You sample $1000$ adult women the questions "how many children does your mother have?" and "how many children did you have?"
Suppose the distribution of the number of children per woman hasn't changed over time and is $p_{i} = 1/3$ for $i = 1, 2, 3.$
(a) What's the average response (approximately) to question (i)?
(b) What's the average response (approximately) to question (ii)?
(c) If we didn't know that the $p_i$ hasn't changed, the results of (a) and (b) might lead us to the conclusion that family sizes are decreasing. Instead, what is there for the explanation for these results? Would increasing sample size help? [There is a subtle assumption needed in (a), what is it?]
I don't quite understand how you're supposed to calculate average response with only this information. I'm sure that the people who are selected are more likely to have mothers who had more kids (it's more probable they were picked) but how would I get an expectation from this? I'm also not so sure how to do it for (b).
I think the answer to (a) is $7/3 = 2.333$ but I do not know a reasonable way how to get this.
I don't think there's any relation to our assumption in (b) so I think (b) is just $2$.
(c) The explanation is that people whos mothers had lots of kids are more likely to get picked, which brings the average up. I don't think increasing the sample size helps but I am not so sure. I am also not sure what this "subtle assumption" is.
I would greatly appreciate some help, particularly for (a) and (c)
You are right that a) gives a greater average. Indeed, the ratio of people in population having parents with 1,2,3 children should be approximately 1:2:3 (the subtle assumption needed there is that the longevity does not depend on the number of children). That said, the average response in a) is $$ \frac 16\cdot 1 + \frac13\cdot 2 + \frac12\cdot 3 = \frac 73, $$ exactly as you estimated.