# Mathematics and Statistics - annotated exemplar Level 3 AS91582

## Use statistical methods to make a formal inference (3.10)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 964KB)

This annotated exemplar is intended for teacher use only. The student work shown does not always represent a complete sample of what is required. Selected extracts are used, focused on the grade boundaries, in order to assist assessors to make judgements at the national standard.

Last updated June 2019.

There is new student evidence for Low Excellence, High Merit, Low Merit, High Achieved and High Not Achieved.

All annotations have been altered to better illustrate the requirements of the standard.

### Low Excellence

Commentary
Student work extract

Student 1 (PDF, 241KB)

For Excellence, the student needs to use statistical methods to make a formal inference, with statistical insight.

This involves integrating statistical and contextual knowledge throughout the statistical enquiry cycle, and may include reflecting about the process; considering other relevant explanations.

The student has demonstrated insight with their integration of statistical and contextual knowledge when posing a comparison question (1) and discussing the centres of the sample distributions (2). There is evidence of reflection about the process (3).

This extract is from a student response which also included the student integrating statistical and contextual knowledge to an appropriate level for the award of Excellence when discussing other aspects seen in the displays of the sample distributions. For example, spread, shape, middle 50%.

For a more secure Excellence, the student needs to provide more depth in the discussion in the inference/conclusion by integrating more fully the contextual knowledge they acquired both while researching the variable prior to posing the question and when explaining their findings.

They could also provide further evidence considering other relevant explanations by investigating other variables, for example the number of bedrooms.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 190KB)

For Merit, the student needs to use statistical methods to make a formal inference, with justification.

This involves linking components of the statistical enquiry cycle to the context and/or to the populations, and referring to evidence such as sample statistics, data values, or features of visual displays in support of statements made.

The student has researched the context and used this to develop a comparison question (1). When discussing sample distributions, the student has used evidence in support of the statements made (2) and there is evidence of linking their discussion to the context and/or population (3).

The student has also made a formal inference which has been supported with evidence (4). In their conclusion, the student has linked the comments made back to findings and to the question posed (5) and commented on sampling variability including the variability of the estimates (6).

To reach Excellence, the student could integrate statistical and contextual knowledge in greater depth throughout the response, for example by reflecting on whether the discussion of the sample distributions is consistent with contextual knowledge acquired through researching the context.

They could also consider other relevant explanations by including a more detailed discussion of the possible reasons for survival other than the cost of the fare.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 174KB)

For Merit, the student needs to use statistical methods to make a formal inference, with justification.

This involves linking components of the statistical enquiry cycle to the context and/or to the populations, and referring to evidence such as sample statistics, data values, or features of visual displays in support of statements made.

This student has researched the context and posed a comparison question (1). When discussing sample distributions, the student has used evidence to support the statements made (2) and there is some evidence of linking their discussion to the context and/or population (3).

The student has also made a formal inference which has been supported with evidence (4). In their conclusion, the student has linked the comments made back to findings and to the question posed (5) and commented on sampling variability including the variability of the estimates (6).

For a more secure Merit, this student could link the investigative question more clearly to research about the difference in fare prices and whether a passenger survives.

In the discussion of sample distributions and in the conclusion, the student could further develop links to the context/population and provide research to support the comments made.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 190KB)

For Achieved, the student needs to use statistical methods to make a formal inference.

This involves showing evidence of using each component of the statistical enquiry cycle: posing a comparison investigative question using a given multivariate data set, selecting and using appropriate displays and summary statistics, discussing sample distributions, discussing sampling variability, including the variability of estimates, making an appropriate formal statistical inference and communicating findings in a conclusion.

The student has researched the context and posed a comparison question (1). They have selected and used appropriate displays and summary statistics (2), provided evidence of discussing sample distributions (3) and linked some of their comments to statistics and visual displays (4).

The student has also made an appropriate formal statistical inference (5), communicated findings in a conclusion (6) and commented on sampling variability (7).

To reach Merit, the student needs to link the comments made in the analysis/inference and conclusion more clearly to the context and the actual population.

They also need to include a discussion on the variability of estimates when discussing sampling variability, and further explanation about the choice of variables when developing their question.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 234KB)

For Achieved, the student needs to use statistical methods to make a formal inference.

This involves showing evidence of using each component of the statistical enquiry cycle: posing a comparison investigative question using a given multivariate data set, selecting and using appropriate displays and summary statistics, discussing sample distributions, discussing sampling variability, including the variability of estimates, making an appropriate formal statistical inference and communicating findings in a conclusion.

This evidence is from a student’s response to the TKI task ‘New Zealand crash statistics’.

The student has researched the context and posed a comparison question (1). They have selected and used appropriate displays and summary statistics (2) and provided evidence of discussing sample distributions (3).

The student has also commented on sampling variability, including the variability of estimates (4), made an appropriate formal statistical inference (5), and communicated findings in a conclusion (6).

For a more secure Achieved, this student needs to ensure that the conclusion refers to the actual population. The student also needs to make the link between the research and the context clearer when developing their question.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 232KB)

For Achieved, the student needs to use statistical methods to make a formal inference.

This involves showing evidence of using each component of the statistical enquiry cycle: posing a comparison investigative question using a given multivariate data set, selecting and using appropriate displays and summary statistics, discussing sample distributions, discussing sampling variability, including the variability of estimates, making an appropriate formal statistical inference and communicating findings in a conclusion.

This evidence is from a student’s response to the TKI task ‘Elite athletes’.

The student has researched the context and posed a comparison question (1). They have selected and used appropriate displays and summary statistics (2) and provided evidence of discussing sample distributions (3).

The student has also discussed sampling variability, including the variability of estimates (4).

To reach Achieved, the student needs to make an appropriate formal statistical inference by using the values from the bootstrap distribution to give an interval estimate for the difference in the population medians.

The student then needs to make a conclusion that goes on to answer the investigative question. This needs to include the direction of the difference.