A random sample of eighty students were asked to vote for their preferred sport, and the results are shown in the table below. If the whole school contains seven hundred students, how many are expected to prefer football?
Now we’ve got thirty-two of the eighty students preferred football, twenty-four of them preferred basketball, eight of them preferred table tennis, and the other sixteen preferred swimming.
Right. Let’s have a look at some of the wording in this question. First of all, we’ve got a random sample. Well a sample is where we take a subset of the students in the school. So rather than interviewing all seven hundred students in this case, we’ve only looked at eighty. And the fact that we selected the students for that sample in a random way, means that we picked them in a nonsystematic way, with each student being equally likely to be chosen. So for example, we could put all of their names on little bits of paper, stick them in a hat, and then just put a hand in and pull one out without looking. And obviously, you’d have to do that eighty times to get eighty different students. Now we’re also told that we picked eighty students out of the seven hundred students who were in the school. And lastly, we need to think about this wording here: How many are expected to prefer football.
Now it’s possible that, in our sample, we picked the only thirty-two people who like football in the entire school. It’s also possible that the only people we left out of the survey, were all the rest of the people who only like football. So we don’t actually know the answer to this question. In order to answer it, we’re gonna have to make an assumption; and that is, that our random sample is representative of the whole school population. And that means that the preferences in our sample are in the same proportion as those for the whole school. And by that, we mean that in our sample, thirty-two out of eighty prefer football; the proportion of students that prefer football is thirty-two eightieths. And if the whole school population’s preferences are in the same proportion, then thirty-two eightieths of them would prefer football. And there are seven hundred students in that group, so we’re trying to calculate thirty-two eightieths of seven hundred.
Now if this is a non-calculator question, I would recommend turning that seven hundred into a simple fraction, seven hundred over one. And then we can try to do some cancelling. I noticed that eighty is divisible by ten and seven hundred is divisible by ten. Eighty divided by ten is eight, and seven hundred divided by ten is seventy. Now I also noticed that thirty-two and eight are both divisible by eight. Eight divided by eight is one, thirty-two divided by eight is four. So the calculation is four over one times seventy over one, or simply four times seventy, which is two hundred and eighty. So if our sample is representative of the population of the whole school, we would expect two hundred and eighty students from the whole school to prefer football.
Now obviously, the actual answer could be very different from two hundred and eighty. But when the question asks how many are expected to prefer football, then we’re being asked to assume that the proportions liking eack- each of the sports in our sample, are the same as the proportions liking each of the sports in the whole school population.