Question

The voltage in a circuit is 115 volts. A particular technique for measuring the voltage gives readings which are normally distributed with mean μ=115 volts and standard deviation 5 volts. Show that the average of four readings has smaller probability of differing from the true value by 3 volts than an individual reading. Hence show that average of several measurement of the same thing is always more accurate than an individual measurement.

Answer 1

Case 1: Probability For Individual reading X differing from true value by 3:

= 115

= 5

Z = 3/5

= 0.60

Table of Area Under Standard Normal Curve gives area = 0.2257

So,

P( For Individual readings differing from true value by 3) = 1 - (2 X 0.2257) = 1 - 0.4514 = 0.5486

EXPLANATION:

0.2257 is the probability that an individual value within true value by 3 on RHS. Same area = 0.2257 on LHS. Thus, 2 X 0.2257 gives the probability that an individual value within true value by 3 on both sides. So, 1 - (2 X 0.2257) gives probability differing from true value by 3.

Case 2: Probability For Average of 4 readings differing from true value by 3:

= 115

= 5

n = 4

SE = /

    = 5/ = 2.5

Z = 3/2.5

= 1.20

Table of Area Under Standard Normal Curve gives area = 0.3849

So,

P( For Average of 4 readings differing from true value by 3) = 1 - (2 X 0.3849) = 1 - 0.7698 = 0.2302

Thus, we note that the Average of 4 readings has smaller probability (0.2302) of differing from the true value by 3 volts than an individual reading (0.5486).

By increasing n from 4 to higher values, we note that average of several measurement of the same thing is always more accurate than an individual measurement.