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Statistical Inference Resource • Years 9 to 13 • Update Nov 26 2013

Lake Taupo Trout Data Sets

for the developing statistical inference at Y 9 to 13

These ten data sets list all the trout weighed in during the Easter fishing competitions in years 1993 to 1999 and then 2006, 2007 and 2011. The early data sets have 600 to 1000 fish and the later ones around 125.

Each fish has length, weight, condition factor, how caught, species, sex, maturity and a fish ID number.Weight is in KgLength is in cm, (minimum for this competition is 45cm measured from nose to the V-in the tail.

Condition Factor is a calculation although I think there is a small alteration for the competition purposes. A good investigation!How Caught is the method used to catch the fish and is DT Deep Trolling, ST Shallow Trolling, FF Fly Fishing, Down Rigging appears in latest datasets as regulations changed. Species is either Rainbow or Brown, oncorhynchus mykiss or salmo trutta. Sex is Male or Female if able to be determined.Maturity is either maiden or spawned.ID Number is a competition generated number showing the order of weighing in as the numbers get bigger.

Jim Hogan 5/20/23


Department of Conservation, (http://www.doc.govt.nz/by-region/central-north-island/) Fishing Officers determined all the measurements and judgements and this data was recorded by competition organisers. This data was used by DOC to give measures of trout in the lake at the time.

Here is a website for research History of Taupo Fishery Link. Google to find out more about fishing methods. Try using “rainbow trout”, “taupo fishing”, “fly fishing” etc. 1998 Home Pages of the competition are available on disc.

KEY IDEAIt is probably a very wise idea to have a theme (of say Fishing or Water) running in your department (or school) so that conceptual knowledge is built over time in anticipation of future learning and assessment opportunities.

Year 9, 10 Teaching Activity Suggestions.These suggestions are for teachers to use to help design a learning programme suitable for their students. See Census at School and www.nzmaths.co.nz for more ideas. Data cards are a good starter. (see NZMATHS, Statistics).

Show a video of fishing to help set the scene. There are u-tube videos (http://www.youtube.com/watch?v=1V8RdK0h1-E) that show a few fishing techniques. This is a fly fishing experience in the South Island. Here is a boy called Lucas http://www.youtube.com/watch?v=24PVjq4jb3Y and a dog called Milo.

Introduce the dataset of 1993, all 618 trout by letting students “fish” and catch 9, 17 or 19 trout each. Have students plot these and make a dot plot and a box and whisker. Why these numbers (gen form 4n + 9 because the middle and quartiles are easy to find, 5 is too small but works).

There is a page of six of these line graphs in the resource. Students can pair up and fish together or do it on their own. Replace the fish if you need and mix them up. There are plenty of fish to catch. The 1999 dataset has over 1000 trout.

Making the Teacher Resource1. Find a box about 20cm by 20cm by 10cm to keep the little fish card sin.2. Decorate with Lake Taupo pictures.3. Print and cut up the dataset required.

1993 Sex and Weight only – Introducing topic and junior yearsData set 1, 1993, 618 fish, weight, length and sex.Data Set 2, 1999, 1000+ fish, weight, length and sex.Data Set 3, 2011, 125 fish, weight, length and sex.

4. Take a sample 17 fish, draw a dot plot and a box and whisker on the grid supplied. Repeat to complete page. Notice variation between samples. [catch and release only…put your fish back!] {Rule for nice sample size 4x + 5]

Jim Hogan 5/20/23

http://www.youtube.com/watch?v=24PVjq4jb3Y

http://www.youtube.com/watch?v=1V8RdK0h1-E

http://www.nzmaths.co.nz/statistical-investigations-units-work?parent_node=

http://www.nzmaths.co.nz/

http://www.tongarirorivermotel.co.nz/trout-fishing/history-of-taupo-fishery/

http://www.doc.govt.nz/by-region/central-north-island/

http://www.doc.govt.nz/by-region/central-north-island/


Things to explore and connect with other areas of mathematics.These suggestions in connect to probability, statistics, geometry, measurement, number and algebra. Group work, investigation and project work.

• Making a dot plot• Making box and whisker• Determining what an outlier could be (3x IQR above or below quartiles)• Comparing Male and Female fish (length or weight)• Estimate the probability of catching a big fish.• Using the Condition Factor formula to find best fish

• Using scale factor and grid to enlarge Trouty

• or other pictures or .• Discuss how to estimate the number of fish in Lake Taupo. [DOC

estimate approximately 1.5 mature fish (>45cm)] Tagging fish simulation.• Estimate the area of Lake Taupo from a map outline [Pics formula, count

squares, approximate shapes]• The average depth of Lake Taupo is about 110m, calculate volume and mass of water in the lake. [Surface area, 616 km2. Average depth, 110 m, about 60 cubic km!..wow]

• Find out how many rivers flow into the lake and how many flow out.

Jim Hogan 5/20/23


• Work out how much energy is stored as hydro-electricity. Name all of the hydro-stations on the Waikato River.

• Draw a cross section diagram showing heights of all stations.• Choose a project to investigate and produce a wall display.

• Go fishing or learn how to tie a fishing fly.

• Discover the National Trout Centre http://www.troutcentre.com/ on the Tongariro River.

• http://www.food.com/recipes/trout, yum• Present your project!

KEY TASKList as many statistical ideas as you can.Outline how statistical inference ideas are developed in your scheme for junior classes.

Jim Hogan 5/20/23

http://www.food.com/recipes/trout

http://www.troutcentre.com/


Year 11 • NCEA Level 1The Achievement Standard:-

AS91035 “Investigate a given multivariate data set using the statistical enquiry cycle” (4cr)

is assessable using the Trout data. There are also possibilities for problems to be solved by students that would provide evidence towards the Work and Study Skills Unit Standards 26623, 26626 and 26627.

Key Learning• taking a random sample to avoid bias• using dot plot and box and whisker to illustrate sample• describing features of a sample distribution• overlap and spread to assist “making a call” for comparisons• point estimates of population parameters• use of contextual information associated with the data

Learning PPDACThe DataEach fish has length, weight, condition factor, how caught, species, sex, maturity and a fish ID number.Weight is in KgLength is in cm, (minimum for this competition is 45cm measured from nose to the V-in the tail.

Condition Factor is a calculation although I think there is a small alteration for the competition purposes.How Caught is the method used to catch the fish and is DT Deep Trolling, ST Shallow Trolling, FF Fly Fishing, Down Rigging appears in latest datasets as regulations changed. Species is either Rainbow or Brown, oncorhynchus mykiss or salmo trutta. Sex is Male or Female if able to be determined.Maturity is either maiden or spawned.ID Number is a competition generated number showing the order of weighing in as the numbers get bigger.

Sample Assessment for AS91035

Jim Hogan 5/20/23


Using the 1993 Lake Taupo Fishing Tournament trout data pose a suitable comparison question and using the PPDAC cycle present your analysis and answer your question.

1. Problem: I wonder if the weight of a female trout is larger than the weight of a male trout caught in the Lake Taupo Fishing Tournament in 1993?

2. Plan: I am going to take a random sample about 30 male and 30 female fish from the 1993 Lake Taupo Fishing Tournament results and computer the mean. This will be my point estimate. I will describe the distributions and use overlap and spread to make my decision.

3. Data: 1993 Lake Taupo Fishing Tournament results

4. Analysis: I used Fathom software to sample 30 random male and 30 random female fish .

Jim Hogan 5/20/23


[Note, iNZight can be used to give the same results and summary data].

In my sample there were 30 of each sex. The male distribution has less spread than the females selected and the weights are grouped around 2kg. The females have 3 big fish but most are grouped around 1.56kg. The mean of the males is 1.9kg and the mean of the females is 1.8kg. From the box and whisker plots it can be seen that 50% of the males are bigger than 75% of the females. The difference in the means is 0.1kg, and the overall spread is 2.19 – 1.44 = UQmales – LQfemales = 0.75kg. The overlap of 0.1kg is much smaller than a third of the overall spread.

5. Conclusion: Based upon this sample it appears that male trout caught in the 1993 Lake Taupo Fishing Tournament are heavier than the female trout caught. The answer to my question is no, female trout do not appear to be heavier than male trout. The mean weight of both male and female fish caught was 1.85kg. This is a point estimate of the weight of any legal fish caught in Lake Taupo in 1993. The minimum length of fish weighed in is 45cm so a legal fish is one that exceeds this length.

I expected female fish to be heavier because at this time of the year they are full of roe ready for spawning up rivers within a month or so.

Broad Answer ScheduleSee above for sample and analysis. A suitable comparison question might be “I wonder if the mean weight of a female trout is bigger than the mean weight of a male trout caught in the Lake Taupo Fishing Tournament in 1993?”

A – PPDAC cycle used, sample taken, graphical evidence used to answer question.M- Features of sample distribution described and overlap shift used to justify answer to question posed.

Jim Hogan 5/20/23


E – Contextual information included describing why the female fish might be larger than the male fish. Female fish in April are preparing for spawning and are full of roe, or eggs. In the following months they migrate up rivers to reproduce.

Note if this context is going to be used to assess AS91035 or AS91036 a more professional examination and more detailed answer schedule should be developed. This example was provided as an illustration only.

KEY TASKSWrite a paragraph explaining why this example meets the standard for assessing AS 91035. Include ideas for improvement.

Develop an assessment for AS91035 or AS91036 with answer schedule task based on this example.

Jim Hogan 5/20/23


Year 12 • NCEA Level 2The internal Achievement Standard:-

AS92064 “Use statistical methods to make an inference (4cr)

could be assessed using this topic.

Key Learning• sample size affects variability• sample variability is inversely proportional to √n• point estimate of population parameters• IQR as a measure of spread• estimate of median based upon median ≠ 1.5xIQR/√n• interpreting informal inference interval• using iNZight (or Fathom) essential

Sample Assessment Using the 1999 Lake Taupo Fishing Tournament trout data pose a suitable comparison question and using the PPDAC cycle present your analysis and answer your question.

Sample AnswerProblem: I wonder if the median condition factor of female trout caught during the Lake Taupo Fishing Tournament were larger than the median condition factor of male trout caught?[Note inclusion of the “median” ref moderator reports 2012/3”.

Plan: I am going to use Fathom software to randomly sample at least 30 female and at least 30 male trout caught during the 1999 Lake Taupo Trout Fishing Tournament. I will make a dot plot and a box and whisker plot of the trout caught. I am choosing 30 of each because from class work a sample of size 30 appears to give a reliable point estimate and interval.

Data: 1999 Lake Taupo Trout Fishing Results database.

Analysis: I sampled 75 trout which included 31 male trout and 43 female trout, There must have been more female trout caught in the tournament because every resample I ended up with more female trout. Displaying all fish caught showed nearly 700 female trout and about 400 male trout. There were almost twice as many female trout caught.

I established an interval based on this sample from which I can be pretty sure that the true population median is contained.

Jim Hogan 5/20/23


Jim Hogan 5/20/23


The median condition factor for the female trout was 45.9 and the mean condition factor for male trout was 42.2. These are point estimates.

Using the formula parameter ≠ 1.5xIQR/√n the interval generated for the female trout for the median condition factor is

= 45.9 ≠ 1.5 x (49.7 – 40.4)/√43 = [48.0, 43.8] (3sf)

and the interval generated for the male trout for the median condition factor is

= 42.2 ≠ 1.5 x (46.7 – 37.2)/√31 = [44.8, 39.6] (3sf)

There was one fish whose sex was unidentified. This has been omitted form the analysis. Some fish are very difficult to sex due to maturity.

The difference in medians is

=median female – median male = 45.9 – 42.2 = 3.7 [mass]/[length]^3

Jim Hogan 5/20/23


Conclusion: The mean condition factor for the females was 3.7 higher but the interval estimates overlapped slightly. I am pretty sure that the mean condition factor for female trout is in the interval 43.8 to 48.0. I am pretty sure that the mean condition factor for males is in the interval 39.6 to 44.8. There is a small overlap 44.8 – 43.8 = 1.0 compared to the overall spread of 48.0 – 39.6 = 8.4. I will not make a call that the condition factor for female fish is bigger than that for male fish because there is an overlap.

A reason to suspect females to have a higher condition factor is that this April competition is just a few months before the spawning period when the fat females deposit abundant quantities of eggs in the rivers around Lake Taupo. The male trout are also fit and healthy but the sperm sacs are not as heavy as the female egg sacs. The fish become fatter but not longer so the condition factor formula should give bigger values for these fatter female fish. This data does not support this hunch.

KEY TASKWrite a paragraph explaining why this example meets the standard for assessing AS 92064. Include ideas for improvement.

Develop an assessment for AS 92064 with answer schedule task based on this example. Use any trout database.

Jim Hogan 5/20/23


Year 13 • NCEA Level 3The internal Achievement Standard:-

AS91581 “Investigate bivariate measurement data” (4cr)AS91582 “Use statistical methods to make a formal inference” (4cr)

could be assessed using this topic. There is time series information but it is not sufficient for assessment.

Key Learning• sample size affects variability• sample variability is inversely proportional to √n• point estimate of population parameters• the bootstrap method of resampling• estimate of parameter interval using bootstrap or other• interpreting formal inference interval• using iNZight (or Fathom) essential• could involve correlation, model and strength of relationship.

Sample Assessment Using the Lake Taupo Fishing Tournament trout data pose a suitable comparison question and using the PPDAC cycle present your analysis and answer your question.

Sample AnswerThe trout data includes years 1993 to 1999, and 2006, 2007 and 2011. I am interested to see if the mean weight of a trout over this time period has changed.

[Usually inference is made about a difference, Hence I wonder if there is difference in the weights of fish in the 1993 tournament and the 2011 tornament? is an alternative question]

Problem: Has the mean weight of trout caught during the Lake Taupo Trout Fishing Tournament over the data periods supplied stayed the same?

Plan: I am going to take a size 20 sample from the 1994, 1998 and 2007 and 2011 data distributions to establish a bootstrapped interval estimate of the median weight for each of these years. A line graph of these will show trends over this period 1994 to 2011.

Data: The data supplied is the fish that were weighed in as legal limit fish. These are fish greater than 45cm in length and “kept” by fishermen who are only allowed to catch a fishing “limit”. This data is really a sample of the whole lake but for this assessment I am going to treat the sample as the population. The time of the tournament is April each year.

I noticed when researching the tournament that there was an Mt Ruapehu situated 40km to the Southwest of Lake Taupo erupted in 1995 and 1996. This deposited ash in the lake. Waikato Regional Council estimate 2.3 million tonnes of ash fell into lake Taupo during this eruption and in the short term “water quality improved”. http://www.waikatoregion.govt.nz/Environment/Natural-resources/Water/Lakes/Lake-Taupo/How-Mount-Ruapehus-eruptions-affect-Lake-Taupo/

Jim Hogan 5/20/23


A thesis on trout in lake taupo by Heeg, 2012 is here http://researcharchive.vuw.ac.nz/bitstream/handle/10063/2046/thesis.pdf?sequence=2

Analysis: I sampled 20 randomly selected trout from the 1994 database. Then I bootrapped by sampling with replacement, 200 samples of size 20. This method mimics the sample 200 times. The diagram shows the original sample, a bootstrap sample and the collection of 200 medians.

The collection of medians is used to establish the interval for true value of the population median. Finding the 5th sample median on the right of the distribution and the 5th sample median on the left establishes a 95 percentile interval containing the median of the population.

Jim Hogan 5/20/23


This works out to be [1.75, 2.28] kg.

This process was repeated for 1998, 2006 and 2011. The results for these years were:-

1998 interval is [1.9, 2.7]

2006 interval is [1.6, 2.25]

Jim Hogan 5/20/23


2011 interval is [0.95, 1.4]

These values were put into a table and graphed.

Jim Hogan 5/20/23


The graph shows an increase in max and min median weight to 1998 and then a decrease through 2006 to 2011. The answer to the problem posed is that the median weight of lake Taupo trout has not stayed the same during the data periods. The median increased to a maximum of around 2kg in 1998 and then decreased to around 1.2kg in 2011.

The eruption noted earlier could have resulted in the lake being cleaned and more suitable for trout. The trout contains some 1.5million catchable trout as researched by DOC and reported in Target magazines. Google “DOC Trout Target NZ” for many reports. The larger trout of earlier years has been replaced by more and smaller trout of today. There is also a catfish issue which could be interfering with food supply chains. The smaller trout of 2011 do have reasonable condition factors so there may be other factors involved.

Answer ScheduleA - The student has used the bootstrap method to establish an answer to part of the question posed.

Jim Hogan 5/20/23


M – The student has justified the sampling method and decisions made as well as answer the question posed.E – A well presented and reasoned report showing contextual knowledge applied appropriately to explain outcomes from the investigation. The bootstrap method is clear.

KEY TASKWrite a paragraph explaining why this example meets the standard for assessing AS 91581. Include ideas for improvement.

Develop an assessment for AS 91581 or AS 91582 with answer schedule task based on this example. Use any trout database. There are some interesting alternatives with the data using methods of fishing.

Author NotesThis document is an attempt to show how early context learning can be used to inform later assessment. Introducing a topic in earlier years provides the opportunity for student interest, learning the context, and being informed prior to assessment in senior levels. The statistical techniques involved are modern and use the computer and in this assessment somewhat artificial as the data is in fact a sample of the whole population of the Lake Taupo trout of which we know little. Estimates of over 1.5million legal fish exist on the DOC website and described in past issues of their publication TARGET.

The software is powerful enough to take a bootstrap sample of size 1018 (1998) and repeat this 10000 times to establish a very tight interval of population parameters. This could be done but only 100 or so resamples are needed for a large samples. It soon becomes very obvious that bootstrapping works, a surprisingly small sample will give a surprisingly good estimate of many parameters. My experience suggests a sample size of between 7 to 18 resampled 200 times will produce a very good inference.

The Central Limit Theorem can be used to assess this standard as no specific reference is made for techniques used. The CLT is being “de-emphasised” however, in an attempt that students understand intervals. The modern computer and available software as shown allows resampling techniques and improved access to understanding the inference process of statistical thinking. The advantage of resampling is any population shape can be analysed in this way as the resample simply mirrors the sample taken. For the population involved here the part of the normal distribution of weights considered is those of trout with lengths above 45cm. This is a truncated distribution. The population of small trout less than 45cm must be enormous and dynamic but finite.

Jim Hogan 5/20/23


A population with a wide variation will produce samples that have wide variations. Hence the IQR in the Year 12 formula. Likewise the larger sample size reduces the variation in the sample hence the 1/√n aspect of the Year 12 formula. Note the 1.5 multiplier in this formula is a guess as to what will work made by John Tukey.

Local situations, strengths and resources must dominate school curriculum and assessment. For the Central Plateau, Rotorua, Napier, Gisborne, Otago regions trout fishing is a significant activity and one that would be popular with may thousands of school students. This topic might not be popular with central city, rural or coastal schools with different priorities.

2013iNZight has become established as the standard software. The VIT modules are excellent teaching resources to show variation in sampling, bootstrapping and randomisation ideas. Google “inzight nz” for finding this FREE software.

See http://schools.reap.org.nz/advisor for all files.

Jim Hogan 5/20/23

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