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Two different groups of test-takers received scores on the [#permalink]
30 Nov 2004, 09:38

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Two different groups of test-takers received scores on the GXYZ standardized test. Group A's scores had a normal distribution with a mean of 460 and a standard deviation of 20. Group B's scores had a normal distribution with a mean of 520 and a standard deviation of 40. If each group has the same number of test-takers, what fraction of the test-takers who scored below 440 belonged to Group B?

i suppose the right answer is A, but actually, the right answer is 5/37, which isn't up there.

This question is based on the known percentages of a normal distribution, and standard deviation. In group A, 16% of the test takers scored below 440, and in group b, 2.5% scored below 440. If we assume 100 students in each class, means 18.5 total scored below 440, so the answer is 2.5/18.5 = 5/37.

In the Princeton Review version of Standard Deviation, they ignore that .5% possibility for convenience sake, so in their world, Class B would have only 2% below 440, making the fraction 2 out of 18, or 1/9.

I don't know why the simplify things like that when it's wrong, but I guess it's a good way to make this stuff taste better to a large portion of their students.

At the same time, I haven't seen a GMAT question that actually required knowing those percentages, and I don't even teach them to my students, so I'm not sure this is even relevant.

The problem says the values are normally distributed. Therefore 68% of all values fall within 1 standard deviation, 95% of all values fall within 2 standard deviations, and 99% of all values fall within 3 standard deviations.

Group A has a mean of 460 and a standard deviation of 20. So 1 standard deviation would be 440, so you are looking for values less than 440. Since you are dealing with only the left side of the curve you use 1/2 of 32 or 16%.

Group B has a mean of 520 and a standard deviation of 40. So it would take 2 standard deviations to get to 440 (520-80). Since you are dealing with the left side of the curve you use 1/2 of 5 or 2.5%.

Since you will be taking both 2.5% and 16% of a number use a number both go into like 400. 2.5% of 400=10 and 16% of 400=64. Since you can't have a fraction of a student and you want below scores of 440(not equal to them), use 9 and 63. Now you have 9/(63+9)=9/72=1/8. So the answer is B.

I had to look up the % tables and do a some calculations to get a number that worked well like 400. At first I got 5/37, but then I figured they wanted scores less than 440 so I decreased my values and ended up with an answer that was there. I really hope there isn't a question like this when I take the gmat.

Since you will be taking both 2.5% and 16% of a number use a number both go into like 400. 2.5% of 400=10 and 16% of 400=64. Since you can't have a fraction of a student and you want below scores of 440(not equal to them), use 9 and 63. Now you have 9/(63+9)=9/72=1/8. So the answer is B.

Todd,

I agree with everything you wrote except this part. You were right on choosing 400, but you could have gone with 10 and 64, getting the right answer. The choice to drop 1 from each is arbitrary - the standard deviation ranges tell us the percentages BELOW a certain point, so while you can't be sure how many got exactly 440, you can know exactly what percentage got BELOW 440, which is how the question is stated. It could be that 2 or 3 or 4 got 440 exactly, but with a normal distribution we're not talking about exacts, we're talking about statistical probabilities.

Therefore, the answer would have been 10/74, or 5/37.

and to answer the other part of your response, I don't think this will be on the test, but if it is, you seem very equipped to deal with it...

Since you will be taking both 2.5% and 16% of a number use a number both go into like 400. 2.5% of 400=10 and 16% of 400=64. Since you can't have a fraction of a student and you want below scores of 440(not equal to them), use 9 and 63. Now you have 9/(63+9)=9/72=1/8. So the answer is B.

Todd,

I agree with everything you wrote except this part. You were right on choosing 400, but you could have gone with 10 and 64, getting the right answer. The choice to drop 1 from each is arbitrary - the standard deviation ranges tell us the percentages BELOW a certain point, so while you can't be sure how many got exactly 440, you can know exactly what percentage got BELOW 440, which is how the question is stated. It could be that 2 or 3 or 4 got 440 exactly, but with a normal distribution we're not talking about exacts, we're talking about statistical probabilities.

Therefore, the answer would have been 10/74, or 5/37.

and to answer the other part of your response, I don't think this will be on the test, but if it is, you seem very equipped to deal with it...

I guess I was right the first time. I couldn't understand why 5/37 was not an answer choice so I figured I must have understood the problem wrong. That's why I tried to fool with the values. Most standard deviation questions I've run into deal with understanding which group has the larger deviation or something of that nature.

The problem says the values are normally distributed. Therefore 68% of all values fall within 1 standard deviation, 95% of all values fall within 2 standard deviations, and 99% of all values fall within 3 standard deviations.

Group A has a mean of 460 and a standard deviation of 20. So 1 standard deviation would be 440, so you are looking for values less than 440. Since you are dealing with only the left side of the curve you use 1/2 of 32 or 16%.

Group B has a mean of 520 and a standard deviation of 40. So it would take 2 standard deviations to get to 440 (520-80). Since you are dealing with the left side of the curve you use 1/2 of 5 or 2.5%.

Since you will be taking both 2.5% and 16% of a number use a number both go into like 400. 2.5% of 400=10 and 16% of 400=64. Since you can't have a fraction of a student and you want below scores of 440(not equal to them), use 9 and 63. Now you have 9/(63+9)=9/72=1/8. So the answer is B.

I had to look up the % tables and do a some calculations to get a number that worked well like 400. At first I got 5/37, but then I figured they wanted scores less than 440 so I decreased my values and ended up with an answer that was there. I really hope there isn't a question like this when I take the gmat.

can you please explain how u got 32(left side of the curve)

68 % of the data is within 1 Standard Deviation...that means

that values between 460 -20 = 440 and 460 + 20 = 480 will fall for all the 68 % of the data. The rest 100-68 = 32 % falls below 440 and above 480.....since we have to calculate below 440 ..we would take 32/2 = 16 %

(Since it is normal distribution , we can divide 32/2 for values below 440)