#24 - A certain airport security scanner designed to detect

[Johnclem]

Thanks Adam,
My apologies . I meant to say a TERM shift in the language , not time shift. I say this because I noticed the author going from SAFE bags to ALL bags. Is this okay to think ? Or there's nothing other than a numbers and percentage issue going on here ?

Thanks very much for your detailed explanation . You helped me see math differently . Lol


Thanks
John

[Nikki Siclunov]

Hello John,

Thanks for the follow-up. I strongly recommend that students resist the urge to look at "keywords" in order to understand an argument As discussed previously, by both Adam and myself, the argument contains a statistical error in reasoning, which has nothing to do with ambiguous use of terminology. Here's the error in a nutshell:

The information in the stimulus deals with statistical measures of sensitivity and specificity. Specificity (also called the true negative rate, TNR) measures the proportion of negatives that are correctly identified as such (e.g., the percentage of safe luggage that is correctly identified as not having an explosive). Specificity tells us nothing about sensitivity. Sensitivity (also called the true positive rate) measures the proportion of positives that are correctly identified as such (e.g., the percentage of dangerous luggage that is correctly identified as having an explosive). The two concepts are completely unrelated.

According to the central premise of the argument, if the scanner examines 100 pieces of luggage containing no explosives, it will erroneously “detect” explosives in one piece. Thus, the true negative rate is 99%, and the false positive rate is 1%. A false positive rate of 1% does not, however, guarantee that explosives are present in 99 out of 100 pieces of luggage that trigger an alert. In fact, the premise tells us nothing about what percentage of the alerts are accurate. Picture an airport with tens of thousands of pieces of luggage being scanned daily. What if not a single one of them contained an explosive? According to the premise, if the scanner examines 10,000 pieces of “safe” luggage, it will erroneously detect explosives in 100 of them (false positive rate of 1%). Are explosives present in 99 of these 100 pieces of luggage? Hardly; explosives weren’t present in any of them.

Since the premise deals with the proportion of “safe” pieces of luggage that erroneously trigger an alert, while the conclusion deals with the proportion of alerts that accurately detect an explosive, the premise and the conclusion deal with proportions based on two different groups. Answer choice (E) is therefore correct.

Please let us know if we can clarify this line of reasoning further

Thanks,

[student987]

Thank you for a great post. This is really helpful! I am following up with a few questions. First, Nikki's post mentions: "In fact, the premise tells us nothing about what percentage of the alerts are accurate." I'm a little confused about this part, because the stimulus says the "scanner...will alert...whenever...contains an explosive." I interpreted this stimulus portion to mean that there is a 100% true positive rate (=0% false negative rate); it's saying that if there's an explosive, it will alarm. This means that there's no chance of it both containing an explosive and not ringing an alarm. How does this sound? If it is correct.. Uh. What does "accuracy" mean? Is it referring to a combination of the true positive rate and the true negative rate...?

Second, does this look correct: "100% - false positive rate = True negative rate" and "100% - false negative rate = true positive rate"? (And vice versa—"100% - true neg = false positive" and "100% - true pos = false neg")?

Last, does the scanner have an "accuracy" rate (in quotations because I'm a bit confused on this term) between 99% and 100%? So an average of 99.5%...? My thought process is that if there are 100 bags with no explosives, there will be 1 false positive alarm. I am assuming that means a 99% accuracy rate. If there are 100 bags with no explosives and 100 bags with explosives, there will be 1 false alarm, 100 true alarms, and 99 safe bags without alarms. That would mean 199 accurate results (alarm + no alarm) and 1 inaccurate result, or 199/200 (99.5%) accuracy rate. If there are 100 bags with explosives, it will alarm 100 times (100% accuracy rate). I'm guessing I'm wrong on this, but I can't figure out where I am off. I would appreciate any help!

[Adam Tyson]

An accurate alert is one where there really are explosives in the bag, student987. We know that every time there are explosives, we will get an alert, but that's only one side of the coin. Go back to the original post in this thread and look at the example of 10,000 bags, none of which have explosives. Is the scanner accurate? Nope! In that case, 100% of the alerts are inaccurate! So having a 100% true positive rating doesn't mean we know the percentage of all alerts that are accurate, because we don't know how many bags (or what proportion of them) have explosives. Play with those numbers however you like, and you'll find the percentage of all alerts that turn out to be accurate varies widely.

I think your formulas are correct, although they may be more work than you need to analyze this question during testing conditions. The clock is ticking, after all!

As for your last question, that's definitely delving into too much math and statistics for me, and also too much for this test. I don't want to get into it with a statistician, and more than that, I absolutely do not have to in order to succeed on the LSAT. Stay away from over-complicating things like this, and remember that this test is designed so that someone with only a rudimentary math education (like a typical liberal arts major) should never have to do any math that is beyond that education. If you want to engage in that sort of analysis for fun, have at it! I would just suggest that you do that with a stats expert, rather than an LSAT expert, and then be sure to leave all that outside when you go to take the test, so as not to clutter your mind up with analyses that won't help you and might get in the way of the real task at hand.

In short, keep it simple, including any math. Keep at it!

[student987]

Ok I see, I won’t try to go into too much technical detail! I have a better question that more accurately sums up my problem here: I actually thought the 10,000 safe luggages example (with 1 erroneous alert) would be too complicated for me to think up at the spot, given that the clock is ticking. That’s why I was trying to come up with another way to think of this, so that I can have something quicker to rely on. A bigger concept to keep in mind, so I can solve these number-related questions without having to resort to playing with numbers.

Do you have any advice here for how to respond quickly to this type of question, other than playing with numbers like the 10,000 safe luggages ex. above? Thinking up examples involving with numbers often takes long and is also from time to time confusing Thanks!

An accurate alert is one where there really are explosives in the bag, student987. We know that every time there are explosives, we will get an alert, but that's only one side of the coin. Go back to the original post in this thread and look at the example of 10,000 bags, none of which have explosives. Is the scanner accurate? Nope! In that case, 100% of the alerts are inaccurate! So having a 100% true positive rating doesn't mean we know the percentage of all alerts that are accurate, because we don't know how many bags (or what proportion of them) have explosives. Play with those numbers however you like, and you'll find the percentage of all alerts that turn out to be accurate varies widely.

I think your formulas are correct, although they may be more work than you need to analyze this question during testing conditions. The clock is ticking, after all!

As for your last question, that's definitely delving into too much math and statistics for me, and also too much for this test. I don't want to get into it with a statistician, and more than that, I absolutely do not have to in order to succeed on the LSAT. Stay away from over-complicating things like this, and remember that this test is designed so that someone with only a rudimentary math education (like a typical liberal arts major) should never have to do any math that is beyond that education. If you want to engage in that sort of analysis for fun, have at it! I would just suggest that you do that with a stats expert, rather than an LSAT expert, and then be sure to leave all that outside when you go to take the test, so as not to clutter your mind up with analyses that won't help you and might get in the way of the real task at hand.

In short, keep it simple, including any math. Keep at it!

[Adam Tyson]

There is something of a shortcut here, and that is to recognize the numbers/percentages issue and to pounce on it. We have evidence about the rate of false positives (1 out of 100 scans of bags with no explosives), and the author draws a conclusion about the rate of true positives (99 out of 100 alerts). Just notice that you cannot draw a conclusion this way, comparing two different types of measurements (scans of bags with no explosives compared to the number of alerts with/without explosives). Alerts vs bags - it's apples to oranges. The numbers may help you see it, or that may be confusing, but that bad comparison is where your analysis should start. Answer E is the only one that deals with a bad comparison. Boom!