Question

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?

Short answer

a. One can expect that an expert correctly identifying a match in the sample is 0.92.

b. The probability that an expert correctly identifies fewer than 900 of the fingerprint matches 0.2297.

Step-by-step solution

Given information

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.

Finding the mean of the sample proportion

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

The mean of the sample proportion is:

$$\begin{gathered} E\left(\hat{p}\right)=p \\ =0.92 \end{gathered}$$

.Since

.

.

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

Finding the standard deviation of the sample proportion

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

Finding the required probability

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.