Coaching Actuaries Math Stat VEE
The net dollar amount in bank transactions for one day follows a normal distribution with mean μ�. An analyst took a random sample of five days and observed the following net amounts: 265, −147, −79, 341, 104
(-165.3, 358.9)
Two teams, Delta and Gamma, handle customer service calls for a corporation. The amount of time a team member spends answering calls on a typical day is normally distributed. Assume both teams are independent and have the same variance for call times. A random sample of four members was taken from each team. Their observed call times are as follows: Delta: 1, 2, 2, 3 Gamma:2, 3, 4, 5 Construct an 80% confidence interval for team Delta's mean call time minus team Gamma's mean call time.
(-2.60, -0.40)
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.
(0.32, 0.56)
An actuary considers a portfolio of 40 policyholders which was taken as a random sample from a block of business. She discovers that 11 of these policyholders were each compensated over $10,000 last year. Construct a 94% confidence interval for the proportion of policyholders who were compensated $10,000 or less last year from this block of business.
(0.59, 0.86)
The following observations were made from a random sample taken from a normal distribution with mean μ�: 2,0,4,4,6,3,1,5,6,9 Construct a 90% confidence interval for μ�.
(2.45, 5.55)
The lifetime of a certain brand of lightbulbs are assumed to be normally distributed. A researcher took a random sample of these lightbulbs and observed the following failure times: 17,19,24,24,25,34,40 Construct the 95% confidence interval for the variance of the lightbulb lifetime.
(27.6, 322.4)
The following summarizes the observations from a random sample of size 10 from a normal distribution with variance σ2�2: ∑i=10(xi−x¯)2=72 Construct the 90% confidence interval for σ2
(4.256, 21.654)
The following summarizes the observations from a random sample of size 10 from a normal distribution with variance σ2�2: ∑i=110(xi−x¯)2=72 Construct the 90% confidence interval for σ2�2.
(4.256, 21.654)
An actuary considers fitting a sample of 200 claims to one of the following models: Gamma distribution with two free parameters, producing a maximized log-likelihood function of -100. Generalized Pareto distribution with three free parameters, producing a maximized log-likelihood function of m�. Using the Bayesian information criterion, calculate m� such that the actuary is indifferent between the two models.
-97.4
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?
0.108
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.
0.11
Claims filed under a group of auto insurance policies have normally distributed amounts with mean 19,400 and standard deviation 5,000. A random sample of 25 claims are taken. What is the probability that the average of the 25 claim amounts exceeds 20,000?
0.27
Among 12 mice housed in a laboratory, three are male. Five mice are chosen using simple random sampling. Find the probability that exactly two male mice are chosen.
0.32
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.
0.369
Claim amounts at an insurance company are independent of one another, modeled by a normal distribution with mean 109 and variance 676. Calculate the probability that a random sample of 25 claim amounts average between 100 and 110.
0.53
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).
0.6
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.
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. What is the probability that X¯−μS�¯−�� is less than 0.51?
0.95
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,ν��,� for specific combinations of p� and ν�: What is the probability that X¯−μS�¯−�� is less than 0.51?
0.95
X1�1, X2�2, X3�3, and X4�4 are independent and normally distributed with common mean 50 and common variance 400. Let Φ(z)Φ(�) be the CDF of the standard normal random variable evaluated at z�. Determine the probability that the sample mean of the four variables is at least 60.
1 - phi(1)
An auto insurer has a random sample of 800 claims. The claim amounts have a mean of 2,000 and a variance of 136,875. Using the Central Limit Theorem, approximate the 99th percentile of the total claim amount from the sample.
1,624,000
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.
1/5
For a random sample of size n� for losses following a Weibull distribution, the log-likelihood function is You use the likelihood ratio test to test the hypothesis that τ=2.75�=2.75 and θ=100 Calculate the test statistic for the test.
12.2
Claim sizes independently follow an exponential distribution with PDF Four claims were observed, having sizes 2, 7, 3, and 4. Determine the maximum likelihood estimate of θ�.
2
You are given the following five independent observations: 521, 658, 702, 819, 1217 You use a single-parameter Pareto distribution to fit the data, having PDF Calculate the maximum likelihood estimate of α�.
2.5
During the noon hour, a bank serves 4 personal line customers, 5 business line customers, and 1 new customer. By simple random sampling, 6 of the customers are chosen to take a survey. What is the probability that exactly 2 personal line customers, 3 business line customers, and 1 new customer take the survey?
2/7
Observations x1�1, x2�2, ..., x10�10 are assumed to come from a distribution with parameters θ� and σ�, having mean 0.5(θ+σ)0.5(�+�) and second moment θ2+σ2�2+�2. ∑10i=1xi=150∑�=110��=150 ∑10i=1x2i=5,000 Estimate θ� using the method of matching moments.
20
The number of claims in a week follows a negative binomial distribution with mean rβ�� and second moment rβ(1+β)+(rβ)2��(1+�)+(��)2. You are given the following information regarding 50 weekly claim counts: Estimate r� using the method of moments.
3.5
For a group of policies, you are given: Losses follow a distribution with CDF F(x)=1−θx,x>θ�(�)=1−��,�>� 20 independent losses resulted in the following: Calculate the maximum likelihood estimate of θ�.
5.5
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?
6.2×10−5
Let X1�1, X2�2, ..., Xn�� denote a random sample from a normal distribution with variance σ2�2. You are given: X¯�¯ is the sample mean. W=∑ni=1(Xi−X¯)2σ2�=∑�=1�(��−�¯)2�2 The first percentile of W� is 1.24. 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�: Determine n�.
8
Losses follow a Pareto distribution with CDF The sample of losses is 100, 200, 500, 1000. Estimate θ� by percentile matching, using the 25th percentile.
808
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.
9
Two independent random samples drawn from normal distributions have equal variances and means μ1�1 and μ2�2. You are given The sample mean from the first sample is 16.9. The sample mean from the second sample is 22.1. The number of observations from both samples combined is 17. The upper bound of the 90% confidence interval for μ1−μ2�1−�2 is 1.44 Determine the lower bound of the 95% confidence interval for μ1−μ2�1−�2.
At least -14, but less than -12
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
At least 0.008
You are given that the distribution of the inter-event times between patients entering a waiting room is exponential with PDF You are testing the following hypotheses: H0:θ=10 H1:θ is estimated by maximum likelihood You are given the following five independent observations of inter-event times: 8,9,9,10,11 Calculate the test statistic of this test.
At least 0.018, but less than 0.019
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
At least 0.050
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.
At least 0.3, but less than 0.4
Let Y1�1, Y2�2, ..., Y10�10 be a random sample with mean μ� and variance σ2�2. Consider the following two unbiased estimators for μ�: Calculate the efficiency of μ^1�^1 relative to μ^2�^2.
At least 0.35, but less than 0.55
You are given the following data for the number of claims during a one-year period from independent policies: A Poisson distribution is fitted to the data using maximum likelihood, having PMF Find the MLE of λ�.
At least 0.4, but less than 0.6
Three claims from a dental insurance plan is given below: 225, 525, 950 Claims are assumed to independently follow a Pareto distribution with PDF
At least 0.6, but less than 0.7
30,500 residents live in Cape City. A random sample consisting of 3% of the residents was taken. The individuals in the sample were asked about their employment status. 758 reported that they were employed. Find the lower bound of the 98% confidence interval for the proportion of employed residents in Cape City.
At least 0.79, but less than 0.80
Two independent random samples drawn from normal distributions have equal variances and means μ1�1 and μ2�2. You are given The first sample of size 3 has a sample mean of 12 and a sample variance of 25.6. The second sample of size 4 has a sample mean of 10 and a sample variance of 19.9.
At least 0.86, but less than 0.92
For a random sample of size 13 from a normal distribution with mean μ�, you are given the following regarding the observations: ∑13i=1(xi−x¯)2=77.8∑�=113(��−�¯)2=77.8 The width of the 100k%100�% confidence interval for μ� is 2.7005. Find the value of k�.
At least 0.91, but less than 0.93
Net profits from movies filmed by Merdeka Studios follow a normal distribution with mean 2.7. A random sample of six movies are examined. 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,ν��,� for specific combinations of p� and ν�: Find the probability that the sample mean net profit is greater than 2.7 minus three-fourths of the sample standard deviation.
At least 0.93, but less than 0.94
The following claim data were generated from a Pareto distribution: 130, 20, 350, 218, 1822 For integer k�, the k�th moment of a Pareto distribution with parameters α� and θ� is Using the method of moments, determine the estimate of θ�.
At least 1,900, but less than 2,100
Two different estimators, γ^1�^1 and γ^2�^2, are available for estimating the parameter, γ�, of a given loss distribution. For a random sample of size 85 with mean 0.702γ0.702� and standard deviation 3.6121, γ^1�^1 is the sample mean. For a different random sample of size 85 with mean 0.0071γ0.0071� and standard deviation 0.0311, γ^2�^2 is the sum. If γ=3�=3, calculate the efficiency of γ^1�^1 relative to γ^2�^2.
At least 1.50, but less than 1.75
You test the following hypotheses by observing one outcome from a normal distribution with mean μ� and standard deviation 1,250: H0:μ=9,000 H1:μ=12,500 Reject H0�0 if the outcome is greater than 11,000. Calculate the probability of a type II error.
At least 10.0%, but less than 12.5%
Losses follow a log-logistic distribution with CDF The sample of losses is: 10,35,80,86,90,120,158,180,200,210,1500 Calculate the estimate of θ� by percentile matching, using the 40th and 80th percentiles.
At least 107
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�.
At least 12.15, but less than 12.30
A sample of ten observed losses has the following summary statistics: ∑i=110x−0.5i=0.34445∑i=110x−1i=0.023999∑i=110x0.5i=488.97∑i=110xi=31,939∑�=110��−0.5=0.34445∑�=110��0.5=488.97∑�=110��−1=0.023999∑�=110��=31,939 You assume that the losses independently come from a Weibull distribution with PDF Determine the maximum likelihood estimate of θ�.
At least 1500, but less than 2500
You are given the following information: A random sample of size 10 is drawn from a normal distribution with variance σ H0:σ2=100 H1:σ2>100 Calculate the smallest critical value in the unit of the sample variance, S2�2, such that the significance level is no more than 5%.
At least 185, but less than 200
Losses follow an exponential distribution with CDF 20 independent losses were recorded as follows: Calculate the maximum likelihood estimate of θ�.
At least 1950, but less than 2100
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.
At least 2, but less than 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.
At least 220
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.
At least 3.1
You observed the following independent losses: 20,30,60,80,120,150,190,23020,30,60,80,120,150,190,230 You fit an exponential distribution to these losses, having PDF and CDF Suppose: θ^1�^1 is the estimate of θ� using percentile matching at the 40th percentile. θ^2�^2 is the estimate of θ� using the maximum likelihood estimation. Determine θ^1−θ^2�^1−�^2.
At least 30, but less than 35
Loss amounts follow a Pareto distribution. You observe the following 10 loss amounts: 1,3,5,6,8,9,10,70,95,120 For integer k�, the k�th moment of a Pareto distribution with parameters α� and θ� is Estimate α� using the method of moments.
At least 4.8, but less than 5.0
You are given two random samples of paid claim amounts for two Hospitals, A and B, which independently follow normal distributions with equal variances and means μA�� and μB��. The results for the paid claim amounts are Hospital A:Hospital B:6.217.897.3410.125.679.717.885.553.894.3312.48Hospital A:6.217.345.677.883.89Hospital B:7.8910.129.715.554.3312.48 The unbiased standard deviations calculated from the observations above are sA=1.56��=1.56 and sB=3.04��=3.04. Calculate the upper bound of the 95% confidence interval for the difference μB−μA��−��.
At least 5.5
Four losses are observed from a lognormal distribution: 200,300,350,450 The k�th moment of a lognormal distribution with parameters μ� and σ2�2 is ekμ+0.5k2σ2���+0.5�2�2. Using the method of matching moments, estimate μ�.
At least 5.6, but less than 5.8
You are given the following hypothesis test: A random sample of size 10 is drawn from a normal distribution with mean μ�. The calculated sample variance is 9. H0:μ≤50 H1:μ>50 The chosen significance level is 5%. Find the critical value for this test in the unit of the sample mean, X¯�¯.
At least 51.50, but less than 51.75
For a random variable X�, you are given: E[X]=α2�[�]=�2 Var[X]=α28���[�]=�28 α^=(k−1)X,k>3�^=(�−1)�,�>3 MSE(α^)=2[Bias(α^)]2���(�^)=2[����(�^)]2 Calculate the value of k�.
At least 7.0, but less than 8.5
Based on a random sample of size 20 from a normal distribution with variance σ2�2, the width of the 95% confidence interval for σ2�2 is 150. Determine the unbiased sample variance calculated from the sample.
At least 75, but less than 100
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.
At least 8%
Based on a random sample of size 7 from a normal distribution with mean μ�, a confidence interval is constructed for μ�. The sample standard deviation is calculated as 5.4763. If we want to be at least 90% confident, determine the smallest possible width of the interval.
At least 8, but less than 10
You are given the following information about a hypothesis test that is being performed on a random sample taken from a normal distribution with mean μ�: H0:μ=80 H1:μ>80 Calculate the highest possible sample mean that would result in the null hypothesis not being rejected at the 1% significance level.
At least 81.50, but less than 82.00
For a random sample from a normal distribution with mean μ�, you record the following observations: 7.6,4.9,8.8,7.6,6.1,14.5,10.3,5.1 The lower bound of a confidence interval for μ� is 6.32. Find the upper bound of this confidence interval.
At least 9.5, but less than 10.5
The mean time between claims for high-risk policyholders is supposedly 2.6. A researcher believes this mean time is incorrect and conducts a hypothesis test. He observed one high-risk policyholder, which resulted in a time of 4.5. Assume these times are normally distributed with variance 2. What is the result of the test?
Do not reject H0�0 at the 0.100 significance level
Which of the following statements is true?
For an unbiased estimator, the mean squared error is always equal to the variance.
For a random sample of size 8 drawn from a Bernoulli distribution with parameter p�, the sum of the observations resulted in 5. Normal percentiles are used to construct confidence intervals for p�. Which of the following statements are true regarding the setup above? A 90% confidence interval for p� is (0.3434, 0.9066). 0.1712 is the calculated standard error used to find a confidence interval for p�, regardless of the confidence level. It is appropriate to construct the intervals using normal percentiles, as all the assumptions are met.
I and II only
For a random sample of size n� drawn from a Pareto distribution with parameter θ=5,000�=5,000, an estimator for θ� is θ^=5,000⋅nn+1�^=5,000⋅��+1 Which of the following statements are true about the estimator θ^�^? θ^�^ is an unbiased estimator of θ�. θ^�^ is a consistent estimator of θ�. When n=10�=10, the mean squared error of θ^�^ is more than 200,000
II and III only
You are given: A random sample of size n� is obtained from an exponential distribution with mean θ� and variance θ2�2. The sample mean, X¯�¯, is used as an estimator for θ�. Determine the mean squared error of the estimator.
Less than -0.5
The number of goals scored in a soccer game follows a negative binomial distribution with mean rβ�� and second moment rβ(1+β)+(rβ)2��(1+�)+(��)2. 20 games produced the following number of goals scored: Use the sample data and the method of moments to estimate the parameter β�.
Less than 0.25
Actuaries speculate that when small vehicles are involved in accidents, the chances of serious injury are higher than that for larger sized vehicles. A random sample of 1,000 accidents is classified according to severity of the injuries and the size of the car. The results are: The following null and alternate hypotheses are created: H0:�0: There is no dependence between size of car and severity of injury. H1:�1: There is dependence between size of car and severity of injury. Calculate the chi-squared test statistic to evaluate the null hypothesis.
Less than 6.6
You are given the following: Claim sizes follow a normal distribution with mean μ� and variance σ2=47,300�2=47,300. H0:μ=10,000 H1:μ<10,000 One claim of 9,600 is observed. Determine the test result.
Reject H0�0 at the 0.05 significance level, but not at the 0.02 level.
You are given the following information on a random sample from a normal loss distribution with mean μ�: The sample mean is 42,000. The sample standard deviation is 8,000. There are 25 loss observations in the sample. To test the hypotheses H0:μ=45,000 versus H1:μ≠45,000, at which value of α� would you reject the null hypothesis?
Reject H0�0 at α=0.10, but not at α=0.05
You are given: A random sample of size n� is obtained from an exponential distribution with mean θ� and variance θ2�2. The sample mean, X¯�¯, is used as an estimator for θ�. Determine the mean squared error of the estimator.
theta squared / n
Paid claim amounts follow a normal distribution with PDF One claim amount, x�, is used to test a set of hypotheses regarding μ�. This test is most powerful when the rejection region is derived from Determine the rejection region of the most powerful test.
x≤7.5+9ln[r]
A single observation, x�, from a normal distribution with mean μ�, variance 4, and PDF H0:μ=1 H1:μ=2 Determine the rejection region of the most powerful α=0.10�=0.10 test.
x≥3.6
