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Audit Tests: Risk, Confidence
and Materiality
Some basic statistics about
Inherent Risk and Control Risk
Audit Testing

Audit testing is done on a single account to test a
hypothesis


H0 : Actual error in the account is less than the
tolerable limit (set in planning / materiality)
Account testing compares:
 Statistical estimates:
x̂  {y% confidence limit}


with
Financial stmt:
A/C balance ± tolerable error

The ‘true’ account value

Suppose you take several samples each of size n
from the population and for each you calculate
x  1.96


n
Then, on average, 95% of the intervals will
contain the true but unknown value µ and 5% will
not.
If you plotted the intervals
vertically they might look like this
Point / Interval

The sample mean provides a point estimate (i.e.
single value approximation) for µ

Confidence limits provide an interval estimate
together with a degree of confidence that the
parameter is in the interval
e.g. with 95% confidence the population mean
height µ is in the interval (164, 166) cms




The width of the interval (i.e. precision of estimate)
depends on sample size.
In the example, the sample size was n=100 so the 95%
confidence interval is
165 ± 1.96 × i.e. (164, 166) cms.
Sample Size

Suppose the sample size had been n=40 but the mean and
standard deviation were still = 165 and s = 5. Then a 95%
confidence interval for µ is



Notice that increasing the sample size increases the precision
of the estimate







165 ± 1.96 × = 165 ± 1.55
which gives (163.5, 166.5).
e.g. width of 95% confidence interval
=U-L
= ( + 1.96 ) - ( - 1.96 )
= 2 × 1.96
So if n = 100, width = = 0.392 s
or if n = 25, width = = 0.784 s.
If you increase the sample size by 4 you decrease the width of
the confidence interval by ½. Precision of the estimate
depends on the term in the standard error SE =
Example - Bolt production

A manufacturer produces bolts with a nominal length of 15
cms. The actual lengths vary slightly. The process is stable
and the population standard deviation is known to be s = 0.3
cms.




A sample of 50 bolts has a mean length of = 14.85 cms. Does
this suggest that the average length of all bolts is not 15 cms?
Sampling distribution of sample mean is
In this case s = 0.3, n = 50, = 14.85 and we want to know if µ =
15 is plausible.
Find a confidence interval for µ. A 95% confidence interval is
given by


14.85 ± 1.96 × i.e. (14.77, 14.93)cms
Interpretation - with 95% confidence the interval (14.77, 14.93)
contains the population mean µ of all bolts produced by the
process. As the interval does not contain 15.0, the data are not
consistent with the hypothesis that µ = 15. That is, the data do
not support the hypothesis that the average length of all bolts is
15cms.
Another Approach

The second approach is to test the hypothesis (i.e. µ = 15cm) more directly
as follows


Assume µ = 15 (that is, assume the null hypothesis is true).
Calculate the probability of getting a sample mean as far away as or further from the
assumed population mean as was observed (i.e. = 14.85)

values as far away as or further from µ = 15 as the observed value = 14.85
This is called the "p-value“

In this case




p-value = P( 14.85 or 15.15)
If ( ~ N(15,) then Z =
Hence P( 14.85 or 15.15).
=
= P(Z < - 3.5 or Z > 3.5) < 0.001 from tables
This probability, p-value <0.001, is very small so we conclude that the sample
data provide evidence against the assumption µ = 15. We reject the
hypothesis that the average length of all bolts was 15 cms.
Hypothesis Testing
 H0 is the assertion that the clients
accounts are correct
N(0,1)
t5
t1
Mean
0
0
0
Variance
1
5/3
Skewness
0
0
Kurtosis
3
9
inf.
0
inf.
TRUTH (unkown)
H0 true
H0 false
Do not reject H0
Correct Decision
Type II Error
Reject H0
Type I error
Correct Decision
DECISION
Power


Type I and Type II errors, and the power of a statistical
test
In hypothesis testing there are two kinds of errors you can
make




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i) Reject H0 (because the p-value is small) when H0 is true
ii) Do not reject H0 (because the p-value is not small) when H0 is
false
power = 1 - P(type II error given H0 is false)
The probability of accepting the null hypthesis when it is false
is conventionally called b ("beta"), so that: power = 1-b.
Ideally studies should be designed so that power, 1-b, is at
least 0.8. This requires using an efficient design and a
sufficiently large sample.
Errors
TRUTH (unkown)
H0 true
H0 false
Do not reject H0
Correct
Decision
Type II Error
Reject H0
Type I error
Correct Decision
DECISION
 H0 is the assertion that the clients
accounts are correct
Risk Measures

Probability of Type I error

Expected loss over entire distribution
of error (Bayes’ Risk)

Willingness to pay for insurance
against a specific risk (in portfolio
theory, Markowitz risk premium)