This is a very poorly designed question, for many reasons. Where is it from?
First, you need to know properties of normal distributions to answer the question. I have never seen a real GMAT question that requires you to know anything about normal distributions, though I have been seeing a lot of test prep company questions about normal distributions, which I think is misleading for test takers. The sets here are also presumably finite, so they can only have an approximately normal distribution; the normal distribution is a continuous (infinite) distribution;
Second, the correct answer to this question is not in the list; the correct answer here will not be a simple fraction; it will be a long decimal, and to answer this question precisely, you would need to consult a statistics table;
Third, they have the wrong answer, probably because the question designer has misunderstood that the 68-95-99.7 rule of normal distributions only gives approximations, not exact values (and they seem to have used 96, not 95, to come up with their answer). If you use exact values here, the answer is very close (correct to three decimal places) to 1/8, not 1/9.
Now that I'm done complaining about the quality of the question, here's how to solve it. I'd note, however, that you won't need to know any of this for the GMAT:
-when data is normally distributed, approximately 68% of values are within one standard deviation of the mean, and approximately 95% of values are within two standard deviations of the mean. These are approximations, rounded to the nearest integer. Note that this is only true if you know that your data is normally distributed; it is not true for an arbitrary set of data;
-One set has a mean of 460 and a standard deviation of 20. Thus approximately 68% of values will be between 440 and 480, and roughly 32% will be outside this range. Normal distributions are symmetric, so roughly 16% of values are below 440. That 16% is only an approximation, however; to three decimal places the percentage is 15.865%, something impossible to work out quickly without consulting a stats table;
-The other set has a mean of 520 and a standard deviation of 40. Thus approximately 95% of values will be between 440 and 600 (within two standard deviations). So approximately 5% of values are outside of this range, and approximately 2.5% are below 440. This is only approximate; again, to three decimal places the percentage is 2.275%.
The sets are of equal size, so to determine the proportion of test takers with scores below 440 who belong to the second set, we simply need to find:
2.275/(2.275 + 15.865) = 2.275/18.14 = 0.1254....
or almost exactly 1/8. So the answer choice which is closest to being correct is 1/8, but that's still just an approximation.
Curious to know where the question is from, but it's not one you should bother with for your prep.
ps5
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Hi Stewart,Ian Stewart wrote:This is a very poorly designed question, for many reasons. Where is it from?
First, you need to know properties of normal distributions to answer the question. I have never seen a real GMAT question that requires you to know anything about normal distributions, though I have been seeing a lot of test prep company questions about normal distributions, which I think is misleading for test takers. The sets here are also presumably finite, so they can only have an approximately normal distribution; the normal distribution is a continuous (infinite) distribution;
Second, the correct answer to this question is not in the list; the correct answer here will not be a simple fraction; it will be a long decimal, and to answer this question precisely, you would need to consult a statistics table;
Third, they have the wrong answer, probably because the question designer has misunderstood that the 68-95-99.7 rule of normal distributions only gives approximations, not exact values (and they seem to have used 96, not 95, to come up with their answer). If you use exact values here, the answer is very close (correct to three decimal places) to 1/8, not 1/9.
Now that I'm done complaining about the quality of the question, here's how to solve it. I'd note, however, that you won't need to know any of this for the GMAT:
-when data is normally distributed, approximately 68% of values are within one standard deviation of the mean, and approximately 95% of values are within two standard deviations of the mean. These are approximations, rounded to the nearest integer. Note that this is only true if you know that your data is normally distributed; it is not true for an arbitrary set of data;
-One set has a mean of 460 and a standard deviation of 20. Thus approximately 68% of values will be between 440 and 480, and roughly 32% will be outside this range. Normal distributions are symmetric, so roughly 16% of values are below 440. That 16% is only an approximation, however; to three decimal places the percentage is 15.865%, something impossible to work out quickly without consulting a stats table;
-The other set has a mean of 520 and a standard deviation of 40. Thus approximately 95% of values will be between 440 and 600 (within two standard deviations). So approximately 5% of values are outside of this range, and approximately 2.5% are below 440. This is only approximate; again, to three decimal places the percentage is 2.275%.
The sets are of equal size, so to determine the proportion of test takers with scores below 440 who belong to the second set, we simply need to find:
2.275/(2.275 + 15.865) = 2.275/18.14 = 0.1254....
or almost exactly 1/8. So the answer choice which is closest to being correct is 1/8, but that's still just an approximation.
Curious to know where the question is from, but it's not one you should bother with for your prep.
First of all thanks for your prompt reply.I will definitely follow your suggestion.
I am naive in GMAT study and its preparation scope.
Well, for me this question was really tough, and since its from one of standard preparation material I could not look over on its solution.
They have provided solution for the same problem. (plz find it's attached)
Can you please explain me, why they have used ratio 34:14:2.
From your last explanation I got vague idea about normal distribution(Need to read it 1s again
).Please help me.Thanks and Regards,
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Thanks a lot Ian for the great explanation.
Even I was not aware of 68-95-99.7 rule, I always used 68:96: xx (~100) rule.
Ket_gmat:
The 34:14:2 rule is same as what Ian described.
Actually 34% lies with in one +ve deviation and 34% within 1 -ve deviation. [total 68% with single deviation to mean]
14% lies between mean-sigma to mean-2*sigma
so including both sides it becomes 28 %
Total becomes 34*2 + 14*2 = 96%
this 96% actually is 95% as described by Ian.
Hope this clarifies.
Once again, Thank you very Ian.
Even I was not aware of 68-95-99.7 rule, I always used 68:96: xx (~100) rule.
Ket_gmat:
The 34:14:2 rule is same as what Ian described.
Actually 34% lies with in one +ve deviation and 34% within 1 -ve deviation. [total 68% with single deviation to mean]
14% lies between mean-sigma to mean-2*sigma
so including both sides it becomes 28 %
Total becomes 34*2 + 14*2 = 96%
this 96% actually is 95% as described by Ian.
Hope this clarifies.
Once again, Thank you very Ian.
