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Re: A survey measures the heights of 900 people, which are found [#permalink]
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The question has given that almost 16.67% (150/900) of the people are in between 5.1 and 5.3. But there is no information provided about the standard deviation. But the question has given us the hint about it to check ourselves.

Initially, let's say, the SD is 0.2. Then we can see that 5.1 is -2SD, 5.3 is -1SD. The area between 5.1 and 5.3 should have contained 14% of the data (but which is not because it contains 16.67% as given in question).
Now, let's say, the SD is below 0.2 (like 0.1). Then we can see that, 5.1 is -4SD, 5.3 is -2SD. This area contains very little amount of data which is definitely not 16.67%

But, if we can take the SD above 0.2 (like 0.3), then we can see that 5.1 is -1.33SD and 5.3 is -0.67SD. There is a possibility of containing 16.67% of the data in the range. (the SD is not exactly 0.3, but definitely above 0.2)

So we can plot our necessary info on a bell diagram (normal curve) and see that 2SD is above 5.9 inch. This give our answer that, number of people above 5.9 is larger than the number of people above 2SD.

Answer: Quantity A
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Re: A survey measures the heights of 900 people, which are found [#permalink]
Carcass wrote:
A survey measures the heights of 900 people, which are found to be normally distributed. The mean height is 5′ 5″, and 150 people in the survey have a height between 5′ 1″ and 5′ 3″.

Quantity A
Quantity B
The number of people in the survey who are taller than 5′ 9″
The number of people in the survey who are more than 2 standard deviations above the mean



A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given

Kudos for the right solution and explanation

pls explain.
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Re: A survey measures the heights of 900 people, which are found [#permalink]
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150 is 16.67% of 900. If it was exactly 16%, A would have been equal to B. But it is not. That means the S.D. is less than 2". Hence A>B.
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Re: A survey measures the heights of 900 people, which are found [#permalink]
Shouldn't the SD be > 2 in these cases?

This is assuming that the starting point of the population with height > 5"9" would only fall before "2 * SD" would Column A > B

please clarify

thanks
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Re: A survey measures the heights of 900 people, which are found [#permalink]
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chacinluis wrote:
We are given a normal distribution of the heights of 900 people with mean 5'5''
and that 150 people fall between the heights of 5'1'' and 5'3''

Since the normal curve is symmetric 150 people also fall between 5'7'' and 5'9''

The question boils down to knowing whether 5'9'' is two standard deviations away from the mean or not. If it is, then both quantities would be the same. If it is closer than two standard deviations away from the mean than we would have more people being taller than 5'9'' than people beyond two standard deviations away from the mean. If 5'9'' is further away than two standard deviations away from the mean then we would have fewer people taller than 5'9'' than beyond two standard deviations away from the mean.

If 5'9'' is two standard deviations away from the mean then the standard deviation would be
(5'9''-5'5'')/2
= 4''/2
=2''

From the normal curve we expect 14% of the mass to be between the first and second standard deviations away from the mean.
That is between 5'7'' and 5'9''

But 900*(0.14)=126
Which does not equal 150

Since we have more people between 5'7'' and 5'9'' than we should have if it were between one and two standard deviations away from the mean, this tells us that the interval 5'7'' and 5'9'' is closer than two standard deviations away from the mean.

So 5'9'' is closer than two standard deviations away from the mean.
Therefore the number of people with heights greater than 5'9'' is more than the number of people with heights greater than two standard deviations greater than the mean.

Final Answer: A


I am not sure if this is the right way to come up with SD like subtracting (5'9 - 5'5)/2. I mean 150/900 is 16.6% which is the portion between 5'7 and 5'9, and why are we assuming that they cannot be beyond the 14% area and be in the 16% area(which is also 2 SDs above the mean), why cant they be in the 16% area instead of being near to the mean
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Re: A survey measures the heights of 900 people, which are found [#permalink]
mrunal2148 wrote:
150 is 16.67% of 900. If it was exactly 16%, A would have been equal to B. But it is not. That means the S.D. is less than 2". Hence A>B.


By 2 SD, do you mean that 5'7 must be at 1 SD and 5'9 must be at 2 SD?
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Re: A survey measures the heights of 900 people, which are found [#permalink]
KarunMendiratta wrote:
Carcass wrote:
A survey measures the heights of 900 people, which are found to be normally distributed. The mean height is 5′ 5′′, and 150 people in the survey have a height between 5′ 1′′ and 5′ 3′′.

Quantity A
Quantity B
The number of people in the survey who are taller than 5′ 9′′
The number of people in the survey who are more than 2 standard deviations above the mean



A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given

Kudos for the right solution and explanation


Points to Remember:
1. In a Normally distributed data, 2% of the data lies above and 2% data lies below 2 S.D from mean
2. In a Normally distributed data, 14% of the data lies between 1 and 2 S.D above and 14% of the data lies between 1 and 2 S.D below the mean
3. In a Normally distributed data, 34% of the data lies 1 S.D above and 34% of the data lies 1 S.D below the mean
4. N.D = Mean ± x(S.D), where 0 ≤ x ≤ 3

Now,
Case I: If 5' 9" lies exactly 2 S.D from the mean, then Col. A = Col. B
Case II: If 5' 9" lies less than 2 S.D from the mean, then Col. A > Col. B
Case III: If 5' 9" lies more 2 S.D from the mean, then Col. A < Col. B

For Case I:
5' 9" = 5' 5" + 2(S.D)
2(S.D) = 4 inches
S.D = 2 inches

This means, 5' 1" is 2 S.D below mean; 5' 3" is 1 S.D below mean; and 5' 7" is 1 S.D above mean
Given, 150 people in the survey have a height between 5′ 1′′ and 5′ 3′′
Therefore, 14% of Data = 150
Data = \(\frac{150}{0.14}\) ≈ 1071 which is far more than 900

This means the S.D of this data is less than 2 inches

Since, S. D < 2 inches, we can say that 5' 9" would lie between 1 S.D and 2 S.D above Mean. So, Case II is applicable

Hence, option A


If 5'72 is at 1 SD (according to the diagram), shouldnt the SD be more than 2?
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Re: A survey measures the heights of 900 people, which are found [#permalink]
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OE

As with all Quantitative Comparison questions, you need the two values to be expressed in comparable terms, so think of Quantity A in terms of standard deviations above the mean. Because this is a standard deviation question, you might not be able to calculate the exact number of people who are taller than a certain height, but you should be able to approximate it as a percentage of the total.

The question stem states that the data are normally distributed, which means that about 68% of the heights of those involved in the study fall within 1 standard deviation of the mean. It also means that about 95% of those heights fall within 2 standard deviations of the mean. The question stem also says that 150 of the 900 people surveyed were between 5′ 1′′ and 5′ 3′′.

You don’t know how great one standard deviation is, but you do know that a certain percentage of the total data lies between two points. Assume for a moment that one standard deviation is 2′′ in height. In that case, the people whose heights are between 5′ 1′′ and 5′ 3′′ would fall between 1 and 2 standard deviations below the mean of 5′ 5′′. Half of 68% is 34%, which is the amount of data that falls between the mean and 1 standard deviation below the mean. Half of 95% is 47.5%, which is the amount of data that falls between the mean and 2 standard deviations below the mean. Use this information to calculate the percentage of data that would fall between 1 and 2 standard deviations below the mean: 47.5% − 34% = 13.5%.

However, the actual number of people whose heights fall between 5′ 1′′ and 5′ 3′′ is 150/ 900, or about 16.7%, which is more than 13.5%. Normal distribution means that data points are grouped more densely near the mean than farther away from it. Because more than 13.5% of the data falls between 5′ 1′′ and 5′ 3′′, you know this data selection is closer to the mean than the data selection between 1 and 2 standard deviations below the mean. In other words, you don’t know exactly what the standard deviation is for this data set, but you do know it is greater than 2′′. That means that two standard deviations above the mean will be greater than 5′ 9′′; therefore, there are more people who are taller than 5′ 9′′ than people who are 2 standard deviations above the mean.

Quantity A is greater, and the correct answer is (A).

Note: If you memorize 68%, 95%, 99.7%, and other common numbers for data sets with normal distribution, standard deviation problems will be much easier. For example, if you have memorized that about 13.5% of the data in a normal distribution falls between the 1 and 2 standard deviations below the mean, you don’t need to calculate that number. Also note that you don’t really need to calculate 150/ 900. 13.5% of 1,000 would be 135, and you can see at a glance that 150/ 900 is going to be larger than 135/ 1000.
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Re: A survey measures the heights of 900 people, which are found [#permalink]
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Let me know if still unclear sir
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Re: A survey measures the heights of 900 people, which are found [#permalink]
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