below, on which of the three tests did the candidate do best?
(^) Raw Score Mean S.D.
Test 1 22 12 4
Test 2 42 30 5
Test 3 110 100 15
The best score is on the numeracy test, (test 1) Z=+2.5
Example 4.9: Setting Probable Limits Using Z
Rather than a tail area we may be interested in both directions of the distribution. If we
wanted to know what are the pulse levels that are likely to be either larger or smaller than
the values, we would expect for 95 per cent of all males sampled? Call these values the
Lower and Upper probable limits. This is equivalent to finding the Z values for 5 per cent
(100 per cent-95 per cent) of cases at the extremes of the distribution. By symmetry half
of the area must be in each tail so we actually need the Z values for p=0.025. The Z
values are −1.96 and +1.96. You should check these values in Table 1, Appendix A.
These values can now be transformed into pulse rates by use of Equation 4.7 to
determine a Z score.
Lower limit=(12)(−1.96)+72=48.48
Upper Limit=(12)(+1.96)+72=95.52
In other words, the upper and lower limits within which 95 per cent of male pulse rates
would fall is between 48 and 96.
Probable Limits and Confidence Intervals
You should not confuse describing probable limits for an observation with the 95 per cent
confidence interval of a sampling distribution. The former is concerned with locating the
limits within which 95 per cent of observations fall and is descriptive. Confidence
intervals are concerned with estimating unknown parameters and are inferential.
4.7 Hypothesis Testing
Another important use of the normal distribution is when testing hypotheses. We can test
hypotheses about observations or statistics. If you have understood how the concepts of
statistics, parameters and sampling distributions are used in estimation you should find
Probability and inference 107