for y = 25, what is the estimated value of value of x, from the following data:
X: 11 12 15 16 18 19 21
y: 21 15 13 12 11 10 9
a) 15 b)7 c)13.588 d) 14.986
To find the estimated value of x for y = 25 using the given data points, you can use a method called linear regression. Linear regression is a statistical technique used to find the relationship between two variables.
In this case, we have pairs of data points (x, y). Let's plot these points on a scatter plot:
X: 11 12 15 16 18 19 21
y: 21 15 13 12 11 10 9
By visually inspecting the scatter plot, we can see that the points seem to follow a linear trend. To estimate x for y = 25, we need to find the equation of the line that best fits the data.
To do this, we can use the least squares method, which minimizes the sum of squared differences between the observed y values and the predicted y values on the line.
First, calculate the mean of the x values and the mean of the y values:
mean of x (x̄) = (11 + 12 + 15 + 16 + 18 + 19 + 21) / 7 = 16
mean of y (ȳ) = (21 + 15 + 13 + 12 + 11 + 10 + 9) / 7 = 13
Next, calculate the difference between each x value and the mean of x (x - x̄), and the difference between each y value and the mean of y (y - ȳ):
X - x̄: -5 -4 -1 0 2 3 5
y - ȳ: 8 2 0 -1 -2 -3 -4
Now, calculate the product of (X - x̄) and (y - ȳ):
(-5) * 8 = -40
(-4) * 2 = -8
(-1) * 0 = 0
0 * -1 = 0
2 * -2 = -4
3 * -3 = -9
5 * -4 = -20
Summing up these product values:
-40 + (-8) + 0 + 0 + (-4) + (-9) + (-20) = -81
Next, calculate the sum of squares of (X - x̄):
(-5)^2 + (-4)^2 + (-1)^2 + 0^2 + 2^2 + 3^2 + 5^2 = 67
Using the formula slope (b) = sum of products / sum of squares, we can calculate the slope:
b = -81 / 67 ≈ -1.209
Now, we can use the equation of the line, y = mx + c, where m is the slope and c is the y-intercept.
Substituting the known values of m and the mean of x̄ and ȳ, we can solve for c:
13 = (-1.209) * 16 + c
13 = -19.344 + c
c = 32.344
Now that we have the equation of the line: y = -1.209x + 32.344
To estimate x for y = 25, we rearrange the equation to solve for x:
25 = -1.209x + 32.344
-7.344 = -1.209x
x ≈ -7.344 / -1.209 ≈ 6.07
So, the estimated value of x for y = 25 from the given data is approximately 6.07.
From the given answer choices, none of them match 6.07. It's possible that there was an error in the given data or there was a mistake in the options.