Answer to Question #168041 in Statistics and Probability for Matt campbell

Question

The number of people looking to change auto insurance companies reached a record low. However, for the population of those people who were looking for a better deal, 45% did change companies. Suppose 110 people looking for a better deal on auto insurance are selected at random and the number who actually switch policies is determined. Find the probability that the sample proportion will be between 0.37 and 0.47.

Answer 1

The standard deviation of all sample proportions is $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.45*0.55}{110}} \thickapprox 0.05$

Hence sample proportion is normally distributed with parameters (0.45; 0.05)

Let's find the probability that the sample proportion will be between 0.37 and 0.47:

$\frac{1}{0.05*\sqrt{2*\pi}} \int^{0.47}_{0.37} e^{\frac{-(x-0.45)^2}{2*0.45^2}} dx \thickapprox 0.6$