Survey: prove 10% of population "Don't know"?

Question

I have a simple question that I expect is a standard situation, but I can't seem to find the right answer (maybe because it's too simple).

I have a survey with the following answers:

(What is your current risk tolerance for your personal investments?)
Very Aggressive                1
Aggressive                    11
Moderate                      29
Conservative                   7
Very Conservative              4
I don't know my risk tolerance 6
(total)                       58

I want to be able to say something about the 6 "Don't know". I have tried to prove that 10% of the population "don't know", by using a chi-square test:

Null hypothesis: 10% of the population "don't know"

           yes    no
expected:  52.2   5.8
observed:  52     6

Chi-square = (52-52.2)^2/52.2 + (6-5.8)^2/5.8 = 0.0077

Chi-square for a two-tailed 1-df test is 3.84

Since 0.0077 < 3.84 I cannot reject the null hypothesis and accept that 10% of the population "don't know".

Is this correct? I have a feeling I'm not barking up the right tree here...

Answer 1

The fact that you failed to reject $10\%$ does not mean that the true value is $10\%$. You could repeat the calculation for $9\%$ and fail to reject that, too. If you do the calculation for $50\%$ I predict you will be able to reject that. What you can do is find a region around $10\%$ that you have a given desired confidence that the correct answer is within. Check various percentages above $10$ until the chi-square test gives $5\%$ chance that the chi-square is worse. Say the break point is $23\%$ (I am making this up-I didn't calculate it) Then do the same for the low end-maybe it is $3\%$. Then you can say that with $90\%$ confidence the true value (under your assumption that the sample is representative, etc.) is in the range $3\%-23\%$