(Achievement Standard 91581)
Richard Arnold and Cushla Thomson
Overview – 3.9 Bivariate Data The most common standard in Level 3 statistics (16 000
students in 2013) 4 credits
Internally assessed
Around 5 weeks spent teaching then assessing the topic (varies across schools and courses – depends on how many standards students are taking)
This Powerpoint, NZQA documents and examples of student work related to 3.9 are on http://padlet.com/cushla_thomson/27oubb6k0yq9
or http://bit.ly/1HnYCNn
Basic idea Students are given a multivariate dataset with some
background information on the variables. They have to choose 2 numeric variables and investigate the relationship between them using linear regression.
They write a report about their investigation
Almost all schools use iNZight for the analysis
At Achieved level: investigate bivariate data
At Merit level: investigate bivariate data with justification
At Excellence level: investigate bivariate data with statistical insight
The investigation cycle - PPDAC In all Level 3 statistics
standards students must show that they’ve used this cycle:
Problem
Plan
Data
Analysis
Conclusion
Requirements for Achieved For Achieved the student is required to investigate bivariate
measurement data. This involves showing evidence of using each component of the statistical enquiry cycle.
This is NZQA-speak for the following skills: The student must: Write a question about the relationship between two quantitative
variables “I want to know what the relationship is between the ____ of ____ (in
m) and the _____ of ______ (in cm)” Identify the explanatory and response variables Use iNZight or Excel to draw a scatterplot Comment on the sign and strength or the relationship
Focus on scatterplot visually before using value of r to back up
Find the line of best fit and use it to make a prediction Form a conclusion about the question
Requirements for Merit Investigate bivariate data with justification
Code for:
All of the above plus links to context throughout the report
Not many stupid statements...
Requirements for Excellence The student must have done what is required for Merit plus
used research to deepen the investigation
Loads of context Residual plot used to look for problems with model
Non-constant variance Non-linear model
Prediction intervals (informally) Students might fit a non-linear model if that makes sense Unusual points will be discussed and possibly removed Students might investigate a second pair of variables
No recipe for Excellence but that doesn’t stop students
trying to find one!
Big gap between Achieved and Excellence See the Padlet link for examples of what is needed for
each of the 3 passing levels http://bit.ly/1HnYCNn
Achieved is mechanical and requires minimal understanding – rote learning will get students there
Excellence requires more communication of understanding and ability to link concepts than many strong first year university students display
Bivariate Data in STAT 193
Correlation (1.5 lectures) Compute Pearson’s r (gcalc)
Rank data (by hand)
Compute Spearman’s r (gcalc: Pearson’s on the ranks)
Plot data, choose appropriate correlation coefficient
Comment on strength and direction
Regression (3 lectures) Draw scatterplot (manually)
Fit regression line a + bX (gcalc)
Draw regression line (manually)
Predictions (gcalc or using formula)
Compute residuals (manually)
Draw residual plot (manually)
Assess regression assumptions using residual plot (linear, constant variance, normal data)
Compute 𝑅2(gcalc) and interpret
Assessment Context is spelled out to a large degree (X and Y are
identified)
Need to consider nature of data and relationship to select correlation coefficient
No inference (no test of slope, or confidence intervals)
No transformation of variables to make relationship linear
Linear relationships only (non-linear in 2nd year)
Statistical Thinking Some consideration of correlation and causation
Need to express confidence in interpolated and extrapolated predictions
Limited requirement to criticise experimental design
What they find easy Calculator work (fitting the line, computing r)
Ranking the data
Drawing a scatterplot
Choosing the correlation coefficient
What they find hard Interpretation of intercept and slope
The meaning of the residual plot: connection of residuals to assumptions
Big picture – correlation and causation
Using appropriate language
Independence
This material is early in the course, so they can practice it – often choose this question on the exam
Richard Arnold and Cushla Thomson
Overview – 3.12 – Statistical Reports Standard taken by relatively few students (4300 in 2013)
E.g. taken by all Level 3 stats students at Onslow and Rongotai but not taken at Wellington Girls’, Tawa
4 credits Externally assessed in the November exams Around 4 weeks spent teaching then assessing the topic
My view: a great standard to wrap around lots of big ideas in
stats – we teach this standard last. Student view: ugh. Too wordy and we have to read the minds of
the examiners. Hard to get Excellence in – 3% E grades last year Heavy reading and writing component – challenging for
international students and local students with poor literacy skills
This Powerpoint, NZQA documents and the 2013 exam paper for 3.12 are on http://padlet.com/cushla_thomson/ymlwvp86vk6z
or http://bit.ly/1qZyca8
3 main parts Inference for poll results – proportions
Survey design; observational studies vs. experiments – critiquing studies and reports of studies
Non-sampling error
Inference for proportions
Inference for proportions in the context of poll results
The only place in Level 3 where students are assessed on classical confidence intervals – BUT…
Students use rules of thumb instead of exact CI (unless a rogue teacher teaches them the exact CIs anyway)
𝑝 ±1
𝑛 for values of 𝑝 between 30% and 70%
Difference of two proportions from one group
Difference of two proportions from 2 groups
Experiments vs observational studies Focus is on critiquing studies and newspaper reports
Difference between observational study and experiment
Can’t prove causation with observational study
Confounding variables
Experimental design principles:
Control and treatment groups
Random allocation
Blinding
Replication
Non-sampling error Focus is on critiquing studies and reports
Sampling methods (good and bad)
Self-selection bias
Interviewer effects
Non-response bias
Questioner effects
Question wording problems
Transfer of findings beyond study population
Statistical reports in STAT 193
Inference for a proportion (1.5 lectures) Taught as part of large sample estimation and testing
Comes after the Central Limit Theorem
Estimates and large sample confidence intervals
𝑝 ± 𝑍𝑝 (1 − 𝑝 )
𝑛
Margin of Error
Sample size calculation
Inference for a proportion (1.5 lectures) Hypothesis test for a single sample 𝐻0: 𝑝 = 𝑝0
Rejection region (inverse Normal: gcalc)
P-value (TEST function in gcalc)
Survey Design/Experimental Design Non-sampling error Polls are used as examples for proportions, teaching
sampling variability
No other explicit survey design material
Informal discussion of poor question design, leading questions, selection bias, observation/intervention: but not examined
This material mostly delayed until 2nd and 3rd year.
Assessment Not required to draw questions out of a large amount
of text
Context is spelled out to a large degree (Hypotheses are identified)
Only limited need for interpretation of findings
Need to know formulae for confidence intervals, margins of error, test statistics
Need to be able to identify and sketch a rejection region, critical value
Statistical Thinking Some consideration of correlation and causation
Limited requirement to criticise experimental design
Appreciation of sampling variability, impact of sample size
What they find easy Calculator work (entering components into the test
function of the calculator)
Substituting into a formula, once they know which one to use
What they find hard Using appropriate language
Rejection regions for one and two-sided tests
Identifying the correct formula (small vs. large sample)
Sampling variability as a concept
Difference between sample and population
This material is in the middle of the course, but the small sample material that follows it (t-tests) makes it harder for them to recall this material. Often skip this question on the exam.