# Probability of a Sample Percentage Compared to Expected Percentage

I have taken a thinking style preference measure with my class of 22 students. According to the manufacturer, each of the five scales has an expected value in the general population. For example, 24% are Realists, however my class showed that 45.45% were realists. I want to determine what the probability is of this happening. I would like to know what test to use to figure this out for the other scales as well. Thank You!

• It's worth bearing in mind that the probability of this happening / the p-value, is the probability of getting these results when drawing a sample from the reference population. The manufacturer (hopefully) normed the measure on a national representative sample, but your class is almost certainly not representative in this sense. Jul 17 '13 at 19:23
• Are the five "scales" different measures of the students on different dimensions, or are they five mutually exclusive categories into which they get one classification each? If the latter, a Chi-square test considering all five at once is more appropriate than picking the most problematic looking single category. Jul 17 '13 at 20:20

In my view, it is more natural to view this problem in terms of the original units, rather than percentages. Essentially, you have set up a binomial problem where the number of students classified as "realists" is the outcome and the probability of success is the "population" measure from the manufacturer, and the number of trials is 22 (the number of students).

The probability of your result, under the assumption that your class is a random sampling of attitudes, is just $\binom{22}{10}0.24^{10}(1-0.24)^{12}=0.0152$.

It's worth considering whether some systemic factor, like students self-selecting to be in the class, produced it.

The more general solution is $Pr(k) = \binom{22}{k}\theta^k(1-\theta)^{22-k}$, where $k$ is the number of students classified as having a particular thinking style.

• Isn't that actually the probability of getting exactly 10/22 Realists, whereas the more usual test would consider the probability of getting "10 or more" in determining a p-value? Jul 17 '13 at 20:17
• Yes. OP asked what the probability was of getting 10/22 successes, not for p-values.
– Sycorax
Jul 17 '13 at 20:19
• Hmm, ok, and +1 for pointing them towards considering the original units rather than just proportions, but I think this might be an overly literal interpretation. On the other hand it's true you have answered what they asked whereas my "10 or more" would be assuming they really meant to ask something slightly different. Jul 17 '13 at 20:28
• I can see where you're coming from, and if I were performing analysis on this problem, that's certainly how I would approach it. But we don't really have the context for why OP wants to know this figure, so I think it's best not to speculate and answer the actual question s/he asked.
– Sycorax
Jul 18 '13 at 12:27