In $200$ families each with $4$ children, we observed the number of boys they had. Summary: $8$ families had $0$ boys, $42$ families had $1$ boys, $67$ had $2$, $70$ had $3$ and $13$ had $4$.
Can we say that families have boys $50$ percent of the time? Test this at $\alpha = 0.05$.
At first glance it seemed as if it were testing a proportion (the proportion of boys), but now it seems to me that this situation is addressing the mean of the number of boys, and we should test whether the mean number of boys in these families is $2$ or not. Am I right in my final reasoning? Currently, while posting this, I'm getting a feeling that we should test two means against each other, the mean number of boys and the mean number of girls, then compare the latter two to each other, so that the hypotheses look like: $H_0$: $\mu_1 = \mu_2$, $H_1: \mu_1 \neq \mu_2$, where each $\mu_i$ refers to a gender's mean. In short, I'm very confused at the moment. Can someone clarify things a little bit?
It looks like you need to do a goodness-of-fit test (i.e. a $\chi^2$ test) to determine if the number of boys follows $Bin(4, 0.5)$ based on your observed data.