How do I calculate the p-value for a (2x2) Chi Square test when an expected value is zero?

Question

For example, a simple Yes/No survey of 500 people:
"Do you think Katie Price will find a cure for cancer in the next decade?"

Expected
No: 500
Yes: 0

Observed
No: 450
Yes: 50

We don't expect anyone to say yes. But actually, 50 people have confused Katie Price for a gifted medical researcher.
How can we statistically prove our observations are significantly different from our expectations?

We can't divide by zero ("the number of yes's we expect"), as it's mathematically undefined, therefore a Chi Square test does not work.

Suggested (but not viable) solutions:
1. Fisher's Exact Test isn't designed for large samples, as it will result in huge numbers (e.g. 500! = a number too large for excel to handle).
2. Yate's correction for continuity does not change the denominator, we must still divide by zero.
3. Randomisation tests are not possible for this sample.

Any help most appreciated!

Answer 1

Tables in a Chi-Square test can have two types of 0 values:

Structural and functional zeroes. Structural zero are impossible case, value in a table number of strokes per hole n hole 1 is necessarily smaller than any table with a score of the whole round.



Functional zeroes are possible but not probable values, if one looks at age of a woman when bearing a child the woman's' ages less than 3 or greater than 80 are possible but improbable. Nonetheless, This is a possible real world solution--outliers as it were.



We can change the test so that the cells with possible values in a dimension are so improbable as to be 0, we add .5 to the cell count and we calculate the chi-square test.



So your data will be Expected 499.5 No .5 Yes, so your chi-square with 1 degree of freedom calculates to:



(450-499.5)^2 + (50-.5)^2 = .25 + 2450.25 = 2450.5 which rejects Ho.



This is a rule of thumb used in modeling tables via chi-square tests. You may balk because it's not a natural part of the theory, but that's the case for the whole test. The whole test uses chi-squre for its test statistic since it's easy ti calculate--but the ddistributionof cells has an ASYMPTOTIC Chi sSquare

distribution, In the professional world, this works quite well.

Answer 2

Chi Square P Value Calculator

Answer 3

Chi Square Calculator 2x2

Answer 4

As stated, the question doesnt make sense. If you really had an expected of 0, there is nothing to analyze - the probability mechanism assigned a value of zero to the likelihood of finding a cure. How was the expected computed? There must have been an underlying probability distribution, and the probability could not have been zero. Are you sure you dont have the Observed and Expected reversed?