Question:Fingerprint expertise. Refer to the Psychological Science (August 2011) study of fingerprint identification, Exercise 4.53 (p. 239). Recall that when presented with prints from the same individual, a fingerprint expert will correctly identify the match 92% of the time. Consider a forensic database of 1,000 different pairs of fingerprints, where each pair is a match.

a. What proportion of the 1,000 pairs would you expect an expert to correctly identify as a match?

b. What is the probability that an expert will correctly identify fewer than 900 of the fingerprint matches?

The proportion of the times a fingerprint expert will correctly identify the fingerprint match is p=92.

A random sample of n=100 different pairs of fingerprints is selected.

Let$\hat{p}$represent the sample proportion of times an expert correctly identify as a match.

The mean of the sample proportion is:

.Since

.

.

Therefore, one can expect that an expert correctly identifying a match in the sample is 0.92.

The standard deviation of the sampling distribution of is obtained as:$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}} \\ =\sqrt{\frac{0.92\times 0.08}{100}} \\ =\sqrt{0.000736} \\ =0.0271 \end{gathered}$$

The probability that an expert will correctly identify fewer than 900 of the fingerprint matches, that is, the sample proportion is less than 0.90, is obtained as:

$$\begin{gathered} P\left(\hat{p}<0.90\right)=P\left(\frac{\hat{p}-p}{{\sigma }_{\hat{p}}}<\frac{0.90-p}{{\sigma }_{\hat{p}}}\right) \\ =P\left(Z<\frac{0.90-0.92}{0.0271}\right) \\ =P\left(Z<\frac{-0.02}{0.0271}\right) \\ =P\left(Z<-0.74\right) \\ =0.2297 \end{gathered}$$

The probability is obtained using the z-table; in the z-table, the value at the intersection of -0.70 and 0.04 is the desired value.

Thus the required probability is 0.2297.