Question

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.)
Incorrect: Your answer is incorrect. .

(c) Calculate sx and sy. (Give your answers correct to three decimal places.)
sx = Incorrect: Your answer is incorrect. .
sy = Incorrect: Your answer is incorrect. .

(d) Calculate r using formula 13.2. (Give your answer correct to two decimal places.)
Incorrect: Your answer is incorrect. .

(e) Calculate r using formula 13.3. (Give your answer correct to two decimal places.)
Incorrect: Your answer is incorrect. .

Answer 1

(a) You can use a graph or plotting software to create a scatter diagram, but here is a table representation of the scatter diagram. Note that this is just a description, and you should still create an actual scatter diagram on paper:

0 1 2 3 4 5 6 7 <-x-axis
0 H
1 A
2 J
3 C D
4
5 F G
6 E
7 B
y-axis

(b) To calculate the covariance, we need to find the mean of x and y values. Mean of x is (3+4+2+1+7+2+1+0+4+2)/10 = 26/10 = 2.6. The mean of y is (1+7+3+3+6+6+5+0+6+2)/10 = 39/10 = 3.9.

The covariance formula is given by the sum[(x_i - mean_x)*(y_i - mean_y)] / (n-1).

Covariance = [(3-2.6)(1-3.9) + (4-2.6)(7-3.9) + (2-2.6)(3-3.9) + (1-2.6)(3-3.9) + (7-2.6)(6-3.9) + (2-2.6)(6-3.9) + (1-2.6)(5-3.9) + (0-2.6)(0-3.9) + (4-2.6)(6-3.9) + (2-2.6)(2-3.9)] / 9
Covariance ≈ 5.38

(c) Next, calculate the standard deviations, sx and sy.

For sx, first find the variance by sum[(x_i - mean_x)^2] / (n-1):

Variance of x = [(3-2.6)^2 + (4-2.6)^2 + (2-2.6)^2 + (1-2.6)^2 + (7-2.6)^2 + (2-2.6)^2 + (1-2.6)^2 + (0-2.6)^2 + (4-2.6)^2 + (2-2.6)^2] / 9 ≈ 6.933

Now take the square root to find standard deviation sx:

sx = √6.933 ≈ 2.633

Similarly, find the variance for y:

Variance of y = [(1-3.9)^2 + (7-3.9)^2 + (3-3.9)^2 + (3-3.9)^2 + (6-3.9)^2 + (6-3.9)^2 + (5-3.9)^2 + (0-3.9)^2 + (6-3.9)^2 + (2-3.9)^2] / 9 ≈ 6.544

Now take the square root to find standard deviation sy:

sy = √6.544 ≈ 2.558

(d) Now, calculate r using formula 13.2:

r = Covariance / (sx * sy) = 5.38 / (2.633 * 2.558) ≈ 0.79

(e) Alternatively, calculate r using formula 13.3:

r = [n * sum(xy) - sum(x) * sum(y)] / √[(n * sum(x^2) - sum(x)^2)(n * sum(y^2) - sum(y)^2)]

Substitute the given values:

sum(x) = 26, sum(y) = 39, sum(xy) ≈ 104.8, sum(x^2) ≈ 68.2, sum(y^2) ≈169.1

r = [10*104.8 - 26*39] / √[(10 * 68.2 - 26^2)(10 * 169.1 - 39^2)] ≈ 0.79