Actuaries Math Stat VEE with
Complete Solutions
A claim department has operated under the following assumptions about expected
automobile claims:
50% of the claims are for cars
20% of the claims are for motorcycles
15% of the claims are for vans
15% of the claims are for trucks
In conducting a chi-squared goodness-of-fit test on the claim department's assumptions,
100 independent claims were recorded as follows:
Calculate the chi-squared test statistic. - ANSWER-At least 2, but less than 5
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
A company wants to determine whether sick days taken by its employees are evenly
distributed throughout the five-day working week. A random sample of the sick days
taken by a sample of 100 employees yielded the following data:
Determine the p�-value for this chi-squared goodness-of-fit test - ANSWER-At least
0.050
A contingency table has been created to test the hypothesis that the number of vehicles
per policyholder is independent of the policyholder's rating classification. There are four
possible rating classifications in the table crossed with the number of vehicles per
policyholder. The number of vehicles per policyholder is limited to a value of 1, 2, 3, or
4.
Calculate the degrees of freedom to be used in this chi-squared test of independence. -
ANSWER-9
A insurance company sends a survey to a group of claimants. Out of 52 responses, 23
indicated that their claim was handled to their satisfaction.
Assuming the responses were independent, construct a 92% confidence interval for the
proportion of claimants who are satisfied with the handling of their claim. - ANSWER-
(0.32, 0.56)
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
, 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 of claims has been drawn from a Burr distribution with CDF
You are given:
75% of the claim amounts in the sample exceed 100.
25% of the claim amounts in the sample exceed 500.
Estimate θ� by percentile matching. - ANSWER-At least 220
A random sample of size 100 from an exponential distribution with PDF
is used to test the hypotheses:
H0:θ=1
H1:θ=2�1:�=2e told that the most powerful test for a certain significance level has a
rejection region derived from
which translates to a critical value of 1.1645 in the unit of the sample mean, X¯ �¯.
Determine the value of r�. - ANSWER-
A random sample of size 21 from a normal distribution with mean μ� yields
observations that are summarized as follows:
You are testing the following hypotheses:
H0:μ=3
H1:μ≠3 - ANSWER-At least 0.008
A random sample of size 9 is drawn from a normal distribution with mean μ�.
The calculated sample standard deviation is 11.1.
H0:μ=7
H1:μ>7
The probability of a type I error is 10%.
Reject H0�0 if X¯≥c, where X¯ is the sample mean.
Calculate c�. - ANSWER-At least 12.15, but less than 12.30
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
Complete Solutions
A claim department has operated under the following assumptions about expected
automobile claims:
50% of the claims are for cars
20% of the claims are for motorcycles
15% of the claims are for vans
15% of the claims are for trucks
In conducting a chi-squared goodness-of-fit test on the claim department's assumptions,
100 independent claims were recorded as follows:
Calculate the chi-squared test statistic. - ANSWER-At least 2, but less than 5
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
A company wants to determine whether sick days taken by its employees are evenly
distributed throughout the five-day working week. A random sample of the sick days
taken by a sample of 100 employees yielded the following data:
Determine the p�-value for this chi-squared goodness-of-fit test - ANSWER-At least
0.050
A contingency table has been created to test the hypothesis that the number of vehicles
per policyholder is independent of the policyholder's rating classification. There are four
possible rating classifications in the table crossed with the number of vehicles per
policyholder. The number of vehicles per policyholder is limited to a value of 1, 2, 3, or
4.
Calculate the degrees of freedom to be used in this chi-squared test of independence. -
ANSWER-9
A insurance company sends a survey to a group of claimants. Out of 52 responses, 23
indicated that their claim was handled to their satisfaction.
Assuming the responses were independent, construct a 92% confidence interval for the
proportion of claimants who are satisfied with the handling of their claim. - ANSWER-
(0.32, 0.56)
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
, 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 of claims has been drawn from a Burr distribution with CDF
You are given:
75% of the claim amounts in the sample exceed 100.
25% of the claim amounts in the sample exceed 500.
Estimate θ� by percentile matching. - ANSWER-At least 220
A random sample of size 100 from an exponential distribution with PDF
is used to test the hypotheses:
H0:θ=1
H1:θ=2�1:�=2e told that the most powerful test for a certain significance level has a
rejection region derived from
which translates to a critical value of 1.1645 in the unit of the sample mean, X¯ �¯.
Determine the value of r�. - ANSWER-
A random sample of size 21 from a normal distribution with mean μ� yields
observations that are summarized as follows:
You are testing the following hypotheses:
H0:μ=3
H1:μ≠3 - ANSWER-At least 0.008
A random sample of size 9 is drawn from a normal distribution with mean μ�.
The calculated sample standard deviation is 11.1.
H0:μ=7
H1:μ>7
The probability of a type I error is 10%.
Reject H0�0 if X¯≥c, where X¯ is the sample mean.
Calculate c�. - ANSWER-At least 12.15, but less than 12.30
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
