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If y = 3x + 4 is the regression line of y on x and the arithmetic mean of x is –1, what is the arithmetic mean of y?
- a)1
- b)–1
- c)7
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If y = 3x + 4 is the regression line of y on x and the arithmetic mean...
Given:
- Regression line of y on x is y = 3x - 4
- Arithmetic mean of x is 1
To find:
- Arithmetic mean of y
Solution:
We know that the regression line of y on x is given by:
y = a + bx,
where a is the y-intercept and b is the slope of the line.
Comparing this with the given equation y = 3x - 4, we see that:
a = -4
b = 3
Now, we know that the arithmetic mean of x is given by:
x̄ = (x1 + x2 + ... + xn) / n,
where x1, x2, ..., xn are the n observations of x.
Since the arithmetic mean of x is 1, we have:
1 = (x1 + x2 + ... + xn) / n.
Multiplying both sides by n, we get:
x1 + x2 + ... + xn = n.
But we don't need to calculate n explicitly, since we only need to find the arithmetic mean of y.
The arithmetic mean of y is given by:
ȳ = (y1 + y2 + ... + yn) / n,
where y1, y2, ..., yn are the corresponding observations of y.
Substituting y = 3x - 4 in this expression, we get:
ȳ = (3x1 - 4) + (3x2 - 4) + ... + (3xn - 4) / n
= 3(x1 + x2 + ... + xn) / n - 4
= 3n / n - 4
= 3 - 4
= -1.
Therefore, the arithmetic mean of y is -1.
Answer: Option A (1) is incorrect. The correct answer is option D (none of these).
- Regression line of y on x is y = 3x - 4
- Arithmetic mean of x is 1
To find:
- Arithmetic mean of y
Solution:
We know that the regression line of y on x is given by:
y = a + bx,
where a is the y-intercept and b is the slope of the line.
Comparing this with the given equation y = 3x - 4, we see that:
a = -4
b = 3
Now, we know that the arithmetic mean of x is given by:
x̄ = (x1 + x2 + ... + xn) / n,
where x1, x2, ..., xn are the n observations of x.
Since the arithmetic mean of x is 1, we have:
1 = (x1 + x2 + ... + xn) / n.
Multiplying both sides by n, we get:
x1 + x2 + ... + xn = n.
But we don't need to calculate n explicitly, since we only need to find the arithmetic mean of y.
The arithmetic mean of y is given by:
ȳ = (y1 + y2 + ... + yn) / n,
where y1, y2, ..., yn are the corresponding observations of y.
Substituting y = 3x - 4 in this expression, we get:
ȳ = (3x1 - 4) + (3x2 - 4) + ... + (3xn - 4) / n
= 3(x1 + x2 + ... + xn) / n - 4
= 3n / n - 4
= 3 - 4
= -1.
Therefore, the arithmetic mean of y is -1.
Answer: Option A (1) is incorrect. The correct answer is option D (none of these).
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Community Answer
If y = 3x + 4 is the regression line of y on x and the arithmetic mean...
Simple just put the value of X in place of
y=3(-1) + 4
y=-3+4
y= 1
y=3(-1) + 4
y=-3+4
y= 1
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