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The following question came to mind while I was playing tennis. Even when I'm taking a break, I can't turn the math brain off. I'm sure this is a simple question for the statisticians here.

Say a tennis player wins 100 out of 200 points in a match. The player faces 30 break points in the match (not serving so well, apparently), and saves 20 of them. Is there evidence that the player plays "better" when facing break point?

My thought (from my limited knowledge of statistics) is to compare the ratio $\frac{20}{30}$ of break-points-against won to the ratio $\frac{80}{170}$ of all other points won, with the expectation for each ratio being 50% $(=\frac{100}{200})$. We could use Pearson's chi-squared test with one degree of freedom. Is this an appropriate approach or am I missing something?

Thanks for your input!

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My thought (from my limited knowledge of statistics) is to compare the ratio 20/30 of break-points-against won to the ratio 80/170 of all other points won, with the expectation for each ratio being 50% (=100/200).

Well, that's what I'd do.

We could use Pearson's chi-squared test with one degree of freedom. Is this an appropriate approach

It is appropriate when checking for a two-sided alternative, though your alternative hypothesis was directional (a two-sample proportions test will take care of that). However, if the idea to test, and what to test for was generated after observing the excess proportion, such tests don't have their nominal properties!

or am I missing something?

Aside from what I've mentioned, and the possibility of failure of the binomial assumptions (e.g. via serial dependence or nonconstant success probability within the two categories), I don't think so.

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