Question

1. One of the major problems faced by an auto insurance company is the filing of fraudulent claims. An insurance company carefully investigated 1000 auto claims filed with it and found 108 of them to be fraudulent. Construct a 96% confidence interval for the corresponding population proportion. Show all work in the space below.

2. Write a full sentence interpreting the numbers you have calculated, i.e. what do these numbers stand for?

3. If you used a 90% Confidence Level instead, the CI would be _____________(narrower/wider). Answer without doing any calculations.

Answer 1

Q1) For 96% confidence level, we have from standard normal tables here:
P(Z < 2.054) = 0.98,

Therefore, due to symmetry, we have here:
P(-2.054 < Z < 2.054) = 0.96

The sample proportion here is computed as:
p = x/n = 108/1000 = 0.108

Therefore the confidence interval here is obtained as:

$p \pm z^*\sqrt{\frac{p(1-p)}{n}}$

0.108 +2.054 本 10.10811-0.108) 1000

$0.108 \pm 0.0202$

This is the required 96% confidence interval for the population proportion here.

Q2) The interpretation of the confidence interval here is that we are 96% confidence that the true population proportion lies in the above obtained confidence interval.

Q3) As the confidence level here is 90%, the critical z value decreases and therefore the margin of error decreases and therefore the confidence interval becomes narrower. Therefore narrower is the required answer here.