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# 3.10 Achievement Objectives

### Excellence

• Use statistical methods to make a formal inference.

• Use statistical methods to make a formal inference, with justification.

• Use statistical methods to make a formal inference, with statistical insight.

Achievement Standard 3.10 #91582 (link to NZQA)

Carry out investigations of phenomena, using the statistical enquiry cycle:
- using existing data sets,
- seeking explanations,
- using informed contextual knowledge, exploratory data analysis, and statistical inference
- communicating findings and evaluating all stages of the cycle

Make inferences from surveys and experiments:
- determining estimates and confidence intervals for differences
- use methods such as resampling to assess the strength of the evidence

Use statistical methods to make a formal inference involves showing evidence of using each component of the statistical enquiry cycle.

Use statistical methods to make a formal inference, with justification 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.

Use statistical methods to make a formal inference, with statistical insight involves integrating statistical and contextual knowledge throughout the statistical enquiry cycle, and may include reflecting about the process; considering other relevant explanations.

Using the statistical enquiry cycle to make a formal inference involves:
- 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
- communicating findings in a conclusion.

Clarifications

Updated December 2014. This document has been updated to address issues that have arisen from moderation.
Students need to provide evidence of each component of the statistical enquiry cycle detailed in Explanatory Note 3 of the standard.

Posing an appropriate comparison investigative question using a given multivariate data set
Sufficient time needs to be allocated for students to research the context and acquire appropriate contextual knowledge. For all grades, students need to identify a purpose and pose an investigative question which is informed by this contextual knowledge. The question needs to be comparative, and needs to refer to the population and the parameter under investigation.
An appropriate question could be ‘is the median number of text messages sent per day by adults in New Zealand greater than the median number of text messages sent per day by teenagers in New Zealand?’

Discussing sample distributions
Students need to discuss, in context, what they see in the displays of the sample distributions. This could include central tendency, spread, shift and unusual values.

Discussing sampling variability including the variability of estimates
Students need to show an understanding that if they were to take another sample from the population this is likely to result in different displays and summary statistics.

Making an appropriate formal statistical inference
Students need to use the bootstrap confidence interval for the difference of the medians/means to answer their investigative question. The inference needs to identify the population and the parameter. Students also need to show an understanding about the nature of the confidence interval.
An appropriate formal inference could be: “I am fairly sure that, in New Zealand, the median number of text messages sent by adults is more than the median number of text messages sent by teenagers and that the difference in the medians is between 12 and 17 text messages per day.”

Required quality of student response
For Merit, students need to justify all findings with reference to evidence from the displays and statistics and link the purpose and findings to their research.
For Excellence, students need to integrate the statistical and contextual knowledge, gained from their research, throughout the response and reflect on the process, which could be shown by considering other relevant explanations.
For Excellence, students need to integrate the statistical and contextual knowledge, gained from their research, throughout the response and also reflect on the process, which could be shown by considering other relevant variables, evaluating the adequacy of the model or showing a deeper understanding of the model.

### Curriculum Achievement Objective

AO: S8-1 Carry out investigations of phenomena, using the statistical enquiry cycle:
-using existing data sets
- seeking explanations
- using informed contextual knowledge, exploratory data analysis and statistical inference
- communicating findings and evaluating all stages of the cycle

S8-2a Make inferences from surveys or experiments
- determining estimates and confidence intervals for differences
- use methods such as re sampling

Covers all:- posing a comparison investigative qn. using a given m.v. 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
- communicating findings in a conclusion