Level 1 Mathematics and Statistics internal assessment ...



Internal Assessment Resource

Mathematics and Statistics Level 1

| |

|This resource supports assessment against: |

|Achievement Standard 91035 |

|Investigate a given multivariate data set using the statistical enquiry cycle |

|Resource title: Sporting success |

|4 credits |

|This resource: |

|Clarifies the requirements of the Standard |

|Supports good assessment practice |

|Should be subjected to the school’s usual assessment quality assurance process |

|Should be modified to make the context relevant to students in their school environment and ensure that submitted |

|evidence is authentic |

|Date version published by Ministry of |November 2012 |

|Education |To support internal assessment from 2013 |

|Quality assurance status |These materials have been quality assured by NZQA. NZQA Approved number |

| |A-A-11-2012-91035-01-4334 |

|Authenticity of evidence |Teachers must manage authenticity for any assessment from a public source, because |

| |students may have access to the assessment schedule or student exemplar material. |

| |Using this assessment resource without modification may mean that students’ work is |

| |not authentic. The teacher may need to change figures, measurements or data sources |

| |or set a different context or topic to be investigated or a different text to read or|

| |perform. |

Internal Assessment Resource

Achievement Standard Mathematics and Statistics 91035: Investigate a given multivariate data set using the statistical enquiry cycle

Resource Reference: Mathematics and Statistics 1.10D

Resource Title: Sporting success

Credits: 4

Teacher guidelines

The following guidelines are designed to ensure that teachers can carry out valid and consistent assessment using this internal assessment resource.

Teachers need to be very familiar with the outcome being assessed by Achievement Standard Mathematics and Statistics 91035. The achievement criteria and the explanatory notes contain information, definitions, and requirements that are crucial when interpreting the standard and assessing students against it.

Context/setting

This activity involves students using the statistical enquiry cycle to make comparisons between groups in a population. The context for this resource is a sample of 100 test match records of the All Blacks rugby team.

This task could be adapted to fit another similar data set with at least two category variables and three numerical variables. The sample must be big enough so that for any category variable there are about 25 - 30 records in each category.

Conditions

This activity requires at least two separate sessions. Confirm the timeframe with your students. Students must work independently.

In the first session, students familiarise themselves with the data and pose two comparison investigative questions. Check students’ investigative questions and, if required, give them time to correct or improve them before they begin the second session. If they are unable to produce at least one suitable investigative question, give feedback of a general nature indicating which of the question criteria have not been met – do not provide the question. If more than minimal feedback is needed, the student is not ready for assessment against this standard.

The second (and subsequent) sessions are for students to carry out the analysis and write conclusions. Any technology is allowed for the analysis.

Resource requirements

Provide students with an electronic and hardcopy (Resource B) version of the data set for their analysis.

Internal Assessment Resource

Achievement Standard Mathematics and Statistics 91035: Investigate a given multivariate data set using the statistical enquiry cycle

Resource Reference: Mathematics and Statistics 1.10D

Resource Title: Sporting success

Credits: 4

|Achievement |Achievement with Merit |Achievement with Excellence |

|Investigate a given multivariate data set |Investigate a given multivariate data set |Investigate a given multivariate data set |

|using the statistical enquiry cycle. |using the statistical enquiry cycle, with |using the statistical enquiry cycle, with |

| |justification. |statistical insight. |

Student instructions

Introduction

This activity requires you to undertake a statistical investigation using a randomly selected sample from the All Blacks test match record. First you will pose two investigative questions that can be answered using the data set. Then you will analyse the sample and form a conclusion for one of your questions.

This assessment is to be completed independently.

The quality of your discussion and reasoning and how well you link this to the context will determine the overall grade.

Task

As you carry out this task you will use the statistical enquiry cycle (Problem, Plan, Data, Analysis, Conclusion).

Problem

Pose two investigative questions that can be explored using the All Blacks test match record. See Resource A for part of the data and a description of the variables.

Your investigative questions must be comparison questions. A suitable comparison investigative question is one that:

• reflects the population

• has a clear variable to investigate

• compares the values of a continuous variable across different categories

• can be answered with the data.

For each question, state the variable you are investigating and the groups you are comparing.

Now choose one of your two questions for investigation using the data found in Resource B.

Plan and data

The sample can be considered representative of all test matches played from 1992.

Analysis

Draw at least two appropriate graphs that show different features of the data in relation to your investigative question.

Give appropriate summary statistics.

Describe features of the distributions comparatively, for example, shape, middle 50%, shift, overlap, spread, unusual or interesting features.

Conclusion

Write a conclusion summarising your findings. The conclusion needs to include an informal inference in response to your investigative question and be supported with relevant evidence.

Resources

Resource A – Sporting success

Details of sample and variables

The table below shows a small part of the data set that you will be using to complete the investigation and a description of the variables. Your teacher will give you a random sample of 100 All Black test match results in electronic and hard copy form.

The sample in the data set was randomly selected from the All Blacks test match results from 1992. The data is from 1992 onwards because the scoring system for tries, conversions, penalties, and drop goals has not changed since then. The data set is sourced from: All_Blacks_Statistics

|Date of match |Test match number |

|Date of match |Day/month/year the match was played. |

|Test match number |Since NZ has started playing test matches, the test match number. Starts at number 267 to |

| |standardise the fact that all the data collected from this point would reflect that all tries were|

| |awarded 5 points. |

|Opposition from northern or|Whether the opposition was from a northern hemisphere or southern hemisphere country. |

|southern hemisphere | |

|Played at home or away |Whether the match was played in NZ (home) or in another country (away). |

|NZ win or loss |Whether NZ won or lost the match. |

|NZ score |Number of points NZ scored at the end of full time. |

|Opposition score |Number of points the opposition scored at the end of full time. |

|Winning margin |The absolute points difference between the two sides. |

Resource B – Sporting success

| |Date of match |Test match number |

|The student shows evidence of investigating a given multivariate data |The student shows evidence of investigating a given multivariate data |The student shows evidence of investigating a given multivariate data |

|set using each component of the statistical enquiry cycle. |set using each component of the statistical enquiry cycle with |set using each component of the statistical enquiry cycle with |

|The student: |justification. |statistical insight. |

|poses an appropriate comparison question |The student: |The student: |

|draws graph(s) and gives summary statistic(s) that allow features of |poses an appropriate comparison question |poses an appropriate comparison question |

|the distributions to be described in relation to the question |draws graph(s) and gives summary statistics that allow features of the |draws graphs and gives summary statistics which allow features of the |

|writes statements which describe comparative features of the |distributions to be described in relation to the question |distributions to be described in relation to the question – at least |

|distributions in context – at least two statements, describing |writes statements, with evidence, which describe comparative features |two different graphs showing different features are expected |

|different features, are expected |of the distributions in context – at least two statements, describing |writes statements with evidence that describe key comparative features |

|answers the comparison question in the context of the investigation or |different features, are expected |of the distributions in context – at least three statements describing |

|makes a correct comparison using an informal inference about the |makes a correct informal inference about the population from the sample|different features are expected, key features include: middle 50%, |

|population. |data which shows an understanding of sampling variability or of the |shift and overlap, shape, spread, any unusual or interesting features |

|For example: |context |makes a correct informal inference about the population from the sample|

|The student poses an appropriate comparison question: |answers the comparison question, with at least one statement of |data which shows an understanding of sampling variability and of the |

|When the All Blacks play at home, do their scores against northern |supporting evidence. |context |

|hemisphere teams tend to be larger than their scores against southern |For example: |answers the comparison question with reference to the population and |

|hemisphere teams? |The student poses an appropriate comparison question: |key supporting evidence summarised in context – what the statistical |

|The student draws at least one graph, for example, a dot plot or a box |When the All Blacks play at home, do their scores against northern |basis for the claim is and the effect of sampling variability or does |

|plot and provides at least one summary statistic for each group, for |hemisphere teams tend to be larger than their scores against southern |the claim make sense considering the actual situation is expected. |

|example, a modal group. |hemisphere teams? |For example: |

|The student describes at least two comparative features of the |The student draws at least one graph, for example, a dot plot or a box |The student poses an appropriate comparison question: |

|distributions in context: |plot and gives summary statistics. |When the All Blacks play at home, do their scores against northern |

|The box of scores against northern hemisphere teams is shifted a little|The student describes, with evidence, at least two comparative features|hemisphere teams tend to be larger than their scores against southern |

|further up the scale than the southern hemisphere box. The |in context: |hemisphere teams? |

|interquartile range is also higher, which suggests that the spread of |I can see in the stem and leaf diagram that the scores between 20 and |The student draws at least two graphs, for example, a dot plot and a |

|the middle 50% of scores is greater. |40 are roughly the same for each group. From the box plot I can tell |box plot and provides summary statistics. |

|The student answers the comparative question in context. |that the spread of the middle 50% of scores for games played at home |The student reflects on at least three features of the distribution |

|Overall, it is not possible to tell if the All Blacks score more points|against southern hemisphere teams (xx points) is slightly bigger than |comparatively. |

|when playing northern hemisphere teams than against southern hemisphere|the spread of the middle 50% of scores against northern hemisphere |The long upper 25% tail for scores against southern hemisphere teams |

|teams. |teams (yy points) and there is quite a large overlap (from … to … |suggests these scores have a positive skew. Every year NZ plays at |

| |points) in the boxes. This tells me that the spread of results is |least two games against South Africa and Australia so I think the |

| |similar for this half of the scores for both northern hemisphere and |majority of southern hemisphere games will be against South Africa and |

| |southern hemisphere teams. |Australia and NZ tends to produce lower scores against these teams. |

| |The scores against the southern hemisphere and northern hemisphere |Games against teams like Argentina and Samoa can produce very high |

| |teams look quite different when comparing the upper and lower |scores for NZ because these teams are usually not as good at rugby. The|

| |quartiles. The boxplot shows that for games against southern hemisphere|test match scores also include games from the World Cup, which includes|

| |teams, the All Blacks are inconsistent. The tail for the top quarter of|some teams that are not very strong and could account for some high |

| |scores against southern hemisphere teams is two-and-half times longer |scores. |

| |than the same tail for northern hemisphere teams. There is a 27-point |There is less difference in the middle 50% of scores against teams from|

| |difference between the best scores against each hemisphere. This is |each hemisphere (… for southern hemisphere and … for northern |

| |equivalent to five tries, which is a big difference in the context of a|hemisphere). Although the medians are different by 6 points (… and …), |

| |rugby game. |the 50% boxes almost completely overlap. |

| |Because there is an overlap of boxes and the median score for each |The overall distribution for scores against northern hemisphere teams |

| |group is in the box of the other group I cannot make a claim that the |is much more symmetrical, with similar distances between each quartile.|

| |All Blacks tend to score more points at home against northern |The stem and leaf plot indicates that there are some scores in the |

| |hemisphere teams than southern hemisphere teams. |higher bracket against northern hemisphere teams (the three scores in |

| | |the 60s) which is not sufficient to claim that the scores against |

| | |northern hemisphere teams tend to be higher than the scores against |

| | |southern hemisphere teams. |

| | |The spread of the lower 25% of scores against southern hemisphere teams|

| | |is about twice as big as against northern hemisphere teams. This |

| | |suggests that when comparing low score games, the All Blacks tend to |

| | |score more points against northern hemisphere teams than southern |

| | |hemisphere teams. This may reflect factors such as the strength of the |

| | |opposition of key southern hemisphere teams such as South Africa and |

| | |Australia. |

| | |I would claim that differences in scores against northern hemisphere |

| | |and southern hemisphere teams (when playing at home) is evident only |

| | |when looking at games in which the All Blacks score either very few or |

| | |very many points. It is not possible to make a claim that NZ scores |

| | |more points when playing at home against northern hemisphere teams than|

| | |southern hemisphere teams, because the middle 50% of the data for each |

| | |group overlaps and the median for each group lies inside the box of the|

| | |other group. |

| | |If I were to investigate further I would reduce the sample to teams NZ |

| | |plays most often: Australia, South Africa, England, France, Scotland, |

| | |Wales, and Ireland. These teams represent both hemispheres and so |

| | |should provide enough data. With the existing data the number of very |

| | |high scores have too much influence – the spread for the southern |

| | |hemisphere data from lowest to UQ is 47 points and from the UQ to the |

| | |max is 43 points. Removing the less competitive teams might help me to |

| | |answer the question. |

| | | |

Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the Achievement Standard.

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