Can some one help me with this one???
Consider the following bivariate data.
Point A B C D E F G H I J
x 3 4 2 1 7 2 1 0 4 2
y 1 7 3 3 6 6 5 0 6 2
(a) Construct a scatter diagram of the given bivariate data. (Do this on paper. Your instructor may ask you to turn in this work.)
(b) Calculate the covariance. (Give your answer correct to two decimal places.)
(c) Calculate sx and sy. (Give your answers correct to three decimal places.)
sx =
sy =
(d) Calculate r using formula 13.2. (Give your answer correct to two decimal places.)
(e) Calculate r using formula 13.3. (Give your answer correct to two decimal places.)
(a) To construct a scatter diagram of the given bivariate data, you need to plot each data pair (x, y) as a point on a graph. The x-values should be plotted on the horizontal axis, and the y-values should be plotted on the vertical axis.
For example, for point A with x = 3 and y = 1, you would locate the point (3, 1) on the graph. Repeat this for all the other points in the data set. Once you have plotted all the points, connect them with a line or use dots to represent the data points.
(b) To calculate the covariance, you can use the following formula:
cov(x, y) = Σ[(xi - mean(x))(yi - mean(y))] / (n - 1)
First, calculate the mean of x and y:
mean(x) = (3 + 4 + 2 + 1 + 7 + 2 + 1 + 0 + 4 + 2) / 10 = 26 / 10 = 2.6
mean(y) = (1 + 7 + 3 + 3 + 6 + 6 + 5 + 0 + 6 + 2) / 10 = 39 / 10 = 3.9
Then, calculate the deviations for each data point:
Deviation for x: (3 - 2.6), (4 - 2.6), (2 - 2.6), (1 - 2.6), (7 - 2.6), (2 - 2.6), (1 - 2.6), (0 - 2.6), (4 - 2.6), (2 - 2.6)
Deviation for y: (1 - 3.9), (7 - 3.9), (3 - 3.9), (3 - 3.9), (6 - 3.9), (6 - 3.9), (5 - 3.9), (0 - 3.9), (6 - 3.9), (2 - 3.9)
Next, calculate the product of the deviations for each pair of data points:
Product of deviations: (−0.6)(−2.9), (1.4)(3.1), (−0.6)(−0.9), (−1.6)(−0.9), (4.4)(2.1), (−0.6)(2.1), (−1.6)(1.1), (−2.6)(−3.9), (1.4)(2.1), (−0.6)(−1.9)
Sum up the products of deviations:
Σ[(xi - mean(x))(yi - mean(y))] = (-0.6)(-2.9) + (1.4)(3.1) + (-0.6)(-0.9) + (-1.6)(-0.9) + (4.4)(2.1) + (-0.6)(2.1) + (-1.6)(1.1) + (-2.6)(-3.9) + (1.4)(2.1) + (-0.6)(-1.9)
Finally, divide the sum by (n - 1), where n is the number of data points:
cov(x, y) = Σ[(xi - mean(x))(yi - mean(y))] / (n - 1)
(c) To calculate sx and sy, you need to find the standard deviation of x and y. The standard deviation is a measure of how spread out the data is from the mean.
To calculate the standard deviation sx of x, you can use the following formula:
sx = sqrt(sum((xi - mean(x))^2) / (n - 1))
To calculate the standard deviation sy of y, you can use the following formula:
sy = sqrt(sum((yi - mean(y))^2) / (n - 1))
In these formulas, sum((xi - mean(x))^2) means you need to calculate the square of the deviation of each data point from the mean of x, and then sum all these squared deviations. Do the same for y.
Finally, divide the sum of squared deviations by (n - 1), where n is the number of data points, and take the square root of the result.
(d) To calculate r using formula 13.2, you can use the following formula:
r = cov(x, y) / (sx * sy)
This formula calculates the correlation coefficient r using the covariance of x and y, and the standard deviations of x and y. Substitute the values you calculated in steps (b) and (c) into this formula to get the final result.
(e) To calculate r using formula 13.3, you can use the following formula:
r = Σ[(xi - mean(x))(yi - mean(y))] / sqrt(Σ(xi - mean(x))^2 * Σ(yi - mean(y))^2)
This formula also calculates the correlation coefficient r, but it uses the deviations of x and y from their means, squared deviations, and sums of squared deviations. Substitute the values you calculated in step (b) into this formula to get the final result.