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A survey measures the heights of 900 people, which are found
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07 Aug 2020, 09:37
07 Aug 2020, 09:37
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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″.
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
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
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Re: A survey measures the heights of 900 people, which are found
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06 Jul 2021, 11:15
06 Jul 2021, 11:15
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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″.
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
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
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General Discussion
Re: A survey measures the heights of 900 people, which are found
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08 Aug 2020, 09:37
08 Aug 2020, 09:37
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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
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
