Hypothesis testing: Fisher's exact test and Binomial test My first post in this community. My knowledge of statistics is limited, and so I am seeking advice on the following problem. (Sorry for the long post, and thank you in advance for your help).
I have a group of 15 children with taste disorders (a not so common condition); 8 are girls, and 7 are boys.
I do the following experiment. Each child in my group tastes a slice of a specially prepared cake, and then I ask the child "does the cake taste good?", and the child can answer either "yes" or "no". Each child tastes the cake separately, and answer my question before meeting any other child in the group.
I collect the following data:





cake tastes good
cake does not taste good





girls
7
1
8


boys
6
1
7



13
2
15





*

*Considering the population of girls with tastes disorders, I do a binomial test with number of success k = 7, number of trials n = 8, and probability of success p = 0.5, to test my null hypothesis H0 = "my cake tastes good for no more than 50% of the population of girls with taste disorders". In python I can run binomtest(7, 8, 0.5, alternative="greater") which gives the following result BinomTestResult(k=7, n=8, alternative='greater', proportion_estimate=0.875, pvalue=0.03515625) and ConfidenceInterval(low=0.5293205913988617, high=1.0). I find that p-value <= 0.05, and therefore I can reject H0, and say that "my cake tastes good for more than 50% of the population of girls with taste disorders".


*Similarly, considering the population of boys with tastes disorders, I can do a binomial test to test my null hypothesis "my cake tastes good for no more than 50% of the population of boys with taste disorders". In python I can run binomtest(6, 7, 0.5, alternative="greater") which gives the following result BinomTestResult(k=6, n=7, alternative='greater', proportion_estimate=0.8571428571428571, pvalue=0.0625). I find that p-value > 0.05, and therefore I cannot reject H0, and I say that "my cake tastes good for no more than 50% of the population of boys with taste disorders".


*Now I run a Fisher's exact test on my contingency table. My null hypothesis is H0 = "there is no significant difference between the proportion of girls with taste disorder who find that my cake tastes good, and the proportion of boys with taste disorders who find that my cake tastes good". In python I can run fisher_exact([[7, 1], [6, 1]], alternative="two-sided") which gives the following result (1.1666666666666667, 1.0), where the fist value (1.17) is the odds ratio, and the second value (1) is the p-value. I find that p-value >= 0.05, and therefore I cannot reject the null hypothesis, and I say that "there is no significant difference between the proportion of girls with taste disorder who find that my cake tastes good, and the proportion of boys with taste disorders who find that my cake tastes good".
The result obtained with the Fisher's exact test ("no significant difference between the proportion of girls and boys who finds that the cake tastes good") seems to contradict the results in (1) and (2), which say that the "more than 50% of the population of girls find that the cake tastes good" (1), and "no more than 50% of boys find that the cake tastes good" (2). How do interpret these results?
[problem re-phrased (hopefully in a better way) according to whuber suggestion]

You are testing two different hypotheses

*

*With the binomial test the hypothesis is that the proportion is 0.5

*With the Fisher exact test the hypothesis is that the proportion is the same for boys as for girls.


which say that the "more than 50% of the population of girls find that the cake tastes good" (1), and "no more than 50% of boys find that the cake tastes good

That is not what the test says.
The tests says that among the girls 87.5% finds that the cake tastes good, and among the boys 85.7% finds that the cake tastes good. The difference between the two is that the effect found in the girls group is statistically significant.
