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. .
(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
(a) To construct a scatter diagram of the given bivariate data, we plot the points with their corresponding x and y values. The scatter diagram is as follows:
Point A: (3, 1)
Point B: (4, 7)
Point C: (2, 3)
Point D: (1, 3)
Point E: (7, 6)
Point F: (2, 6)
Point G: (1, 5)
Point H: (0, 0)
Point I: (4, 6)
Point J: (2, 2)
(b) To calculate the covariance, we will use the formula:
cov(X, Y) = Σ[(x - mean of X) * (y - mean of Y)] / (n - 1)
First, we need to calculate the mean of X and Y. The mean of X is:
mean of X = (3 + 4 + 2 + 1 + 7 + 2 + 1 + 0 + 4 + 2) / 10 = 26 / 10 = 2.6
The mean of Y is:
mean of Y = (1 + 7 + 3 + 3 + 6 + 6 + 5 + 0 + 6 + 2) / 10 = 39 / 10 = 3.9
Next, we calculate the covariance:
cov(X, Y) = [(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
cov(X, Y) = [-0.4 * -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] / 9
cov(X, Y) = (1.16 + 4.34 + 0.54 + 1.44 + 9.24 + -1.26 + 1.76 + 10.14 + 2.94 + 1.14) / 9
cov(X, Y) = 30.94 / 9 = 3.438
Therefore, the covariance is 3.438 (correct to two decimal places).
(c) To calculate sx and sy, we will use the formulas:
sx = √[Σ(x - mean of X)² / (n - 1)]
sy = √[Σ(y - mean of Y)² / (n - 1)]
First, we need to find the squares of the differences between each x value and the mean of X:
(3 - 2.6)² = 0.16
(4 - 2.6)² = 1.44
(2 - 2.6)² = 0.36
(1 - 2.6)² = 2.56
(7 - 2.6)² = 14.44
(2 - 2.6)² = 0.36
(1 - 2.6)² = 2.56
(0 - 2.6)² = 6.76
(4 - 2.6)² = 1.44
(2 - 2.6)² = 0.36
The sum of these squares is:
Σ(x - mean of X)² = 0.16 + 1.44 + 0.36 + 2.56 + 14.44 + 0.36 + 2.56 + 6.76 + 1.44 + 0.36 = 30.36
Next, we divide the sum by (n - 1) and take the square root to find sx:
sx = √(30.36 / 9) = √3.373 = 1.836 (correct to three decimal places)
Similarly, we calculate the squares of the differences between each y value and the mean of Y:
(1 - 3.9)² = 15.21
(7 - 3.9)² = 12.96
(3 - 3.9)² = 0.81
(3 - 3.9)² = 0.81
(6 - 3.9)² = 4.41
(6 - 3.9)² = 4.41
(5 - 3.9)² = 1.21
(0 - 3.9)² = 15.21
(6 - 3.9)² = 4.41
(2 - 3.9)² = 6.76
The sum of these squares is:
Σ(y - mean of Y)² = 15.21 + 12.96 + 0.81 + 0.81 + 4.41 + 4.41 + 1.21 + 15.21 + 4.41 + 6.76 = 66.94
Dividing the sum by (n - 1) and taking the square root gives us sy:
sy = √(66.94 / 9) = √7.438 = 2.728 (correct to three decimal places)
Therefore, sx = 1.836 and sy = 2.728 (both correct to three decimal places).
(d) To calculate r using formula 13.2, we use the formula:
r = cov(X, Y) / (sx * sy)
Substituting the values calculated earlier:
r = 3.438 / (1.836 * 2.728) = 0.675 (correct to two decimal places)
Therefore, r = 0.675 (correct to two decimal places).
(e) To calculate r using formula 13.3, we use the formula:
r = Σ[(x - mean of X) * (y - mean of Y)] / √[Σ(x - mean of X)² * Σ(y - mean of Y)²]
Substituting the values calculated earlier:
r = 30.94 / √(30.36 * 66.94) = 0.897 (correct to two decimal places)
Therefore, r = 0.897 (correct to two decimal places).
To construct a scatter diagram of the given bivariate data, you will need to plot each data point on a graph using the x and y coordinates provided. Here is how you can do it:
1. Draw a set of axes on a piece of graph paper, with the x-axis labeled on the horizontal axis and the y-axis labeled on the vertical axis.
2. Plot each data point on the graph according to their x and y values. For example, point A has x=3 and y=1, so you would mark a point on the graph at coordinates (3,1).
3. Repeat this step for all the other data points, plotting each one on the graph.
4. Once all the points have been plotted, connect them with a line or a curve to create a scatter diagram. The pattern formed by the points can help you identify any trends or relationships in the data.
Now let's move on to the calculations:
(b) To calculate the covariance, you can use the following formula:
cov(X,Y) = Σ((X - X̅)(Y - Ȳ))/(n - 1)
where X and Y are the data sets, X̅ and Ȳ are their respective mean values, and n is the number of data points.
First, calculate the mean of the x-values (X̅) and the mean of the y-values (Ȳ):
X̅ = (3 + 4 + 2 + 1 + 7 + 2 + 1 + 0 + 4 + 2)/10 = 2.6
Ȳ = (1 + 7 + 3 + 3 + 6 + 6 + 5 + 0 + 6 + 2)/10 = 3.9
Next, calculate the covariance using the formula:
cov(X,Y) = [((3-2.6)(1-3.9)) + ((4-2.6)(7-3.9)) + ... + ((2-2.6)(2-3.9))]/(10-1)
Performing the calculations, you will get the covariance value.
(c) To calculate the standard deviation of x (sx) and the standard deviation of y (sy), you can use the following formulas:
sx = sqrt(Σ((x - x̅)²)/(n-1))
sy = sqrt(Σ((y - ȳ)²)/(n-1))
where x and y are the individual data points, x̅ and ȳ are the mean values calculated earlier, and n is the number of data points.
To calculate sx and sy, apply the formulas using the given data points and the mean values computed earlier. Round your answers to three decimal places.
(d) To calculate the correlation coefficient (r) using formula 13.2, you can use the following formula:
r = cov(X, Y)/(sx*sy)
where cov(X, Y) is the covariance calculated earlier, and sx and sy are the standard deviations of x and y respectively.
Plug in the values and calculate r. Round your answer to two decimal places.
(e) To calculate the correlation coefficient (r) using formula 13.3, you can use the following formula:
r = Σ(xy)/sqrt(Σ(x²) * Σ(y²))
where Σ(xy) is the sum of the products of x and y for each data point, Σ(x²) is the sum of the squares of x, and Σ(y²) is the sum of the squares of y.
Calculate the sums and perform the division to find r. Round your answer to two decimal places.