Question

#1 Suppose the true proportion of voters in the county who support a restaurant tax is 0.52. Consider the sampling distribution for the proportion of supporters with sample size n = 157. What is the mean of this distribution? What is the standard error of this distribution?

#2

A city has just added 80 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions:
(i) Each new recruit has a 0.37 probability of remaining with the police force until retirement.
(ii) Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.33.
(iii) The number of pensions that the city will provide on behalf of each new hire is independent of the number of pensions it will provide on behalf of any other new hire.
Determine the probability that the city will provide at most 57 pensions to the 80 new hires and their husbands. Enter your answer as a number accurate to 4 decimal places

Answer 1

1)

Solution

Given that,

p = 0.52

1 - p = 1 - 0.52 = 0.48

n = 157

The mean of the sampling distribution of proportion is ,

= p = 0.52

The standard error of the sampling distribution is ,  

=  p ( 1 - p ) / n

=   (0.52 * 0.48) / 157 = 0.0399

=