Question

In: Statistics and Probability

Calculate the covariance between variables X and Y. Is it a positive or negative relationship between...

Calculate the covariance between variables X and Y. Is it a positive or negative relationship between the two variables?

b. Calculate correlation coefficient between X and Y. Is it a positive or negative relationship? Is it a strong linear, weak linear or nonlinear relationship between X and Y?

c. Use the Y data to calculate mean, range, standard deviation and variance.

d. Use the first Y value to calculate the Z-score. Is it an outlier?

e. Calculate the 60th percentile for the Y data.

1 -22 22
2 -33 49
3 2 8
4 29 -16
5 -13 10
6 21 -28
7 -13 27
8 -23 35
9 14 -5
10 3 -3
11 -37 48
12 34 -29
13 9 -18
14 -33 31
15 20 -16
16 -3 14
17 -15 18
18 12 17
19 -20 -11
20 -7 -22

Solutions

Expert Solution

Let us first calculate the covariance between X and Y. This is calculated with the formula:

where are means of x and y values.

First we form a table as below for further calculations.

X Y X^2 Y^2 XY
-22 22 484 484 -484
-33 49 1089 2401 -1617
2 8 4 64 16
29 -16 841 256 -464
-13 10 169 100 -130
21 -28 441 784 -588
-13 27 169 729 -351
-23 35 529 1225 -805
14 -5 196 25 -70
3 -3 9 9 -9
-37 48 1369 2304 -1776
34 -29 1156 841 -986
9 -18 81 324 -162
-33 31 1089 961 -1023
20 -16 400 256 -320
-3 14 9 196 -42
-15 18 225 324 -270
12 17 144 289 204
-20 -11 400 121 220
-7 -22 49 484 154
Total -75 131 8853 12177 -8503

From the data, we have n=20

  

Now

Since the Covaraiance is negative, the relationship between X and Y is negative.

b). To find out the linear relationship between X and Y, we use the formula.

  

  

  

. Since the Correlation coefficient is -0.8134, it is a negative relationship between X and Y. Since the Value is more than 0.75, it is a string negative linear relationship.

c). For the Y data,

The mean

Variance

Therefore the sample variance of Y is

The range is Maximum-minimum of all values

d). Z score for first value of Y ie 22

The Z score is given by:

   Since we have calculated , s=24.4077

  

. It doesn't seem to be an outlier.

e). To calculate, 60th percentile, we need to order the data in ascending order and locate the 20*60/100 th value.

ie 12th value. The sorted data is presented below:

-29
-28
-22
-18
-16
-16
-11
-5
-3
8
10
14
17
18
22
27
31
35
48
49

The 12th value in the sorted data is the 60th percentile which is 14.

  

orchestra answered 1 year ago
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