Q 2981

[arash.nouraee]

Can anyone explain why the assumption is related to equal numbers of people rather than hours slept??

P: Those who take regular naps are more likely to sleep less than 7 hours a night than those who don't.

(A): (There are equal numbers of people who take regular naps and those who don't.)
(A): (Those who do not take regular naps tend to sleep more than 7 hours a night.)
C: People who sleep for more than 7 hours tend to not take regular naps.

Answer:

(A): (There are equal numbers of people who take regular naps and those who don't.)

[christine.defenbaugh]

Thanks for your question arash.nouraee!

What's going on here is a classic problem of switching what groups are the focus in an analysis of likelihoods.

Let's use a different example to illustrate the problem.

Astronauts are more likely than non-astronauts to like strawberry ice cream. Therefore, people who like strawberry ice cream tend to be astronauts.

This conclusion is crazy! There are probably millions of people in the world who like strawberry ice cream, and the vast majority of them are not astronauts! The error made in this argument it assumes that just because there is a smaller proportion of one group (the non-astronauts) that prefers strawberry ice cream, that must mean there is a smaller real number of people. Maybe only 10% of the world's 7 billion people like strawberry ice cream (700 million people), while 50% of the astronauts like it (maybe 50 people?).

In order to make any conclusion about what most strawberry-ice-cream lovers are (astronauts or not), we need to know something about how the size of the two groups compare.

Let's apply this to the arcade question!

The premise tells us that regular nappers are more likely to sleep less than 7 hours than non-nappers. So, the percent breakdown might look like this:

................................Regular Nappers............Non-Nappers
more than 7 hours...............25%..........................75%
less than 7 hours..,..............75%..........................25%

The conclusion we want to get to is that the more than 7 hours group is more likely to be non-nappers than nappers. In other words, we need "75% of non-nappers" to be a bigger real number than "25% of nappers". If the two groups were the same size, then that would be true!

So, the assumption that the two groups are equal would enable us to get to the conclusion about the 'more than 7 hours' group!

The wrong answer ("those who do not take regular naps tend to sleep more than 7 hours") doesn't help get us there, because it still just gives us information about the napping/non-napping groups. In other words, more than half of the non-nappers sleep more than 7 hours, but we still don't know how many non-nappers there are total! More than half of a very small group is still a very small number!

These switches between comparing different groups can be very tricky to navigate. Take a look at PT44-S2-Q21 for another great example of this issue.

Please let me know if this helps clear things up a bit!

[vincent.m]

Shouldn't the conclusion be phrased differently? Shouldn't it be those who do not take regular naps tend to sleep for more than 7 hours?

[ohthatpatrick]

Let’s put some real numbers to it. Let’s say there are 100 total people in the world.

We’ll use the correct answer "equal # of ppl who take regular naps and those who don’t".

50 - regular-nappers
50 - no-nappers

Let’s add in the original premise that "those who take regular naps are more likely to sleep less than 7 hrs. a night than those who don’t".

That means that if we break up the regular nappers into "more than 7" / "less than 7" and do the same to the no-nappers, we should have a bigger number of "less than 7" for the regular nappers.

Let’s make up some applicable numbers:
REGULAR NAPPERS
40 - less than 7
10 - more than 7

NO NAPPERS
35 - less than 7
15 - more than 7

Is the original conclusion true?

C: People who sleep for more than 7 hours tend to not take regular naps.

It is.
There are 25 people who sleep for more than 7.
15 of them are No-Nappers. 10 of them are Nappers.
So "most of the more-than-7 crowd are No-Nappers."

Saying that "ppl who are X tend to be Y" is the same as "most X’s are Y’s".

Is your alternative conclusion also true?

C: Those who do not take regular naps tend to sleep for more than 7 hours?

It is not. There are 50 No-Nappers. Most of them (35 of them) sleep less than 7. So with these numbers I made up, your conclusion would not be true.

You COULD make up different numbers that still fit the original parameters and DO make your conclusion true. But your conclusion does not HAVE to be true. Meanwhile, the conclusion in the actual problem DOES always have to be true.

Hope this helps.

[DavidS899]

I don't understand this at all.

You were using equivalent samples to prove the assumption true but the assumption is that the samples are equivalent.

You could equally say in the regular naps group there are 40 who sleep less than seven hrs and 1 who sleeps more. And in the no nap group there could be 1 who sleeps less than seven and 100 that sleep more than seven. This would prove our conclusion but disprove that assumption.

Either this is just incomprehensible to me or the language is misleading.

[WendyH528]

I had to read it many times over to understand the explanations given.
In order for the Conclusion to be true, based on the premise — there has to be an equal number of Nappers & Non-Nappers.
I think I understand the ‘re-framing’ of what the correct ‘Answer’ confirms.