3 Probabilities and the standard normal distribution

Since the standard normal distribution is used so frequently a table of values has been produced to help us calculate probabilities - located at the end of the Workbook. It is based upon the following diagram:

Figure 3

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Since the total area under the curve is equal to 1 it follows from the symmetry in the curve that the area under the curve in the region Z > 0 is equal to 0.5 . In Figure 3 the shaded area is the probability that Z takes values between 0 and z 1 . When we ‘look-up’ a value in the table we obtain the value of the shaded area.

Example 3

What is the probability that Z takes values between 0 and 1.9? (Refer to the table of normal probabilities at the end of the Workbook.)

Solution

The row beginning ‘1.9’ and the column headed ‘0’ is the appropriate choice and its entry is 4713. This is to be read as 0.4713 (we omitted the ‘0.’ in each entry for clarity.) The interpretation is that the probability that Z takes values between 0 and 1.9 is 0.4713.

Example 4

What is the probability that Z takes values between 0 and 1.96?

Solution

This time we want the row beginning 1.9 and the column headed ‘6’.

The entry is 4750 so that the required probability is 0.4750.

Example 5

What is the probability that Z takes values between 0 and 1.965?

Solution

There is no entry corresponding to 1.965 so we take the average of the values for 1.96 and 1.97. (This linear interpolation is not strictly correct but is acceptable.)

The two values are 4750 and 4756 with an average of 4753. Hence the required probability is 0.4753.

Task!

What are the probabilities that Z takes values between

  1. 0 and 2
  2. 0 and 2.3
  3. 0 and 2.33
  4. 0 and 2.333?
  1. The entry is 4772; the probability is 0.4772.
  2. The entry is 4893; the probability is 0.4893.
  3. The entry is 4901; the probability is 0.4901.
  4. The entry for 2.33 is 4901, that for 2.34 is 4904.

    Linear interpolation gives a value of 4901 + 0.3 ( 4904 − 4901 ) i.e. about 4902; the

    probability is 0.4902.

Note from Table 1 that as Z increases from 0 the entries increase, rapidly at first and then more slowly, toward 5000 i.e. a probability of 0.5. This is consistent with the shape of the curve.

After Z = 3 the increase is quite slow so that we tabulate entries for values of Z rising by increments of 0.1 instead of 0.01 as in the rest of Table 1.