraint: k must be a positive integer

p is the given probability

Constraint: 0 ≤ p < 1

Semantic

CHISQINV(p, k) returns the value x, such that CHISQDIST(x, k,TRUE()) = p.

Example

CHISQINV(0.5, 9)

returns approximately 8.342832692

.

Remarks

If you need CHISQINV(p,k) for a non interger parameter k, then use GAMMAINV(p,k/2,2) instead

Application:

Testing the Fairness of a Six-Sided Dice

A game company, "Polyhedral Games," claims that its six-sided dice are perfectly fair. To test this claim, an independent auditor, "AuditWorks," rolls one of their dice 600 times.

The Hypothesis

Null Hypothesis (H0): The die is fair. The probability of rolling each number (1, 2, 3, 4, 5, 6) is equal, which is 1/6.
Alternative Hypothesis (Ha): The die is not fair. At least one of the probabilities is different from 1/6.

The Data

The auditor records the observed frequencies of each roll.

	Number Rolled	Observed Frequency (O)	Expected Frequency (E)
	A	B	C
1	1	95	100
2	2	105	100
3	3	98	100
4	4	102	100
5	5	90	100
6	6	110	100
7	Total	600	600

Observed Frequency (O): The actual number of times each number was rolled.
Expected Frequency (E): If the die were perfectly fair, we would expect each number to be rolled 600×(1/6)=100 times.

The Chi-Squared Calculation

The chi-squared statistic (χ2) is calculated using the formula:

$χ^{2} = \sum \frac{( O - E ) ^{2}}{E}$

Using the data from the table:

For the number 1: (95−100)2/100=0.25
For the number 2: (105−100)2/100=0.25
For the number 3: (98−100)2/100=0.04
For the number 4: (102−100)2/100=0.04
For the number 5: (90−100)2/100=1.00
For the number 6: (110−100)2/100=1.00

Calculated χ2=0.25+0.25+0.04+0.04+1.00+1.00=2.58

Using CHISQINV to find the Critical Value

To make a decision about the null hypothesis, we need to compare our calculated χ2 statistic to a critical value. The critical value is the threshold for a given significance level (α) and degrees of freedom.

Significance Level (α): Let's choose a standard significance level of 0.05 (or 5%). This means we are willing to accept a 5% chance of rejecting a true null hypothesis.
Degrees of Freedom (df): For a goodness-of-fit test, the degrees of freedom are the number of categories minus 1. In this case, there are 6 categories (the numbers 1 through 6), so df=6−1=5.

Now, we can use the CHISQINV function to find the critical value:

CHISQINV(probability, degrees_freedom)

The probability argument in the CHISQINV function is 1−α. In this case, 1−0.05=0.95.

CHISQINV(0.95, 5)

This function returns approximately 11.07. This is our critical value.

Conclusion

Critical Value: 11.07
Calculated χ2: 2.58

Since our calculated χ2 statistic (2.58) is less than the critical value (11.07), we fail to reject the null hypothesis.

Final Result: Based on the evidence, there is not enough statistical support to conclude that the die is unfair at a 5% significance level. The results from the 600 rolls are within the range of what would be expected from a fair die.

Result for χ2:

2.58

Result for CCHISQINV:

11.07