Question

A researcher measures the relationship between the number of interruptions during a class and time spent "on task" (in minutes). Answer the following questions based on the results provided. Number of Interruptions Time Spent "On Task" 11 15 5 40 9 18 4 31 Part (a) Compute the Pearson correlation coefficient. (Round your answer to three decimal places.) Part (b) Multiply each measurement of interruptions times 3 and recalculate the correlation coefficient. (Round your answer to three decimal places.) Part (c) Divide each measurement in half for time spent "on task" and recalculate the correlation coefficient. (Round your answer to three decimal places.) Part (d) True or false: Multiplying or dividing a positive constant by one set of scores (X or Y) does not change the correlation coefficient. Note: Use your answers in (a) to (c) to answer true or false. True False

Answer 1

Let the X indicate Number of interruptions and Y indicate the Time spent on "Task".

Then X values are -- 11, 15, 5, 40. and Y values are -- 9, 18, 4, 31

(a)

Pearson correlation coefficient between X and Y is given by,

Cor(X,Y) = ,

r =9 COV(X,Y) 0x<oy

Where Cov (X,Y) is the sample covariance given by, Cov(X,Y) =

ΣΕ (X-2) (Y- η-1

– sample standard deviation of X =
ΣΩ,(-2): 1-1 Ν

– sample standard deviation of Y =
ΣΩ,(Yi-y)2- 2-1 Ν

Here, n – sample size = 4

X-sample mean of x = 481Xi
and
Ý - sample mean of Y =

X = 17.75
and
Y = 15.5

Cov(X,Y) =
528.5 - = 176.1667 3
,
Ox = 710.75 6 -= 15.3921
,  
421 -118462

Cor(X,Y) = =
176.1667 15.3921-19462 = 0.966

The answer is 0.966

(b)

If X values are multiplied by 3 the new values will be X values are -- 33, 45, 15, 120. and Y values are -- 9, 18, 4, 31

Then we have

X = 53.25
and
Y = 15.5

Cov(X,Y) =
158.9 = 528.S.
,
Ox= 6396.75 V 3 = 46.1763
,  
421 -118462

Cor(X,Y) = =
| 528.5 46.1763x11.8462 = = 0.966

The answer is 0.966

(c)

If Y values are divided by half the new values will be X values are -- 11, 15, 5, 40. and Y values are -- 4.5, 9, 2, 15.5

X-sample mean of x = 481Xi
and
Ý - sample mean of Y =

X = 17.75
and
Y = 7.75

Cov(X,Y) =
264.25. == 88.0833 3
,  
Ox = 710.75 6 -= 15.3921
,  
| 105.25 = 5.9231

Cor(X,Y) = =
88.0833 15.3921-5.52j = 0.966

The answer is 0.966.

(d)

True or false: Multiplying or dividing a positive constant by one set of scores (X or Y) does not change the correlation coefficient. --- TRUE.