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The mean and the standard deviation (s.d.) of 10 observations are 20 and 2, respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal to
- a)-10
- b)-20
- c)-5
- d)10
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The mean and the standard deviation (s.d.) of 10 observations are 20 a...
Let's assume the 10 observations are denoted by x1, x2, ..., x10.
According to the given information, the mean of these observations is 20. This means that the sum of these observations is 10 * 20 = 200.
We are also given that the standard deviation of these observations is 2. The standard deviation is calculated using the formula:
s.d. = sqrt((Σ(xi - mean)^2) / n)
where Σ represents the sum, xi represents each observation, mean represents the mean of the observations, and n represents the number of observations.
Using this formula, we can calculate the sum of the squared differences from the mean:
2^2 * 10 = (Σ(xi - 20)^2)
4 * 10 = Σ(xi^2 - 40xi + 400)
40 = Σ(xi^2 - 40xi + 400)
40 = Σxi^2 - 40Σxi + 400
Now, let's consider the new set of observations obtained by multiplying each observation by p and then reducing it by q. The new observations can be denoted by y1, y2, ..., y10.
yi = (xi * p) - q
The mean of the new observations can be calculated as follows:
mean of yi = (Σ((xi * p) - q)) / 10
mean of yi = (p * Σxi - 10q) / 10
Since the mean of the new observations is also given as 20, we can equate the above equation to 20:
(p * Σxi - 10q) / 10 = 20
p * Σxi - 10q = 200
p * Σxi = 200 + 10q
Σxi = (200 + 10q) / p
Now, let's consider the sum of the squared differences from the mean of the new observations:
Σ(yi - mean of yi)^2 = Σ(((xi * p) - q) - mean of yi)^2
Σ(yi - mean of yi)^2 = Σ(((xi * p) - q) - ((p * Σxi - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (p * Σxi - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (p * (200 + 10q) / p - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (200 + 10q - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - 200 / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - 20))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - (q + 20)))^2
Σ(yi - mean of yi)^2 = Σ(xi^2 * p^2
According to the given information, the mean of these observations is 20. This means that the sum of these observations is 10 * 20 = 200.
We are also given that the standard deviation of these observations is 2. The standard deviation is calculated using the formula:
s.d. = sqrt((Σ(xi - mean)^2) / n)
where Σ represents the sum, xi represents each observation, mean represents the mean of the observations, and n represents the number of observations.
Using this formula, we can calculate the sum of the squared differences from the mean:
2^2 * 10 = (Σ(xi - 20)^2)
4 * 10 = Σ(xi^2 - 40xi + 400)
40 = Σ(xi^2 - 40xi + 400)
40 = Σxi^2 - 40Σxi + 400
Now, let's consider the new set of observations obtained by multiplying each observation by p and then reducing it by q. The new observations can be denoted by y1, y2, ..., y10.
yi = (xi * p) - q
The mean of the new observations can be calculated as follows:
mean of yi = (Σ((xi * p) - q)) / 10
mean of yi = (p * Σxi - 10q) / 10
Since the mean of the new observations is also given as 20, we can equate the above equation to 20:
(p * Σxi - 10q) / 10 = 20
p * Σxi - 10q = 200
p * Σxi = 200 + 10q
Σxi = (200 + 10q) / p
Now, let's consider the sum of the squared differences from the mean of the new observations:
Σ(yi - mean of yi)^2 = Σ(((xi * p) - q) - mean of yi)^2
Σ(yi - mean of yi)^2 = Σ(((xi * p) - q) - ((p * Σxi - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (p * Σxi - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (p * (200 + 10q) / p - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - (200 + 10q - 10q) / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - 200 / 10))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - q - 20))^2
Σ(yi - mean of yi)^2 = Σ((xi * p) - (q + 20)))^2
Σ(yi - mean of yi)^2 = Σ(xi^2 * p^2
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Community Answer
The mean and the standard deviation (s.d.) of 10 observations are 20 a...
μ = 20, σ2 = 2
And pμ - q = 20/2 ⇒ 20p - q = 10
Also,
|p| σ2 = |p| × 2 = 1 ⇒
If p = 1/2 ⇒ q = 0 (rejected)
And pμ - q = 20/2 ⇒ 20p - q = 10
Also,
|p| σ2 = |p| × 2 = 1 ⇒
If p = 1/2 ⇒ q = 0 (rejected)
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The mean and the standard deviation (s.d.) of 10 observations are 20 and 2, respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal toa)-10b)-20c)-5d)10Correct answer is option 'B'. Can you explain this answer?
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The mean and the standard deviation (s.d.) of 10 observations are 20 and 2, respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal toa)-10b)-20c)-5d)10Correct answer is option 'B'. 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 The mean and the standard deviation (s.d.) of 10 observations are 20 and 2, respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal toa)-10b)-20c)-5d)10Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The mean and the standard deviation (s.d.) of 10 observations are 20 and 2, respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal toa)-10b)-20c)-5d)10Correct answer is option 'B'. Can you explain this answer?.
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