ACTUARIES MATH STAT VEE
An agency has ten left-handers and twenty right-handers. Five individuals are chosen
for a survey using simple random sampling.
Calculate the probability all five have the same handedness.
ANSWER 0.11
There are 97 men and 3 women in an organization. A simple random sample of 5
people is taken to form a committee.
What is the probability that the committee includes all 3 women?
ANSWER 6.2×10−5
A marine scientist has six ocean water samples that are evenly split between the Pacific
and Atlantic oceans. Four of the ocean water samples are chosen using simple random
sampling.
Find the probability that all the Pacific ocean samples are chosen.
ANSWER 1/5
A random sample X1�1, X2�2, ..., X10�10 is drawn from a normal distribution with
variance 12. Let X¯�¯ be the sample mean.
Let χ2p,ν��,�2 be the 100p�th percentile of a chi-squared random variable with ν�
degrees of freedom. The following table lists values of χ2p,ν��,�2 for specific
combinations of p� and ν�:
Find the probability that ∑10i=1(Xi−X¯)2∑�=110(��−�¯)2 is greater than 120.
ANSWER At least 0.3, but less than 0.4
A manufacturer makes golf balls with a mean weight of 1.62 ounces and a standard
deviation of 0.05 ounces. A random sample of 100 balls are taken.
Using the Central Limit Theorem, approximate the probability that the sample mean
weight of the 100 balls is between 1.615 and 1.625 ounces.
ANSWER 0.68
IPO returns from the real estate sector are normally distributed with mean μ�. An
analyst takes a random sample of 12 IPOs from the real estate sector.
Let X¯�¯ and S2�2 be the sample mean and sample variance of the 12 IPO returns,
respectively.
Let tp,ν��,� be the 100p�th percentile of a t� random variable with ν� degrees of
freedom. The following table lists values of tp,�,� for specific combinations of p�
and ν�:
What is the probability that X¯−μS�¯−�� is less than 0.51?
ANSWER 0.95
You are given the following information from a random sample:
, Xi�� is normally distributed with mean 1 and variance 2.
Sample size is 20.
S2�2 is the unbiased sample variance.
Let χ2p,ν��,�2 be the 100p�th percentile of a chi-squared random variable with ν�
degrees of freedom. The following table lists values of χ2p,ν��,�2 for specific
combinations of p� and ν�:
Calculate c�, such that Pr(S2≤c)=0.95Pr(�2≤�)=0.95.
ANSWER At least 3.1
A random sample X1�1, X2�2, ..., Xn�� is drawn from a normal distribution with
mean μ�.
You are given:
X¯�¯ is the sample mean.
S� is the unbiased sample standard deviation.
V=X¯−μS/n−−√�=�¯−��/�
The 20th percentile of V� is −0.8583−0.8583.
Calculate Pr(|V|<0.8583)Pr(|�|<0.8583).
ANSWER 0.6
A class has 8 boys and 7 girls. Using simple random sampling, the teacher selects 3 of
the children for a special project.
Calculate the probability that two girls and one boy are selected.
ANSWER 0.369
Out of 50 basketball players trying to make a team, 5 will be chosen. 36 of the 50 are
seniors and the remaining players are juniors. The manager wants 2 seniors and 3
juniors to make the team.
What is the probability of meeting the manager's objectives if players were selected by
simple random sampling?
ANSWER 0.108
You are given the following information about a random sample of claim amounts:
Claim severity follows a Pareto distribution with α=3�=3 and θ=50�=50, with mean
and variance given by
mean=θα−1,variance=2θ2(α−2)(α−1)−(θα−1)2mean=��−1,variance=2�2(�−2)(�−1)
−(��−1)2
The sample size is 100.
Using the Central Limit Theorem, calculate the probability that the sample mean claim
amount will lie between 30 and 35.
ANSWER At least 8%
An agency has ten left-handers and twenty right-handers. Five individuals are chosen
for a survey using simple random sampling.
Calculate the probability all five have the same handedness.
ANSWER 0.11
There are 97 men and 3 women in an organization. A simple random sample of 5
people is taken to form a committee.
What is the probability that the committee includes all 3 women?
ANSWER 6.2×10−5
A marine scientist has six ocean water samples that are evenly split between the Pacific
and Atlantic oceans. Four of the ocean water samples are chosen using simple random
sampling.
Find the probability that all the Pacific ocean samples are chosen.
ANSWER 1/5
A random sample X1�1, X2�2, ..., X10�10 is drawn from a normal distribution with
variance 12. Let X¯�¯ be the sample mean.
Let χ2p,ν��,�2 be the 100p�th percentile of a chi-squared random variable with ν�
degrees of freedom. The following table lists values of χ2p,ν��,�2 for specific
combinations of p� and ν�:
Find the probability that ∑10i=1(Xi−X¯)2∑�=110(��−�¯)2 is greater than 120.
ANSWER At least 0.3, but less than 0.4
A manufacturer makes golf balls with a mean weight of 1.62 ounces and a standard
deviation of 0.05 ounces. A random sample of 100 balls are taken.
Using the Central Limit Theorem, approximate the probability that the sample mean
weight of the 100 balls is between 1.615 and 1.625 ounces.
ANSWER 0.68
IPO returns from the real estate sector are normally distributed with mean μ�. An
analyst takes a random sample of 12 IPOs from the real estate sector.
Let X¯�¯ and S2�2 be the sample mean and sample variance of the 12 IPO returns,
respectively.
Let tp,ν��,� be the 100p�th percentile of a t� random variable with ν� degrees of
freedom. The following table lists values of tp,�,� for specific combinations of p�
and ν�:
What is the probability that X¯−μS�¯−�� is less than 0.51?
ANSWER 0.95
You are given the following information from a random sample:
, Xi�� is normally distributed with mean 1 and variance 2.
Sample size is 20.
S2�2 is the unbiased sample variance.
Let χ2p,ν��,�2 be the 100p�th percentile of a chi-squared random variable with ν�
degrees of freedom. The following table lists values of χ2p,ν��,�2 for specific
combinations of p� and ν�:
Calculate c�, such that Pr(S2≤c)=0.95Pr(�2≤�)=0.95.
ANSWER At least 3.1
A random sample X1�1, X2�2, ..., Xn�� is drawn from a normal distribution with
mean μ�.
You are given:
X¯�¯ is the sample mean.
S� is the unbiased sample standard deviation.
V=X¯−μS/n−−√�=�¯−��/�
The 20th percentile of V� is −0.8583−0.8583.
Calculate Pr(|V|<0.8583)Pr(|�|<0.8583).
ANSWER 0.6
A class has 8 boys and 7 girls. Using simple random sampling, the teacher selects 3 of
the children for a special project.
Calculate the probability that two girls and one boy are selected.
ANSWER 0.369
Out of 50 basketball players trying to make a team, 5 will be chosen. 36 of the 50 are
seniors and the remaining players are juniors. The manager wants 2 seniors and 3
juniors to make the team.
What is the probability of meeting the manager's objectives if players were selected by
simple random sampling?
ANSWER 0.108
You are given the following information about a random sample of claim amounts:
Claim severity follows a Pareto distribution with α=3�=3 and θ=50�=50, with mean
and variance given by
mean=θα−1,variance=2θ2(α−2)(α−1)−(θα−1)2mean=��−1,variance=2�2(�−2)(�−1)
−(��−1)2
The sample size is 100.
Using the Central Limit Theorem, calculate the probability that the sample mean claim
amount will lie between 30 and 35.
ANSWER At least 8%
