Are my results statistically significant on their own despite unapplicable Chi-square test? I have the results of a survey and I want to see if there are significative differences between age groups (young, middle and senior) regarding to an answer. For instance, in the table below, we see that 95 % of young people said "Yes" (268 surveyed people) vs 5 % who said no (13 people).

Usually I would run a Chi-square test to test for statistical significance but in this case I obviously cant because I only have 1 senior person who said "No" making up the 7 % (and I need at least 5 answers in each case for a Chi-square test).
My question is; can I still use the other results? I mean, we just established that we cannot test statistical significance between age groups for this answer. But does this mean that the other answers (the 95% etc...) are not significant on their own ? Can I still use the results for the Young people for instance?
I have in mind that I need at least 30 values because of normality ; is this for each cell (which means that I cant use the 13 answers of young people saying "No")? For each row (again, I can only use the "Yes"s because I only have 24 "No"s)? Or the whole sample (I have 403 answers in total), in which case every result is significant on their own (even the 7% for the only senior who said "no") ?
Thank you very much
 A: Whether it makes sense to try to distinguish between
the the three age groups has been discussed in Comments.
Whether or not age categories make sense, a traditional chi-squared test (based on counts, never percentages) gives a P-value of questionable
accuracy because you have an extremely small expected
count (less than $1)$ for Seniors saying No. [It is small expected counts that matter; looking at observed counts can be misleading.]
TBL = rbind(c(268,101,9), c(13,11,1))
TBL
     [,1] [,2] [,3]
[1,]  268  101    9
[2,]   13   11    1

chisq.test(TBL)

        Pearson's Chi-squared test

data:  TBL
X-squared = 4.2454, df = 2, p-value =  0.1375

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

chisq.test(TBL)$exp  # EXPECTED COUNTS
         [,1]       [,2]      [,3]
[1,] 263.56824 105.052109 9.3796526
[2,]  17.43176   6.947891 0.6203474

As implemented in R, a more useful P-value for this test can be
simulated. But even then the P-value is nowhere near significance at the 5% level:
chisq.test(TBL, sim=T)

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

data:  TBL
X-squared = 4.2454, df = NA, p-value = 0.1119

Not even the extended version of Fisher's Exact Test
(for a $2 \times 3$ table) implemented in R shows
significance at the 5% level:
fisher.test(TBL)$p.val
[1] 0.09588885

Whether or not the age categories make sense for your
purposes, it seems clear that the percentages of No
opinions are too nearly the same for all three age
groups to show anything worth reporting. (As in @whuber's Comment,
these tests are not invalid; the evidence for differences
among groups is very weak; significant at 10% or 12%, not 5%.)
If you want to know whether the percentage of Yes votes
is significantly above half, the answer--for all subjects
taken together--the answer is that the null hypothesis
is strongly rejected in favor of a right-sided alternative.
That is to say, for all 402 subjects 69% Yes is
significantly above 50% Yes.
        Exact binomial test

data:  278 and 378 + 24
number of successes = 278, number of trials = 402, 
 p-value = 5.523e-15
alternative hypothesis: 
 true probability of success is greater than 0.5
95 percent confidence interval:
  0.651437 1.000000
sample estimates:
 probability of success 
              0.6915423 

Note: If you want to do similar binomial tests for the the three age
groups separately, you could do so. However, in order to avoid
'false discovery' with multiple tests on the same data, you
should test at the 1% level rather than the 5% level.
Perhaps results for Seniors are unpersuasive--because you have
only ten of them--regardless of P-value. (That has nothing to do
with the fact that $10 < 30,$ but with the fact that ten is just very small, and it will be hard to believe they are truly representative of any particular population. Certainly, readers would wonder how they were selected and from what population.)
Also, any reports should make it clear that the original
intent was to see if there might be significant differences among age
groups, which there are not.
