Previous Page Table of Contents Next Page


8. Exercises

8.1 EXERCISE 1

Group I - Small samples

During the year 2000, a sample of Portuguese landings of hake (Merluccius merluccius) were recorded. Catches, in tonnes were the following:

Catches (in tonnes) of a sample of Portuguese landings of hake
48372174165130473148
39474155288176277349
301508339114211477170
304255409267274166211
299353192    
  1. Calculate:
    1. The sample size.
    2. The range of the sample values.
    3. The median.
    4. The mean.
    5. The total value.
    6. The sample variance.
    7. The sample standard deviation.
    8. The sample coefficient of variation.

Group II - Large samples

A sample of 195 individuals was extracted from the catch of a trawler. The individual total lengths were measured to the cm below. Registered values were the following:

Total length (in cm) of a catch
241516171819203026232224272221
242021171819203026282224272221
192021171819203026282224232221
192021291819203027281924232221
192021183019203027281924232621
252021252019192627281924232621
252021252021192627282924232621
252021252021192227282924232618
252023182021192231282924232218
192323182021192231282924232218
192323182021192226202926212722
192223182021191922202020212722
262623272327231921252525252522
  1. From this data calculate the mean and the variance.
  2. Choose an adequate class interval and build up a table with the length frequencies distribution.
  3. From the table built in 2., calculate:
    1. The sample mean and sample variance. Compare these results with the ones obtained in 1.
    2. Three statistics of location.
    3. Three statistics of dispersion.
    4. Number of individuals with a length less than 20 cm.
    5. Percentage of individuals with a length equal to or greater than 20 cm.
    6. Percentage of observations between 23 and 25 cm.
    7. The value that corresponds to a length equal to or greater than 45% of all the observations.
    8. The value that corresponds to a length smaller than 21% of all the observations.
    9. The quantile of order 96%.

8.2 EXERCISE 2

Group I - Relative frequencies

In certain ports, the fishing gears used by the vessels were classified into “purse seines”, “trawls”, “handlines”, “longlines” and “trammel nets”. We also know the types of the fishing vessels, that is, “small boats without engine”, “small boats with engine”, “purse seiners” and “stern trawlers”. The information about the numbers of vessels according to boat type and gear used is summarized in Table 8.1.

Table 8.1

Numbers of vessels according to the type of boat and gear used

Type of vessel  Fishing gear  Total
Purse seinesTrawlsHandlinesLonglinesTrammel nets 
Small boats without engine1502220865
Small boats with engine15423401799
Purse seiners50000050
Stern Trawlers05100051
Total8055456025265
  1. Calculate:
    1. Relative frequencies of number of boats by fishing gear.
    2. Percentage of vessels that operate handlines.
    3. Percentage of vessels that operate trammel nets.
  2. Calculate:
    1. Relative frequency of boats not operating trammel nets.
    2. Percentage of boats operating purse seines or longlines.
    3. Relative frequency of boats that do not operate with handlines, nor trammel nets, nor longlines.
    4. Proportion of the total fleet that are small boats without engine.
    5. Proportion of the total fleet that are small boats with engine.
    6. Proportion of the total fleet that are small boats.
    7. Check that the proportion of 2.f) is the sum of 2.d) plus 2.e).
  3. Calculate:
    1. Proportion of small boats without engine that operate handlines.
    2. Proportion of the total fleet that are small boats without engine.
    3. Proportion of the total fleet that are small boats without engine operating handlines.
    4. Check that the proportion of 3.c) is the product of 3.a) times 3.b).
  4. Calculate:
    1. The percentage of small boats without engine operating handlines or longlines.
    2. The percentage of purse seiners operating purse seines.
    3. The relative frequency of vessels that are not purse-seiners.
    4. The percentage of small boats with engine that operate trawls.
    5. The percentage of the fleet that fishes with traps.

Group II - Properties of probabilities

Consider Table 8.1 presented in Group I.

We want to choose one boat, randomly, out of the 265 boats. Every boat has the same probability of being selected.

  1. What is the probability that the boat operates:
    1. Handlines.
    2. Trammel nets.
    3. Does not operate trammel nets.
    4. Operates purse-seines or longlines.
    5. Does not operate handlines, nor longlines neither trammel nets.
  2. What is the probability that the boat will be:
    1. A small boat without engine.
    2. A small boat with engine.
    3. A small boat.
    4. Show that the probability 2.c) is equal to the sum of probability 2.a) plus the probability 2.b).
  3. Calculate the probability of the boat being:
    1. A purse seiner.
    2. A stern trawler.
    3. Neither a purse seiner nor a stern trawler.
    4. Show that probability 3.c) is equal to probability 2.c).
  4. Calculate:
    1. If we choose a boat from the small boats without engine, what is the probability that she operates with handline?
    2. If we choose a boat out of the total fleet what is the probability that she is a small boat without engine?
    3. If we choose a boat out of the total fleet what is the probability that she is a small boat without engine operating with handline?
    4. Check that the probability of 6.c) is equal to the product of the probability of 4.a) times the probability of the 4.b)

Group III - Normal distribution

A random variable X is normally distributed with a mean μ = 20.6 and a standard deviation σ= 2.

  1. Calculate:
    1. The probability of X being less than or equal to 18.
    2. The probability of X being greater than 18.
    3. The probability of X being less than 25.
    4. The probability of X being between 18 and 25.
  2. Calculate x such that:
    1. Prob {X ≤x} = 0.8413.
    2. Prob {X ≥x} = 0.9772.
    3. Prob {X < x} = 0.9986.
    4. x is the 95% quantile of the distribution of X.
    5. x is the median of the distribution of X.

Group IV - The standard normal distribution

The random variable Z has a standard normal distribution.

  1. Calculate:
    1. The probability of the values of Z being between -1 and 1.
    2. The probability of the values of Z being between -2 and 2.
    3. The probability of the values of Z being between -3 and 3.
  2. Calculate:
    1. The z1 value for which the probability that the values of the variable Z will be smaller than z1 is 2.5% (Prob{Z<z1}=0.025).
    2. The z2 value for which the probability of the variable Z being smaller than z2 is 97.5% (Prob{Z<z2}=0.975).
  3. Consider the interval limited by z1 and z2 of the previous exercises 2.a) and 2.b).
    1. Compute the probability that the variable Z will be within the interval (z1, z2).
    2. Repeat exercise 3.a) but with (Prob{Z<z1}=0.004) and (Prob{Z<z2}=0.954).
    3. Repeat exercise 3.a) but with probabilities 0.012 and 0.962.
    4. Note that the Prob{z1≤Z ≤ z2} is equal to 0.950 in the exercises 3.a), 3.b) and 3.c). Verify that the smallest of these intervals is the interval with symmetrical values, z1 and z2.

Group V - t-student distribution

Using the t-student distribution:

  1. Calculate:
    1. Prob {t(10) >1.812}.
    2. Prob {t(19) <1.729}.
    3. Prob {-1.34 < t(15) < +2.602}.
  2. Calculate the value of a that makes the following expressions true:
    1. Prob {t(8) < a} = 0.95.
    2. Prob {t(26) > a} = 0.99.
    3. Prob {-a < t(20) < +a} = 0.95.

The general result of Group IV 3.d) is also true for the t-student distribution.

  1. Calculate the a value such that:
    1. Prob {t < a} = 0.95, with 40, 60, 120 and infinite degrees of freedom.
    2. Compare the values a obtained in 3.a) with the corresponding a values if the probability distribution of 3.a) was replaced with the Z-distribution, i.e., Prob {Z < a}= 0.95.

Group VI - Bernoulli distribution

Consider the discrete variable X which takes the value 1 with probability P= 0.18 and the value 0 with probability Q= 0.82.

  1. Show that the expected value of the random variable X is equal to P.
  2. Show that the variance of the random variable X is equal to PQ.

8.3 EXERCISE 3

An important element in fish stock assessment is the knowledge of the total catch landed for each of the main fisheries and species. The total catch per fishing trip is an element in this assessment, but given the irregular pattern of port calls and the low numbers of port samplers, it is difficult to get many records.

In a given fishing port, the port samplers have been instructed to record the total shrimp catch of at least three landings every week, which they do using a pre-determined and fixed sampling strategy.

Group I - Sample

In the first week of May 2000, they recorded the catch of three shrimp trawlers. The data collected is shown in the following table.

Shrimp landings recorded in three landings
(first week of May 2000)
Landing no.123
Shrimp landing (Kg)5384351352
  1. Calculate, from the sample and the knowledge of total number of landings:
    1. The mean shrimp landing per sampled fishing trip.
    2. The variance of the sampled landings.
    3. The standard deviation of the sampled landings.
  2. By the end of the week, and from the port records, the scientist in charge of the sampling programme learned that a total of 10 landings of shrimp were done during that week. Guess the total amount of shrimp landed in that week.

Group II - Population

A few months later, for the purpose of this exercise, an agreement with the fishing companies gave the scientist access to the data from all landings actually done on that week (table below). These data represent thus the population of shrimp landings done during that week.

Population of all shrimp landings (in kg)

Landing no.12345678910
Shrimp landing538090644259804358591352711
  1. Using these data, calculate the following parameters:
    1. The population mean, that is, the average shrimp landing per fishing trip during that week.
    2. The population variance and the modified variance of the landings.
    3. The standard deviation of the landings.
    4. The total amount of shrimp landed.
    5. The proportion of all landings below 400 Kg.
    6. The relative frequency of landings between 400 and 800 Kg.
  2. Build at least 10 samples of 3 landings each that could have been selected from that population.
  3. Repeat the calculations done on number 1. a) to d), for each of these samples.
  4. Compare the values of the statistics obtained in the previous item with the values of the corresponding population parameters.

Group III - Sampling

Table 8.2 is presented at the end of this exercise with data from the 120 different samples of 3 landings that could have been taken from the 10 landings that actually took place during that week.

Using these data, and adopting the sample mean landing as the estimator, , of the population mean landing.

  1. Plot the histogram of the sampling distribution of the estimator, , using an appropriate class interval.
  2. Calculate from the data of the sampling distribution of the samples presented in Table 8.2:
    1. The 120 values of the estimator.
    2. The expected value of the estimator,E[].
    3. The sampling variance of the same estimator, V[].
    4. The error, σ of the estimator.
  3. Compare the expected value obtained in 2.b) with the population calculated in Group I-1.a).
  4. Using the results obtained in previous exercises, check the theoretical expression:
  5. Calculate:
    1. The percentiles of the sampling distribution of the estimator with the following orders:
      1. 1.0%.
      2. 2.5%.
      3. 3.5%.
      4. 50.0%.
      5. 95.0%.
      6. 96.0%.
      7. 97.5%.
      8. 98.5%.
    2. Four intervals that encompass 95% of all possible sample means.
    3. The width of the four intervals.
  6. What is the shortest of these intervals that holds 95% of all possible sample means.
  7. Considering that the port samplers could have taken any of the possible samples, calculate the probability of getting a sample of 3 landings with an average landing.
    1. Below or equal to 600 Kg.
    2. Above 600 Kg.
    3. Between 199 and 953 Kg.
  8. Calculate:,
    1. The value l such that there is a probability of 95% of getting a sample with an average landing smaller than l.
    2. Two values l1 and l2 that there is a probability of 95% of getting a sample with an average landing between l1 and l2.

Table 8.2
All possible samples of 3 landings that could have been taken from the 10 landings that actually took place in that week

No.Landing 1Landing 2Landing 3
15380906
25380230
35380598
4538020
55380435
65380859
753801123
85380711
9538906230
10538906598
1153890620
12538906435
13538906859
145389061123
15538906711
16538230598
1753823020
18538230435
19538230859
205382301123
21538230711
2253859820
23538598435
24538598859
255385981123
26538598711
2753820435
2853820859
29538201123
3053820711
31538435859
325384351123
33538435711
345388591123
35538859711
365381123711
370906230
380906598
39090620
400906435
410906859
4209061123
430906711
440230598
45023020
460230435
470230859
4802301123
490230711
50059820
510598435
520598859
5305981123
540598711
55020435
56020859
570201123
58020711
590435859
6004351123
610435711
6208591123
630859711
6401123711
65906230598
6690623020
67906230435
68906230859
699062301123
70906230711
7190659820
72906598435
73906598859
749065981123
75906598711
7690620435
7790620859
78906201123
7990620711
80906435859
819064351123
82906435711
839068591123
84906859711
859061123711
8623059820
87230598435
88230598859
892305981123
90230598711
9123020435
9223020859
93230201123
9423020711
95230435859
962304351123
97230435711
982308591123
99230859711
1002301123711
10159820435
10259820859
103598201123
10459820711
105598435859
1065984351123
107598435711
1085988591123
109598859711
1105981123711
11120435859
112204351123
11320435711
114208591123
11520859711
116201123711
1174358591123
118435859711
1194351123711
1208591123711

8.4 EXERCISE 4

Group I - Estimation of the population mean

Consider a Population with = 40,S2 = 25 and N = 2000. A sample of 21 elements was selected from the population, with a simple random sampling design, without replacement.

Table below presents the values selected:

Sample data

304238384142424636423435403539383940374645
  1. Compute the following statistics from the sample:
    1. The mean.
    2. The variance s2.
    3. The standard deviation s.
  2. Adopt as an estimator of the population mean, µ, and estimate:
    1. The population mean.
    2. The estimator is biased?
    3. The sampling variance of .
    4. The error of .
    5. A 95% confidence interval of m.

Group II - Estimation of the population total

Consider a population of 20 purse seiners landing their catches at a certain port during one day.

The vessels are numbered from 1 to 20 according to the arrival time.

  1. Describe a procedure to select a simple random sample of 4 numbered vessels.
  2. Consider the following possible sample:
    Vessels number 3, 15, 10 and 6
    At the arrival of the vessels it was verified that their corresponding landings were:
    Landings (Kg) 17.9, 2.8, 6.5 and 3.5
    1. Choose an estimator of the total amount of fish landed in the port during this day.
    2. Indicate the sampling distribution of that estimator and present the formulae to obtain the expected value and the expected sampling variance.
  3. Based on the sample presented in 2., estimate:
    1. The total landing.
    2. The sampling variance.
    3. The error of the estimate.
    4. A 95% confidence interval for the population total landings.
  4. Estimate the approximate size of the sample necessary if one would like to have an error 10% smaller than the one previously calculated in 3.c).

Group III - Proportions

Consider a population of 100 shrimps in a box.

The aim is to estimate the proportion, P, of females within the box and the total number of females within the box. It was decided to select a simple random sample of size n = 30, and to adopt as estimator of P, the proportion, p, of females in the sample. The number of females in the sample was 12.

  1. Estimator of the proportions.
    1. Calculate the proportion of females in the sample.
    2. Calculate the sample variance.
    3. Write, according to the sampling theory, the expressions for the expected value of p, and for the sampling variance of p.
    4. Estimate the sampling variance of p.
    5. Estimate the error of p.
    6. Estimate the 95% confidence interval for the proportion P applying the binomial distribution.
    7. Estimate the 95% confidence interval for the proportion P applying the normal approximation to the binomial distribution.
  2. Estimator of total number with proportions.
    1. Estimate the total number of females in the population.
    2. Estimate the sampling variance and the error of the total number.
    3. Estimate the 95% confidence interval for the total number of females of the population applying the binomial distribution.
    4. Estimate the 95% confidence interval for the total number of females of the population applying the normal approximation to the binomial distribution.

8.5 EXERCISE 5

Group I - Landing ports

Consider a purse seiner fleet landing sardines in a given fishing port. A stratified sampling design is to be applied in order to estimate the total landings from these vessels. The composition of the fleet is given by Table 8.3.

Table 8.3
Number of vessels by power classes of a purse seiner fleet

Power class
(HP)
Number of vessels
100-10
200-50
>30020
Total80
  1. A random stratified sampling design will be applied considering the 3 HP categories as strata. The total size of the sample will be 16 vessels allocated proportionally to the number of vessels in each stratum. Calculate the number of vessels to be sampled in each stratum.
  2. The landing of sardines of each sampled vessel was registered. Table 8.4 summarises the sample values of total landings and coefficients of variation

Table 8.4
Total landings of sardines and coefficient of variation by vessel power classes

Power class
(HP)
Total landings
(tonnes)
CV
100-40.98
200-600.73
>300200.68
  1. Calculate the average landing per vessel in each category.
  2. Calculate the variance between total landings within each stratum.
  1. Present estimates for each stratum of:
    1. Mean landing.
    2. Expected sampling variance of the estimator of the mean.
    3. Error of the estimator of the mean.
    4. Total landing.
    5. Expected variance of the estimator of total landing.
    6. Error of the estimator of total landing.
  2. Present estimates for the total fleet of:
    1. Mean landing.
    2. Expected variance of the estimator of the mean.
    3. Error of the estimator of the mean.
    4. Total landing.
    5. Expected variance of the estimator of total landing.
    6. Error of the estimator of total landing.

Group II - Surveys

A research vessel has carried out a demersal trawl survey on the continental shelf and on the slope off Libya. The goal of the survey was to estimate the biomass of the European hake (Merluccius merluccius) in the area.

The survey was designed as a stratified random survey. The study area was divided into 10strata, according to two geographical areas and five depth levels. Each haul was done at a speed of 3 knots, with one-hour duration. The trawl net had a horizontal opening of 50 m. It is assumed that the vertical opening was enough to catch all the hakes that occur in the trawling area.

Table 8.5 presents the two areas and their respective depth zones, the area of each stratum in square nautical miles (nm2), the number of hauls carried out, the average catch and the standard deviation within each stratum. The sampling fraction is negligible.

Table 8.5
Characteristics of the survey area

Depth
(m)
Area
(mn2)
Number of trawlsAverage catches
(kg)
Standard deviation of catches
(Kg)
Area 1
100–20020851251.4
200–300755132810.5
300–400660913447.8
400–500540104314.7
500–60088011133.6
Area 2
100–2001252114914.5
200–3005001412227.7
300–40035085514.0
400–500445106415.7
500–60045095716.4
  1. For each stratum estimate:
    1. An index of total biomass of European hake.
    2. The error of the estimator.
    3. The coefficient of variation of the estimator.
  2. For the total area estimate:
    1. Total biomass of European hake.
    2. The error of the estimator.
    3. The 95% confidence limits of the total biomass in the area.
  3. Consider that only 100 trawls can be carried out during the next year's survey. Under these conditions:
    1. Calculate the proportional allocation of the total 100 trawls to the strata areas.
    2. Calculate the strata allocation that gives the maximum precision in the estimation of the total abundance.

8.6 EXERCISE 6

Group I - Selection of the clusters

Along the coast of a region, divided into 5 provinces, 35 landing places were identified. The landing places and their number of vessels are presented in Table 8.6. The sizes of the landing places were considered to be the number of vessels in each place.

With the objective of estimating the total landing of the region, it was decided to select 15 landing places.

Table 8.6
Number of vessels of each province, by landing place

Landing PlaceNumber of Vessels
- Province 1 -
19
230
312
49
59
64
75
810
  
- Province 2 -
930
10150
1141
1218
138
1427
154
  
- Province 3 -
1625
175
1815
- Province 4 -
1928
2060
2116
2224
2336
2420
2552
2613
2735
- Province 5 -
2813
2948
3014
3116
3212
3313
3411
3538
Total860
  1. Select the 15 clusters with equal probabilities.
  2. Select the 15 clusters with probabilities proportional to the cluster sizes.
  3. Considering the 5 provinces as strata, select 3 clusters from each province with probabilities proportional to the sizes of the clusters of each stratum.

Group II - Selection with equal probabilities

Consider a population divided into 23 clusters. Aiming at estimating the total value of the population, it was decided to select 5 clusters using the simple random criteria with replacement. Table 8.7 presents a summary of the obtained data.

Table 8.7
Sample data

ClustersSizes of the clustersTotal values of the clusters
11501244
7501324
2501335
14501300
9501270
Total2506473
  1. Indicate:
    1. The number of clusters in the population.
    2. The number of clusters in the sample.
    3. The number of elements in cluster 14.
  2. Calculate:
    1. The sample mean value per cluster.
    2. The sample mean value per element.
    3. The sample variance between clusters.
  3. Choose an estimator of the total value of the population and estimate:
    1. The expected value of the estimator.
    2. The sampling variance of the estimator.
    3. The error of the estimator.

Group III - Selection with probabilities proportional to sizes, with replacement

Consider a population of fishing vessels divided into 23 clusters with an unequal number of vessels, which are taken as cluster sizes. With the aim of estimating the total landings, Y, of the population, a sample of 5 sites was selected using a random criterion with probabilities proportional to the size of the cluster and with replacement. Table 8.8 summarises the sample data.

Table 8.8
Sample data

Clusters sampledNumber of vesselsMean landings per vessel,
y
13023.78
43224.46
82025.05
132024.15
182723.70
Total fleet822--
  1. Adopting the Hansen-Horowitz estimator of the total landing, estimate:
    1. The total value of the population.
    2. The error of your estimator.
  2. Adopting the Horvitz-Thompson estimator of the total landing, estimate:
    1. The total landing of all the vessels.
    2. The sampling variance of your estimator and its error.
    3. An approximate 95% confidence interval of the total landing.

8.7 EXERCISE 7

Group I -Selection with simple random sampling at both stages

A two-stage sampling has been carried out in order to estimate the total landings from the demersal longline fleet. During the first stage 5 vessels out of 58 have been sampled with a simple random criteria without replacement. During the second stage a sample of 50 fish boxes was drawn (by simple random criteria without replacement) from each selected vessel. The sample information of this two-stage sampling is summarized in Table 8.9.

  1. 1. Estimate:
    1. The total weight of fish landed.
    2. The error of the estimation.
  2. Proportions
    Consider that in the 5 vessels sampled, 10, 15, 7, 5 and 20 boxes of fish were observed among the boxes sampled in vessels 1, 2, 3, 4 and 5, respectively (Table 8.7).
    1. Estimate the proportion of boxes in the total landings.
    2. Estimate the error of the estimation.

Table 8.9
Sample data

VesselTotal number of boxes in the vesselsNumber of boxes sampledTotal weight of the sample
(Kg)
SD of box weight in each vessel
(Kg)
1200509902.02
21005014051.90
32505014402.14
4905013302.21
52305011053.24

Group II - First selection -unequal probabilities with replacement. Second stage -simple random sampling with replacement

A two-stage sampling has been undertaken with the aim of estimating the total weight of shrimp landed.

During the first stage, 5 out of 58 trawlers were randomly sampled with replacement, and unequal probabilities. During the second stage, a sample of 50 boxes was simple randomly drawn from each of the vessels selected in the first stage. The sample information is summarized in Table 8.10.

Table 8.10
Sample data

Sampled vessel numberProbability of the vessel being sampledTotal number of boxesNumber of boxes sampledTotal weight of the sample
(Kg)
SD of box weight in each vessel
(Kg)
10.022505024.801.20
20.033005026.481.19
30.011005026.701.32
40.041505026.001.44
50.102005025.402.18
Fleet total 12000   
  1. Estimate the total weight of shrimps landed.
  2. Estimate the error of the estimation.

Previous Page Top of Page Next Page