Question

1.

A city has just added 105 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 0.5 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 0.21.
(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 103 pensions to the 105 new hires and their husbands. Enter your answer as a number accurate to 4 decimal places.

2. A population has parameters μ=238.9  and σ=74.7. You intend to draw a random sample of size n=153n=153.

What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)

0x =

3.

CNNBC recently reported that the mean annual cost of auto insurance is 1035 dollars. Assume the standard deviation is 226 dollars. You take a simple random sample of 85 auto insurance policies.

Find the probability that a single randomly selected value is more than 977 dollars.
P(X > 977) =

Find the probability that a sample of size n=85n=85 is randomly selected with a mean that is more than 977 dollars.
P(M > 977) =

4.

Based on a sample of 35 people, the sample mean GPA was 3.05 with a standard deviation of 0.03

The test statistic is :

The critiical vaule is :

You wish to test the following claim (HaHa) at a significance level of α=0.02α=0.02.

      Ho:μ=76.5
      Ha:μ>76.5

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=6n with a mean of M=81.2M and a standard deviation of SD=7.7

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

Thank You

Answer 1

2:

The standard deviation of the distribution of sample means is

3:

Given that

The z-score for X = 977 is

The  probability that a single randomly selected value is more than 977 dollars is

P(X > 977) = P(z > -0.26) = 1 - P(z <= -0.26) = 1 - 0.3974 = 0.6026

The z-score for M = 977 is

The  probability that a sample of size n=85 is randomly selected with a mean that is more than 977 dollars is

P(M > 977) = P(z > -2.37) = 1 - P(z <= -2.37) = 1 - 0.0089= 0.9911