Question

Based on past experience, a bank believes that 8 % of the people who receive loans will not make payments on time. The bank has recently approved 200 loans.

What are the mean and standard deviation of this model?

Answer 1

To solve this problem, we can model the situation using a binomial distribution since we are dealing with a fixed number of trials (loans), each with two possible outcomes (making payments on time or not).

Let:
- \( n \) = number of trials (loans approved) = 200
- \( p \) = probability of success (not making payments on time) = 0.08
- \( q \) = probability of failure (making payments on time) = \( 1 - p = 0.92 \)

### Mean

The mean \( \mu \) of a binomial distribution can be calculated using the formula:

\[
\mu = n \cdot p
\]

Substituting in the values:

\[
\mu = 200 \cdot 0.08 = 16
\]

### Standard Deviation

The standard deviation \( \sigma \) of a binomial distribution can be calculated using the formula:

\[
\sigma = \sqrt{n \cdot p \cdot q}
\]

Substituting in the values:

\[
\sigma = \sqrt{200 \cdot 0.08 \cdot 0.92}
\]
\[
\sigma = \sqrt{200 \cdot 0.0736} \approx \sqrt{14.72} \approx 3.84
\]

### Conclusion

Thus, the mean number of people who will not make payments on time is \( 16 \), and the standard deviation is approximately \( 3.84 \).