Question

Indicate whether each of the following statements is true (T) or false (F) and explain your answer:

(a) The covariance between two variables is the same irrespective of the unit of measurements.

(b) The expressions “rule of thumb” and “conventionally” are synonyms.

(c) If the covariance between Y and itself is 16, then the standard deviation of Y is 16. 2

(d) If there is no relationship between two variables Y and X, then their correlation coefficient is zero.

(e) For each of the last 10 years, we recorded whether a company has made profit (1), loss (–1), or broke even (0), then the sum of the absolute value of the data is the same as the number of years in which the company has made profit.

(f) For any set of numbers y1, y2, · · · , yn, ∑n j=1 (yj − y¯) = 0.

Answer 1

a) False

Let X and Y are measured in metre and the covariance between them is cov(X,Y).

Now let the unit is changed to centimetres. Let U,V are values of Xand Y in cm.

That is, U =100*X, V =100*Y.

Then cov(U,V) = cov(100*X,100*Y) = 10000* cov(X,Y) which is generally different from cov(X,Y). Thus the given statement is false.

b) True

This is true because the meanings are same.

c) False

Cov(Y,Y) is actually Var(Y). And standard deviation of Y is sd(Y) =√ [var(Y)]

Now it is given that Var(Y)=16

So sd(Y)=√16= 4 . So sd is 4 not 16.2. Hence it is false.

d)True.

This is by definition of covariance. If not relation exists at all, between the variables then ofcourse the covariance is 0.

e) False

Let in 1st profit is made, 2nd year loss is made and for rest of the 8 years, broke even occurred.

So sum of absolute values of the data is 2 which is not same as the no.of years when profit is made(which is 1).

f) True.

See,

Hope the solution helps. Thank you.