1. Audit Tests: Risk, Confidence and Materiality Some basic statistics about Inherent Risk and Control Risk

2. Audit Testing Audit testing is done on a single account to test a hypothesis  H 0 : Actual error in the account is less than the tolerable limit (set in planning / materiality) Account testing compares:  Statistical estimates: {y% confidence limit} with  Financial stmt: A/C balance ± tolerable error

3. The ‘true’ account value Suppose you take several samples each of size n from the population and for each you calculate Then, on average, 95% of the intervals will contain the true but unknown value µ and 5% will not.

4. If you plotted the intervals vertically they might look like this

5. 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.

6. 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  165 ± 1.96 × = 165 ± 1.55  which gives (163.5, 166.5). Notice that increasing the sample size increases the precision of the estimate  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 =

7. 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.

8. 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) < 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.

9. Hypothesis Testing N(0,1)t5t5 t1t1 Mean000 Variance15/3inf. Skewness000 Kurtosis39inf. TRUTH (unkown) DECISION H 0 trueH 0 false Do not reject H 0 Correct DecisionType II Error Reject H 0 Type I errorCorrect Decision H 0 is the assertion that the clients accounts are correct

10. 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 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.

11. Errors H 0 is the assertion that the clients accounts are correct TRUTH (unkown) DECISION H 0 trueH 0 false Do not reject H 0 Correct Decision Type II Error Reject H 0 Type I errorCorrect Decision

12. 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)