What test to use to compare proportions of answers "Yes", "No" and "I don't know" in two different researches I am trying to recreate the results of one research I found, because authors did not mention what test they used. The situation is as follows:
They gathered the following counts of answers for their question: 29 yes, 2 no, 72 I don't know.
They compared their answers with the answers of other people, they had the following counts: 110 yes, 4 no, 92 I don't know.
So they compared their answer counts with other answer counts and wrote that the p-value is 0.00012.
I would like to find out what test might be used here, because I was thinking about simple proportion tests, but there we have to compare only yes with yes, no with no and idk with idk, each in 1 test, but that gives 3 different p-values.
 A: Probably a chi-squared test of homogeneity, using
the following table:  In R, it might have been as
shown.
x1 = c(29, 2, 72);  x2 = c(110, 4, 92)
TBL= rbind(x1, x2);  TBL
   [,1] [,2] [,3]
x1   29    2   72
x2  110    4   92

chisq.test(TBL)

        Pearson's Chi-squared test

data:  TBL
X-squared = 17.971, df = 2, p-value = 0.0001252

Warning message:
In chisq.test(TBL) : 
Chi-squared approximation may be incorrect

The warning message calls attention to the very small
counts in the second column, which may mean that the
'chi-squared' statistic does not have an approximate chi-squared distribution, possibly making the P-value unreliable.

Addendum, per Comment: Showing table of expected counts for
this chi-squared test.  Chi-squared statistic (sum of six 'contributions') is
$\sum_i\sum_j\frac{(X_{ij}- E_{ij})^2}{E_{ij}},$ where $X_{ij}$ are in data TBL and $E_{ij}$ are displayed using $-notation as below:
chisq.test(TBL)$exp
       [,1] [,2]      [,3]
x1 46.33333    2  54.66667
x2 92.66667    4 109.33333

Warning message:   # as above


In R, it is possible to obtain a more reliable P-value by simulation, as demonstrated below:
chisq.test(TBL, sim=T)

        Pearson's Chi-squared test 
        with simulated p-value 
        (based on 2000 replicates)

data:  TBL
X-squared = 17.971, df = NA, p-value = 0.0004998

Another possible path, given the error message, would be to use Fisher's exact test, as extended to a $2\times 3$ table. [The original Fisher test was only for $2\times 2$ tables.]
fisher.test(TBL)

        Fisher's Exact Test for Count Data

data:  TBL
p-value = 4.758e-05
alternative hypothesis: two.sided

By any reasonable test, it seems clear that the
the two researchers are using different instruments
or methods.
