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 | ||||||
| 48 | 372 | 174 | 165 | 130 | 473 | 148 |
| 39 | 474 | 155 | 288 | 176 | 277 | 349 |
| 301 | 508 | 339 | 114 | 211 | 477 | 170 |
| 304 | 255 | 409 | 267 | 274 | 166 | 211 |
| 299 | 353 | 192 | ||||
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 | ||||||||||||||
| 24 | 15 | 16 | 17 | 18 | 19 | 20 | 30 | 26 | 23 | 22 | 24 | 27 | 22 | 21 |
| 24 | 20 | 21 | 17 | 18 | 19 | 20 | 30 | 26 | 28 | 22 | 24 | 27 | 22 | 21 |
| 19 | 20 | 21 | 17 | 18 | 19 | 20 | 30 | 26 | 28 | 22 | 24 | 23 | 22 | 21 |
| 19 | 20 | 21 | 29 | 18 | 19 | 20 | 30 | 27 | 28 | 19 | 24 | 23 | 22 | 21 |
| 19 | 20 | 21 | 18 | 30 | 19 | 20 | 30 | 27 | 28 | 19 | 24 | 23 | 26 | 21 |
| 25 | 20 | 21 | 25 | 20 | 19 | 19 | 26 | 27 | 28 | 19 | 24 | 23 | 26 | 21 |
| 25 | 20 | 21 | 25 | 20 | 21 | 19 | 26 | 27 | 28 | 29 | 24 | 23 | 26 | 21 |
| 25 | 20 | 21 | 25 | 20 | 21 | 19 | 22 | 27 | 28 | 29 | 24 | 23 | 26 | 18 |
| 25 | 20 | 23 | 18 | 20 | 21 | 19 | 22 | 31 | 28 | 29 | 24 | 23 | 22 | 18 |
| 19 | 23 | 23 | 18 | 20 | 21 | 19 | 22 | 31 | 28 | 29 | 24 | 23 | 22 | 18 |
| 19 | 23 | 23 | 18 | 20 | 21 | 19 | 22 | 26 | 20 | 29 | 26 | 21 | 27 | 22 |
| 19 | 22 | 23 | 18 | 20 | 21 | 19 | 19 | 22 | 20 | 20 | 20 | 21 | 27 | 22 |
| 26 | 26 | 23 | 27 | 23 | 27 | 23 | 19 | 21 | 25 | 25 | 25 | 25 | 25 | 22 |
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.
Numbers of vessels according to the type of boat and gear used
| Type of vessel | Fishing gear | Total | ||||
| Purse seines | Trawls | Handlines | Longlines | Trammel nets | ||
| Small boats without engine | 15 | 0 | 22 | 20 | 8 | 65 |
| Small boats with engine | 15 | 4 | 23 | 40 | 17 | 99 |
| Purse seiners | 50 | 0 | 0 | 0 | 0 | 50 |
| Stern Trawlers | 0 | 51 | 0 | 0 | 0 | 51 |
| Total | 80 | 55 | 45 | 60 | 25 | 265 |
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.
Group III - Normal distribution
A random variable X is normally distributed with a mean μ = 20.6 and a standard deviation σ= 2.
Group IV - The standard normal distribution
The random variable Z has a standard normal distribution.
Group V - t-student distribution
Using the t-student distribution:
Prob {-a < t(20) < +a} = 0.95.
The general result of Group IV 3.d) is also true for the t-student distribution.
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.
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. | 1 | 2 | 3 |
| Shrimp landing (Kg) | 538 | 435 | 1352 |
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. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Shrimp landing | 538 | 0 | 906 | 442 | 598 | 0 | 435 | 859 | 1352 | 711 |
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.
, using an appropriate class
interval.
].
].
of the estimator.
| No. | Landing 1 | Landing 2 | Landing 3 |
| 1 | 538 | 0 | 906 |
| 2 | 538 | 0 | 230 |
| 3 | 538 | 0 | 598 |
| 4 | 538 | 0 | 20 |
| 5 | 538 | 0 | 435 |
| 6 | 538 | 0 | 859 |
| 7 | 538 | 0 | 1123 |
| 8 | 538 | 0 | 711 |
| 9 | 538 | 906 | 230 |
| 10 | 538 | 906 | 598 |
| 11 | 538 | 906 | 20 |
| 12 | 538 | 906 | 435 |
| 13 | 538 | 906 | 859 |
| 14 | 538 | 906 | 1123 |
| 15 | 538 | 906 | 711 |
| 16 | 538 | 230 | 598 |
| 17 | 538 | 230 | 20 |
| 18 | 538 | 230 | 435 |
| 19 | 538 | 230 | 859 |
| 20 | 538 | 230 | 1123 |
| 21 | 538 | 230 | 711 |
| 22 | 538 | 598 | 20 |
| 23 | 538 | 598 | 435 |
| 24 | 538 | 598 | 859 |
| 25 | 538 | 598 | 1123 |
| 26 | 538 | 598 | 711 |
| 27 | 538 | 20 | 435 |
| 28 | 538 | 20 | 859 |
| 29 | 538 | 20 | 1123 |
| 30 | 538 | 20 | 711 |
| 31 | 538 | 435 | 859 |
| 32 | 538 | 435 | 1123 |
| 33 | 538 | 435 | 711 |
| 34 | 538 | 859 | 1123 |
| 35 | 538 | 859 | 711 |
| 36 | 538 | 1123 | 711 |
| 37 | 0 | 906 | 230 |
| 38 | 0 | 906 | 598 |
| 39 | 0 | 906 | 20 |
| 40 | 0 | 906 | 435 |
| 41 | 0 | 906 | 859 |
| 42 | 0 | 906 | 1123 |
| 43 | 0 | 906 | 711 |
| 44 | 0 | 230 | 598 |
| 45 | 0 | 230 | 20 |
| 46 | 0 | 230 | 435 |
| 47 | 0 | 230 | 859 |
| 48 | 0 | 230 | 1123 |
| 49 | 0 | 230 | 711 |
| 50 | 0 | 598 | 20 |
| 51 | 0 | 598 | 435 |
| 52 | 0 | 598 | 859 |
| 53 | 0 | 598 | 1123 |
| 54 | 0 | 598 | 711 |
| 55 | 0 | 20 | 435 |
| 56 | 0 | 20 | 859 |
| 57 | 0 | 20 | 1123 |
| 58 | 0 | 20 | 711 |
| 59 | 0 | 435 | 859 |
| 60 | 0 | 435 | 1123 |
| 61 | 0 | 435 | 711 |
| 62 | 0 | 859 | 1123 |
| 63 | 0 | 859 | 711 |
| 64 | 0 | 1123 | 711 |
| 65 | 906 | 230 | 598 |
| 66 | 906 | 230 | 20 |
| 67 | 906 | 230 | 435 |
| 68 | 906 | 230 | 859 |
| 69 | 906 | 230 | 1123 |
| 70 | 906 | 230 | 711 |
| 71 | 906 | 598 | 20 |
| 72 | 906 | 598 | 435 |
| 73 | 906 | 598 | 859 |
| 74 | 906 | 598 | 1123 |
| 75 | 906 | 598 | 711 |
| 76 | 906 | 20 | 435 |
| 77 | 906 | 20 | 859 |
| 78 | 906 | 20 | 1123 |
| 79 | 906 | 20 | 711 |
| 80 | 906 | 435 | 859 |
| 81 | 906 | 435 | 1123 |
| 82 | 906 | 435 | 711 |
| 83 | 906 | 859 | 1123 |
| 84 | 906 | 859 | 711 |
| 85 | 906 | 1123 | 711 |
| 86 | 230 | 598 | 20 |
| 87 | 230 | 598 | 435 |
| 88 | 230 | 598 | 859 |
| 89 | 230 | 598 | 1123 |
| 90 | 230 | 598 | 711 |
| 91 | 230 | 20 | 435 |
| 92 | 230 | 20 | 859 |
| 93 | 230 | 20 | 1123 |
| 94 | 230 | 20 | 711 |
| 95 | 230 | 435 | 859 |
| 96 | 230 | 435 | 1123 |
| 97 | 230 | 435 | 711 |
| 98 | 230 | 859 | 1123 |
| 99 | 230 | 859 | 711 |
| 100 | 230 | 1123 | 711 |
| 101 | 598 | 20 | 435 |
| 102 | 598 | 20 | 859 |
| 103 | 598 | 20 | 1123 |
| 104 | 598 | 20 | 711 |
| 105 | 598 | 435 | 859 |
| 106 | 598 | 435 | 1123 |
| 107 | 598 | 435 | 711 |
| 108 | 598 | 859 | 1123 |
| 109 | 598 | 859 | 711 |
| 110 | 598 | 1123 | 711 |
| 111 | 20 | 435 | 859 |
| 112 | 20 | 435 | 1123 |
| 113 | 20 | 435 | 711 |
| 114 | 20 | 859 | 1123 |
| 115 | 20 | 859 | 711 |
| 116 | 20 | 1123 | 711 |
| 117 | 435 | 859 | 1123 |
| 118 | 435 | 859 | 711 |
| 119 | 435 | 1123 | 711 |
| 120 | 859 | 1123 | 711 |
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
| 30 | 42 | 38 | 38 | 41 | 42 | 42 | 46 | 36 | 42 | 34 | 35 | 40 | 35 | 39 | 38 | 39 | 40 | 37 | 46 | 45 |
.
as an estimator of the population mean, µ, and estimate:
.
.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.
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.
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 |
| >300 | 20 |
| Total | 80 |
Table 8.4
Total landings of sardines and
coefficient of variation by vessel
power classes
| Power class (HP) | Total landings (tonnes) | CV |
| 100- | 4 | 0.98 |
| 200- | 60 | 0.73 |
| >300 | 20 | 0.68 |
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 trawls | Average catches (kg) | Standard deviation of catches (Kg) |
| Area 1 | ||||
| 100–200 | 2085 | 12 | 5 | 1.4 |
| 200–300 | 755 | 13 | 28 | 10.5 |
| 300–400 | 660 | 9 | 134 | 47.8 |
| 400–500 | 540 | 10 | 43 | 14.7 |
| 500–600 | 880 | 11 | 13 | 3.6 |
| Area 2 | ||||
| 100–200 | 1252 | 11 | 49 | 14.5 |
| 200–300 | 500 | 14 | 122 | 27.7 |
| 300–400 | 350 | 8 | 55 | 14.0 |
| 400–500 | 445 | 10 | 64 | 15.7 |
| 500–600 | 450 | 9 | 57 | 16.4 |
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 Place | Number of Vessels |
| - Province 1 - | |
| 1 | 9 |
| 2 | 30 |
| 3 | 12 |
| 4 | 9 |
| 5 | 9 |
| 6 | 4 |
| 7 | 5 |
| 8 | 10 |
| - Province 2 - | |
| 9 | 30 |
| 10 | 150 |
| 11 | 41 |
| 12 | 18 |
| 13 | 8 |
| 14 | 27 |
| 15 | 4 |
| - Province 3 - | |
| 16 | 25 |
| 17 | 5 |
| 18 | 15 |
| - Province 4 - | |
| 19 | 28 |
| 20 | 60 |
| 21 | 16 |
| 22 | 24 |
| 23 | 36 |
| 24 | 20 |
| 25 | 52 |
| 26 | 13 |
| 27 | 35 |
| - Province 5 - | |
| 28 | 13 |
| 29 | 48 |
| 30 | 14 |
| 31 | 16 |
| 32 | 12 |
| 33 | 13 |
| 34 | 11 |
| 35 | 38 |
| Total | 860 |
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.
| Clusters | Sizes of the clusters | Total values of the clusters |
| 11 | 50 | 1244 |
| 7 | 50 | 1324 |
| 2 | 50 | 1335 |
| 14 | 50 | 1300 |
| 9 | 50 | 1270 |
| Total | 250 | 6473 |
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.
| Clusters sampled | Number of vessels | Mean landings per vessel, y |
| 1 | 30 | 23.78 |
| 4 | 32 | 24.46 |
| 8 | 20 | 25.05 |
| 13 | 20 | 24.15 |
| 18 | 27 | 23.70 |
| Total fleet | 822 | -- |
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.
| Vessel | Total number of boxes in the vessels | Number of boxes sampled | Total weight of the sample (Kg) | SD of box weight in each vessel (Kg) |
| 1 | 200 | 50 | 990 | 2.02 |
| 2 | 100 | 50 | 1405 | 1.90 |
| 3 | 250 | 50 | 1440 | 2.14 |
| 4 | 90 | 50 | 1330 | 2.21 |
| 5 | 230 | 50 | 1105 | 3.24 |
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.
| Sampled vessel number | Probability of the vessel being sampled | Total number of boxes | Number of boxes sampled | Total weight of the sample (Kg) | SD of box weight in each vessel (Kg) |
| 1 | 0.02 | 250 | 50 | 24.80 | 1.20 |
| 2 | 0.03 | 300 | 50 | 26.48 | 1.19 |
| 3 | 0.01 | 100 | 50 | 26.70 | 1.32 |
| 4 | 0.04 | 150 | 50 | 26.00 | 1.44 |
| 5 | 0.10 | 200 | 50 | 25.40 | 2.18 |
| Fleet total | 12000 |