Write the properties of correlation coefficient. The time it takes to transmit a file always depends on the file size. Suppose you transmitted 30 files, with the average size of 126 Kbytes and the standard deviation of 35 Kbytes. The average transmitted time was 0.04 seconds with the standard deviation 0.01 seconds. The correlation coefficient between the time and size was 0.86. Based on these data, fit a linear regression model and predict the time it take to transmit a 400Kbyte file.

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The properties of corelation cofficient are:

  1. Correlation cofficient liea between -1 to +1
    i.e. -1 ≤ x ≤ +1
  2. Correlation cofficient is symmetrical
    i.e. rxy = ryx = r
  3. Correlation cofficient is independent of change of origin and scale.
  4. Correlation cofficient is the geometric mean of the two regression cofficient \(r = \pm \sqrt{b_{yx} \times b_{xy}}\)
  5. Correlation cofficient has no unit because of relative measure.

Solution:

In the given problem we have two variables: the transmission time and average size of file.

Y = transmission time

X = average file size

The linear regression model is given by

Y = a + bX

The slope b is given by

b = correlation coefficient × \(\frac{SD_{y}}{SD_{x}}\)

Where SDy is the standard deviation of average transmittance time and SDx is the standard deviation of average file size.

b = \(0.86 \times \frac{0.01}{35}\)

b = 0.0002457

The y-intercept a is given by

a = y – bx

a =  0.04 – (0.0002457) × 126

a = 0.04 – 0.030958

a = 0.009042

Therefore, the linear regression model is

Y = 0.009042 + 0.0002457X

Part II:

Predict the time it will take to transmit a 400 Kbyte file.

Substitute X = 400 in the regression model

Y = 0.009042 + 0.0002457 × (400)

Y = 0.009042 + 0.09828

Y = 0.1073 seconds

Therefore, the predicted time to transmit a 400 Kbyte file is 0.1073 seconds.

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