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In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:
- a)increases by 1
- b)decreases by 1
- c)decreases by 2
- d)increases by 2
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
In a set of 2n distinct observations, each of the observation below th...
Mean value of the set of 2n observations.
Substitute the values in the above equation.
Then the mean value is,
Substitute the values in the above equation.
Then the mean value is,
Most Upvoted Answer
In a set of 2n distinct observations, each of the observation below th...
Explanation:
Let's assume that the original set of observations is sorted in ascending order. Since there are 2n observations, the median will be the average of the n-th and (n+1)-th observations.
Let's say the n-th observation is x. Then, the (n+1)-th observation will be (x+1) since there are 2n distinct observations.
Step 1: Increasing observations below the median by 5
In the original set, there are n observations below the median. When we increase each of these observations by 5, the new set will have the following observations below the median:
x+5, x+5+5, x+5+5+5, ..., x+5n.
Step 2: Decreasing observations above the median by 3
In the original set, there are n observations above the median. When we decrease each of these observations by 3, the new set will have the following observations above the median:
(x+1)-3, (x+1)-3-3, (x+1)-3-3-3, ..., (x+1)-3n.
Step 3: Calculating the mean of the new set
To calculate the new mean, we need to find the sum of all the observations in the new set and divide it by the total number of observations.
The sum of the observations below the median can be calculated using the arithmetic series formula:
Sum_below = (n/2) * (2(x+5) + (n-1) * 5) = n(x+5) + (5n(n-1))/2.
The sum of the observations above the median can also be calculated using the arithmetic series formula:
Sum_above = (n/2) * (2((x+1)-3) + (n-1) * (-3)) = n(x-2) - (3n(n-1))/2.
The total sum of the new set will be:
Total_sum = Sum_below + Sum_above
= n(x+5) + (5n(n-1))/2 + n(x-2) - (3n(n-1))/2
= 2nx + 5n - 3n(n-1)/2.
The total number of observations in the new set is 2n.
Therefore, the mean of the new set will be:
Mean = Total_sum / (2n)
= (2nx + 5n - 3n(n-1)/2) / (2n)
= x + 5/2 - (3/4)(n-1).
Since x is the n-th observation, it is less than the (n+1)-th observation. Therefore, x < (x+1)="" and="" x="" />< />
Hence, the mean of the new set (x + 5/2 - (3/4)(n-1)) will be greater than the mean of the original set (x + 1/2).
Therefore, the mean of the new set increases by 1, which is option A.
Let's assume that the original set of observations is sorted in ascending order. Since there are 2n observations, the median will be the average of the n-th and (n+1)-th observations.
Let's say the n-th observation is x. Then, the (n+1)-th observation will be (x+1) since there are 2n distinct observations.
Step 1: Increasing observations below the median by 5
In the original set, there are n observations below the median. When we increase each of these observations by 5, the new set will have the following observations below the median:
x+5, x+5+5, x+5+5+5, ..., x+5n.
Step 2: Decreasing observations above the median by 3
In the original set, there are n observations above the median. When we decrease each of these observations by 3, the new set will have the following observations above the median:
(x+1)-3, (x+1)-3-3, (x+1)-3-3-3, ..., (x+1)-3n.
Step 3: Calculating the mean of the new set
To calculate the new mean, we need to find the sum of all the observations in the new set and divide it by the total number of observations.
The sum of the observations below the median can be calculated using the arithmetic series formula:
Sum_below = (n/2) * (2(x+5) + (n-1) * 5) = n(x+5) + (5n(n-1))/2.
The sum of the observations above the median can also be calculated using the arithmetic series formula:
Sum_above = (n/2) * (2((x+1)-3) + (n-1) * (-3)) = n(x-2) - (3n(n-1))/2.
The total sum of the new set will be:
Total_sum = Sum_below + Sum_above
= n(x+5) + (5n(n-1))/2 + n(x-2) - (3n(n-1))/2
= 2nx + 5n - 3n(n-1)/2.
The total number of observations in the new set is 2n.
Therefore, the mean of the new set will be:
Mean = Total_sum / (2n)
= (2nx + 5n - 3n(n-1)/2) / (2n)
= x + 5/2 - (3/4)(n-1).
Since x is the n-th observation, it is less than the (n+1)-th observation. Therefore, x < (x+1)="" and="" x="" />< />
Hence, the mean of the new set (x + 5/2 - (3/4)(n-1)) will be greater than the mean of the original set (x + 1/2).
Therefore, the mean of the new set increases by 1, which is option A.
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In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:a)increases by 1b)decreases by 1c)decreases by 2d)increases by 2Correct answer is option 'A'. Can you explain this answer?
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In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:a)increases by 1b)decreases by 1c)decreases by 2d)increases by 2Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:a)increases by 1b)decreases by 1c)decreases by 2d)increases by 2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:a)increases by 1b)decreases by 1c)decreases by 2d)increases by 2Correct answer is option 'A'. Can you explain this answer?.
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