Self-study
Nearly all mail that is correctly addressed arrives at its destination within two business days of being sent. ██ █████ █████████ █████████ ████ █████ ██████ ████ ████ ████ ████ ██ ██ ███████ ██ ████████ ████████ ████████ ████ ████ ███████ █████ ████████ ████ ██ ████ █████ █████ █████
Stimulus Summary
This is a tough question. Working with some example numbers can help you understand what's going on. We'll use ✓ for correctly addressed mail and ✗ for incorrectly addressed mail, and think through the stimulus one fact at a time.
Fact 1: Nearly all correctly addressed mail arrives within 2 business days.
Let's say there are 5 correctly addressed pieces of mail. Most end up in the 2-day group; the rest take 3 or more days.
✓✓✓
2 days
✓✓
3+ days
Fact 2: If correctly addressed mail takes longer than 2 business days, it must be damaged in transit.
So the ✓ marks in the 3+ day group must be damaged.
✓✓✓
2 days
✓✓
3+ days
↓
damaged
Fact 3: Most mail arrives more than 2 business days after being sent.
If most correctly addressed mail arrives within 2 days, what's causing most mail overall to take 3+ days? There must be a bunch of incorrectly addressed mail dragging the totals past 2 days. Let's add ✗ marks to represent incorrectly addressed mail:
✓✓✓
2 days
✓✓
3+ days
↓
damaged
✗✗
3+ days
✓ = correctly addressed ✗ = incorrectly addressed
We needed at least 2 ✗ to tip the balance: 3 pieces arrive within 2 days, but 4 arrive in 3+ days. If we'd added only 1 ✗, it would be a 3-to-3 tie, and we need 3+ days to be the majority. So at minimum, 2 out of 7 total pieces of mail are incorrectly addressed.
The Impact of "Nearly All"
Is
✓✓✓✓✓✓
2 days
✓
3+ days
✗✗✗✗✗✗
3+ days
Here, 6 out of 7 correctly addressed pieces arrive within 2 days (instead of just 4 out of 7 that we started with in our first example). To make most mail arrive in 3+ days, we need at least 6 incorrectly addressed pieces in the 3+ day group. That's 6 out of 13 total pieces of mail. The proportion of incorrectly addressed mail only gets larger as we push more correctly addressed mail into the 2-day group.
As we go through the answers, keep in mind which parts of the visual are fixed and which are variable. What's fixed: there must be enough ✗ in the 3+ day group to make it the majority. What's variable: the exact number of damaged ✓ pieces, and whether any ✗ happen to land in the 2-day group. We're looking for something guaranteed by the stimulus, so the correct answer will describe something that must be true regardless of how we visualize the variable parts.
19.
If the statements above are █████ █████ ███ ██ ███ █████████ ████ ██ █████
a
A large proportion ██ ███ ████ ████ ██ █████████ █████████ ██ ███████ ██ ████████
In some versions of our diagram, it looks like a decent amount of correctly addressed mail ends up damaged (the ✓ marks in the 3+ day group). But that proportion is variable. Since the stimulus says nearly all correctly addressed mail arrives within 2 days, the damaged portion could be tiny:
✓✓✓✓✓✓✓✓✓
2 days
✓
3+ days
↓
damaged
Here, only 1 out of 10 correctly addressed pieces is damaged. That doesn't seem like a "large proportion." But even if you think it's still large, we could make the ratio even more extreme. Just imagine only 1 out of 1 million correctly addressed pieces is damaged.
b
No incorrectly addressed ████ ███████ ██████ ███ ████████ ████ ██ █████ █████
In our stimulus visual, we placed ✗ marks in the 3+ day group because that's what we needed to make 3+ days the majority. But nothing in the stimulus says incorrectly addressed mail can't arrive within 2 days. We could just as easily draw some ✗ marks in the 2-day group:
✓✓✓✗✗
2 days
✓✓✗✗✗✗
3+ days
So we can't conclude that no ✗ marks land in the 2-day group.
c
Most mail that ███████ ██████ ███ ████████ ████ ██ █████ ████ ██ █████████ ██████████
(C) is trying to reverse the relationship in the first sentence. That's the cleanest way to understand why it's wrong. "Nearly all A are B" does not prove "Most B are A." Unfortunately, if that's not enough to convince you that (C) doesn't have to be true, we do have to get into some example numbers.
The stimulus tells us that of all correctly addressed mail, nearly all arrives within 2 days. But knowing that nearly all ✓ marks land in the 2-day group doesn't mean ✓ marks dominate that group. There could be plenty of ✗ marks in the 2-day group too:
✓✓✓✓✗✗✗✗✗
2 days
✓✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗
3+ days
In this example, nearly all correctly addressed mail (4 out of 5 ✓) is in the 2-day group. But the 2-day group has 9 pieces total, and 5 of them are ✗. Most of the 2-day arrivals are actually incorrectly addressed. (C) doesn't have to be true because a large volume of incorrectly addressed mail can flood the 2-day group, as long as we add enough 3+ day mail to offset the additional number added to the 2-day group.
d
A large proportion ██ ████ ██ ███████████ ██████████
See the stimulus analysis to understand why (D) must be true. In order for most (over half) mail to arrive in 3+ days despite almost all correctly addressed mail arriving within 2 days, there has to be a large proportion of incorrectly addressed mail in the total pool. Otherwise, we wouldn't have enough 3+ day mail to say that over half of mail arrives in 3+ days. And as we saw, the stronger the phrase "nearly all" is (the closer it gets to "all"), the larger the proportion of incorrectly addressed mail has to be. No matter the example numbers we use, a large proportion of the total mail must be incorrectly addressed.
e
More mail arrives ██████ ███ ████████ ████ ██ █████ ████ ████ ███████ ███████ ███ ███ █████ ████████ ████ █████ █████ █████
Most mail arrives three business days or more after being sent. But (E) is comparing two different groups: mail arriving within 2 days and mail arriving between 2 and 3 days. We don't have any information restricting how much mail falls in the 2-to-3-day window. A large amount of incorrectly addressed mail could arrive between 2 and 3 business days, making that group bigger than the within-2-days group.