Anna teaches introductory-level statistics at the University of Auckland. She enjoys facilitating workshops to support professional development of statistics teachers and thinks teaching statistics (and mathematics) is awesome. Anna is also undertaking a PhD in statistics and data science education.
I wrote a guest post earlier this week for Allan Rossman’s excellent blog Ask Good Questions. If you aren’t already subscribed to Allan’s blog you should be! He spent a year writing a new post every week, so there are so many very good advice and ideas for teaching statistics on his blog. Allan’s work with Beth Chance on teaching simulation-based inference has influenced a lot of what we teach in New Zealand, so you’ll also recognise some of the activities (and look for the shout out to New Zealand!)
If cats aren’t your thing, then the guest post the week before features ladybugs and lizards and was written by none other than Christine Franklin – another amazing US-based statistics educator and researcher who fortunately loves visiting us here in New Zealand whenever she can!
And just by total coincidence, but if counting spots is your thing then check out this new app I developed called 101 dalmatians!
Catch a random sample of dogs and use the sample to estimate the mean number of spots per dog for the small population of 101 dalmatians. For a bit of context, you could re-watch some of the original version of the movie first!
While we’re talking about dogs, awesome educator Julia Crawford (Cognition Education) shared this video on the Stats teachers NZ Facebook page as a good discussion starter for experiments.
I don’t have a dog, so have tried out experiments with my cat Elliot. The video below was made for my students when we first went into lockdown (I also tried the cat and square challenge a few years ago). I’ve also added snippets from the videos the super awesome Dr Michelle Dalyrymple and Emma Lehrke sent me of their dogs more successfully engaging with the activity!
Of course, don’t forget you can also contribute to the It’s raining cats and dogs (hopefully) project, by making a data card about each of your dogs or cats. I’m going to create the next set of cards soon, and include a digital platform to work with the cards (similar to Stickland).
And, 😺🐶😺🐶, how could we forget about emoji? Pip Arnold has been making and sharing a bunch of videos and resources for using CODAP with younger statistic students. Did you know you can use emoji in sampler plugin for CODAP? Just copy them from a web age and paste them into tool. When you use the emojis as values in formulae, just make sure to put “” quotes around them. You see emoji in action below, and check out how this all was set up in CODAP here.
For more modelling activities, this time using TinkerPlots, check out Anne Patel‘s presentation for the Auckland Mathematical Association. Her presentation covers a wide variety of important teaching ideas and resources, with lots of practical advice based on her nearly-finished PhD research. Sure, there’s nothing about cats or dogs but she does talk about Census At School, which doesn’t yet ask questions about dogs and cats but maybe could!
Don’t forget, if you’ve got a question about teaching statistics, then feel free to submit this question anonymously using the form below. Who knows? Your question might even inspire a new post 🙂
While I continue to decide whether to quit Facebook, I’ve been trying to keep on top of my admin responsibilities for the Stats Teachers NZ Facebook group while keeping an eye on any stats-related posts on the NZ Maths teachers Facebook group. Since not everyone is on Facebook, I thought I’d do a quick post sharing some of the ideas for teaching stats I’ve recently shared within these groups.
How is the bootstrap confidence interval calculated?
The method of bootstrap confidence interval construction we use at high school level in NZ is to take the central 95 percentile of the bootstrap distribution (the 1000 re-sampled means/medians/proportions/differences etc). There are other bootstrap methods (but we don’t cover these at high school level) and because of the approach we use you can get non-symmetrical confidence intervals.
Here are a couple of videos featuring Chris Wild talking about bootstrap confidence intervals:
Does shape of the bootstrap distribution tell us anything about whether some values in a confidence interval are more likely to be the true value of the parameter?
All the values in the confidence interval are plausible in terms of the population parameter (well, except for the case of impossible values e.g. a negative value when estimating the mean length of a piece of string, or 0% when estimating a population proportion when your sample proportion was not 0%!). As an extra note, we often see skewness in the bootstrap distribution when using small samples whose distributions are skewed (since we resample from the original sample). Small samples are not that great at getting a feel for what the shape of the underlying/population distribution is.
Is a bootstrap confidence interval a 95% confidence interval?
Sample size is a key consideration here With large sample sizes, the bootstrap method does “work” about 95% of the time, hence giving us 95% confidence. But, just like norm-based methods (e.g. using 1.96 x se), with small samples our confidence level will not be as high using the “central 95% percentile” approach.
Can students use both the CL5 and CL6 rules when making a call? Why can’t the CL5 rulebe used with sample sizes bigger than 40?
The rules are designed to scaffold student understanding of what needs to be taken into account when comparing samples to make inferences about populations. Once students learn and can use a higher level rule, they should use this rule by itself. The two rules use different features of the sample distributions and do not give the same “results”. If you use both at the same time, you are encouraging an approach where you select the method that will give you the result you want!
In terms of whether the rule “works” we have to consider not just the cases of “making a call” when we should, but also the cases of “not making a call” when we should. Yes, the CL5 “works” when applied to data from bigger samples than 40, in terms of “evidence of a true difference”. The problem is that for larger sample sizes, when using the CL5 rule, you become much more likely to think the data provides “no evidence of a true difference” when really it does. In this respect, the rule does not “get better” as you increase sample size It’s too stringent, which is why we move to higher curriculum level “rules” or approaches, ones where we learn to take sample size (among other things) into account.
If a sample size is larger, does that mean it is more representative of the population?
Let’s say you have access to 5000 people who voted for the national party in the last election and ask them whether they support Judith Collins as the next PM, and obtain a sample proportion. If you used this sample proportion to construct a confidence interval, it would have small margin of error (narrow interval, high precision), BUT the confidence interval would probably “miss the target” if you were wanting to infer about the proportion of all NZers who support Judith Collins as the next PM because of high bias/inaccuracy
It is important to know the “target” population for the inference you want to make using the same, and check if the sample you are using was taken from this population. In terms of teaching sample to population inference, we need to use a random sample from this population. Our inference methods only model sampling error (how do random samples from populations behave) not nonsampling error (everything else that can go wrong, including the method used to select the sample). If we can’t use a random sample (which in practical terms is pretty difficult to obtain when your sampling frame is not a supplied dataset), then we need to consider how the sample was obtained and also be prepared to assume/indicate even more uncertainty for our inference, in addition to what we are modelling based on sampling variation 🙂
Watch out for a common student misconception that larger populations require larger samples. The population size is not important or relevant (unless you want to get into finite population corrections), it’s the size of the sample that is important in terms of quantifying sampling error. Hence why it was a question in my first stage stats test a couple of weeks ago!
How can you find good articles for 2.11 Evaluate a statistically based report?
I’ve written a little bit about finding and adapting statistical reports here. To summarise, I find newspaper articles are often not substantial enough, since 2.11 requires the report to be based on a survey and students need to be given enough info about how the survey-based study was carried out to be able to critique it. Often the executive summary from a national NZ-based survey works better (with some trimming, adaption). I like NZ on Air based surveys, as this recent one looks do-able with some adaption: Children’s Media Use Survey 2020 – it even mentions TikTok!
Can you create a links to iNZight Lite and VIT online with data pre-loaded?
Yes – I made a video about setting up data links to iNZight Lite here:
If you want to use the Time Series module with your data, just chance the “land=visualize” part of the URL to “land=timeSeries”.
What should a student do if they get negative forecasts from their time series model, when the variable being modelled can’t take on negative values?
You want the student to go back and take a look at the data! And then the model. And ask themselves – what’s gone wrong here? Is it how I’m modelling the trend? Or is it how I’m modelling the seasonality? Or both? Is the trend even “nicely behaved” enough to model? Same with the seasonality
Often the data shows why the model fitted will not do a good job, even before looking at the forecasts generated. We should be encouraging students to look at the data that was used to build the model, particularly for time series when we are focusing on modelling trend and seasonality. Students should be encouraged to ask – why is the model generating negative values for the forecast? How did it learn to do this from the data I used? Can I develop a better model?
Do you have more questions? Chuck them in the Google form below and I’ll see what I can do 🙂
On Tuesday, my good friend Dr Michelle Dalrymple won this year’s Prime Minister’s Science Teacher award. It was so great to be able to fly down to Christchurch with Maxine Pfannkuch to watch the live streaming of the award ceremony with Michelle, her family and her colleagues at Cashmere High School. Michelle was the first mathematics and statistics teachers to win the prize, and it couldn’t have gone to a more deserving teacher!
You can read more about the awesomeness of Michelle in the links below:
In her acceptance speech, Michelle thanked me for being her “statistics hero”. Well, turns out she’s also mine and here’s just one example of why!
After a year or so after I moved from teaching high school statistics to teaching a very large introductory statistics course, I had conversation with Michelle where I complained about how much I missed doing the kinds of hands-on interactive activities that are so important for teaching statistics. I told her what I was being told by others at the university level: that you just can’t do those kinds of things with large lectures, there’s too many students, it won’t work, things could go wrong, not all the students will want to do this, etc.
Michelle listened to me first and then suggested that I try doing something small initially. She told me about one of her activities – comparing how long it takes to eat M&Ms using a plastic fork vs chopsticks – and suggested doing this with just 10 of my 500 students. She explained that I could ask for volunteers, bring them down to the front of the lecture theatre, record the data live, and then use this within the same lecture. I tried this activity out and it worked brilliantly – just imagine a whole lecture theatre of students cheering on students eating M&M’s!
In her pragmatic way, Michelle helped me remember that there’s always a way to do what you know is best for teaching and learning. Her encouragement and attitude to “make it happen” inspired the first of many interactive activities I have since developed to use in my teaching of intro stats. It’s natural to focus on the limitations that a teaching environment or system presents, especially for very large introductory statistics classes of over 300 students. But what Michelle helped me re-affirm in terms of my teaching approach for “large scale teaching” is that it can be more helpful and rewarding to think of the opportunities that working with such a large group of students offers.
Which is one of the reasons why we (Rhys Jones, Emma Lehrke and I) have set up a new sub blog that focuses specifically on teaching large introductory statistics courses. It’s called “Go big or go home!“. In this blog we will share our experiences with trying to build more interactivity and engagement within our very large lecture-based classes. I know that many people reading this blog are statistics teachers based at the school level, so I haven’t assumed you will want to receive emails about new posts for this sub blog. Check out the Go big or go home! blog if you’re interested in reading more and subscribing to this new blog.
I developed the probability distribution explorer as part of my Masters research into teaching probability distribution modelling. The proposed teaching framework and the tool were developed in response to use of data for distribution modelling for AS91586, in particular the need for students to demonstrate use of methods related to the distribution of true probabilities versus distribution of model estimates of probabilities versus distribution of experimental estimates of probabilities.
The tool was developed primarily to support comparisons of the “distribution of experimental estimates of probabilities” and “distribution of model estimates of probabilities”. When reviewing research literature, I found limited examples of how to teach this comparison using an informal approach i.e. not using a Chi-square goodness-of-fit test. Consequently, I also found a lack of statistically sound criteria to enable drawing of conclusions in such resources as textbooks, workbooks and assessment exemplars.
This led to my research, which involved a small group of New Zealand high school statistics teachers. Focusing on the Poisson distribution, the criteria used by ten Grade 12 teachers for informally testing the fit of a probability distribution model was investigated. I found that criteria currently used by the teachers were unreliable as they could not correctly assess model fit, in particular, sample size was not taken into account.
After exploring the goodness-of-fit using my visual inference tool, teachers reported a deeper understanding of model fit. In particular, that the tool had allowed them to take into account sample size when testing the fit of the probability distribution model through the visualisation of expected distributional shape variation. I’ve re-developed the tool this year to support NZQA as they explore opportunities for assessment within a digital environment. A team of teachers are developing prototype assessment activities for AS91586 and these will be trialled with students in schools later in the year.
The video below gives a general introduction to the tool, using data on how many times I say “um” when I’m teaching. The video itself provides another source of data because, um … well, you’ll see if you watch!
Just a quick post to let you know that the mathstatic.co.nz site is hopefully only temporarily down, and I am working with my hosting company to get it back online ASAP. This affects the random redirect tool, the BYOP sampler tool and the experiment lab page, which will not be available until this gets sorted. I’ll update this post soon with a progress update!
It seems the issue is that some overseas dodgy folk have been using the random redirect tool for fraudalent things like phishing scams. So, I’m going to restrict the URLs that can be used – which means analysis time to identify which sites/URL patterns to accept e.g. Google forms, survey monkey etc. 🙂
mathstatic.co.nz is back up and running! It probably was a couple of hours ago, but I have been rewriting the code that processes the random redirect requests. Below are the main changes to the random redirect tool to better prevent issues in the future.
Due to abuse of this tool by dodgy folk, only links with domains on the approved list will now be accepted! Please complete this form to request a domain to be added to the approved list, but don’t expect any new additions to happen any time soon (this is a free tool remember and was created for simple classroom-based randomised experiments with Google forms).
Any random redirect URLs created using this tool can be disabled at any time. If this has happened to you and you are a legitimate teacher, educator or researcher, then send me an email and I might be able to help you.
After emailing me this morning to say everything was sorted with mathstatic.co.nz, my webhosting company then decided to set my site to “maintenance” mode this afternoon and remove some crucial code used to redirect the URLs to the right locations on my website 🙁 I’m trying to get things rest back to what they were now.
Well, I had been meaning to retire the old mathstatic.co.nz website anyway! I’m not sure when mathstatic will be online again, so:
Today I demonstrated some in-class interactive activities that I had developed for my super large intro statistics lectures at a teaching and learning symposium. I’ve shared a summary of the activities and the data below.
Quick summary of the activity
1. Head to how-old.net upload or take a photo of yourself and record the age given by the #HowOldRobot
2. Complete a Google form (or similar) with your actual age and the age given by the #HowOldRobot
I also get students to draw things in class and use their drawings as data. Below are all the drawings of cats made from the demonstration today, and also from the awesome teachers who helped me out last night. If you click/touch and hold a drawing you will be able to drag it around. How many different ways can you sort the drawings into groups?
Last week I was down in Wellington for the VUW NZCER NZAMT16 Mathematics & Statistics Education Research Symposium, as well as for the NZAMT16 teacher conference. It was a huge privilege to be one of the keynote speakers and my keynote focused on teaching data science at the school level. I used the example of following music data from the New Zealand Top 40 charts to explore what new ways of thinking about data our students would need to learn (I use “new” here to mean “not currently taught/emphasised”).
It was awesome to be back in Wellington, as not only did I complete a BMus/BSc double degree at Victoria University, I actually taught music at Hutt Valley High School (the venue for the conference) while I was training to become a high school teacher (in maths/stats and music). I didn’t talk much in my keynote about the relationship between music and data analysis, but I did describe my thoughts a few years ago (see below):
All music has some sort of structure sitting behind it, but the beauty of music is in the variation. When you learn music, you learn about key ideas and structures, but then you get to hear how these same key ideas and structures can be used to produce so many different-sounding works of art. This is how I think we need to help students learn statistics – minimal structure, optimal transfer, maximal experience. Imagine how boring it would be if students learning music only ever listened to Bach.
Due to some unforeseen factors, I ended up ZOOMing my slides from one laptop at the front of the hall to another laptop in the back room which was connected to the data projector. Since I was using ZOOM, I decided to record my talk. However, the recording is not super awesome due to not really thinking about the audio side of things (ironically). If you want to try watching the video, I’ve embedded it below:
You can also view the slides here: bit.ly/followthedataNZAMT. I’m not sure they make a whole lot of sense by themselves, so here’s a quick summary of some of what I talked about:
Currently, we pretty much choose data to match the type of analysis we want to teach, and then “back fit” the investigative problem to this analysis. This is not totally a bad thing, we do it in the hope that when students are out there in the real world, they think about all the analytical methods they’ve learned and choose the one that makes sense for the thing they don’t know and the data they have to learn from. But, there’s a whole lot of data out there that we don’t currently teach students about how to learn from, which comes from the computational world our students live in. If we “follow the data” that students are interacting with, what “new” ways of thinking will our students need to make sense of this data?
Album covers are a form of data, but how do we take something we can see visually and turn this into “data”. For the album covers I used from one week of 1975 and one week of 2019, we can see that the album covers from 1975 are not as bright and vibrant as those from 2019, similarly we can see that people’s faces feature more in the 1975 album covers. We could use the image data for each album cover, extract some overall measure of colour and use this to compare 1975 and 2019. But what measure should we use? What is luminosity, saturation, hue, etc.? How could we overfit a model to predict the year of an album cover by creating lots of super specific rules? What pre-trained models can we use for detecting faces? How are they developed? How well do they work? What’s this thing called a “confusion matrix”?
An intended theme across my talk was to compare what humans can do (and to start with this), with what we could try to get computers to do, and also to emphasise how important human thinking is. I showed a video of Joy Buolamwini talking about her Gender Shades project and algorithmic bias: https://www.youtube.com/watch?v=TWWsW1w-BVo and tried to emphasise that we can’t teach about fun things we can do with machine learning etc. without talking about bias, data ethics, data ownership, data privacy and data responsibility. In her video, Joy uses faces of members of parliament – did she need permission to use these people’s faces for her research project since they were already public on websites? What if our students start using photos of our faces for their data projects?
I played the song that was number one the week I was born (tragedy!) as a way to highlight the calendar feature of the nztop40 website – as long as you were born after 1975, you can look up your song too. Getting students to notice the URL and how it changes as you navigate a web page is a useful skill – in this case, if you navigate to different chart weeks, you can notice that the “chart id” number changes. We could “hack” the URL to get the chart data for different weeks of the years available. If the website terms and conditions allow us, we could also use “web scraping” to automate the collection of chart data from across a number of weeks. We could also set up a “scheduler” to copy the chart data as it appears each week. But then we need to think about what each row in our super data set represents and what visualisations might make sense to communicate trends, features, patterns etc. I gave an example of a visualisation of all the singles that reached number one during 2018, and we discussed things I had decided to do (e.g. reversing the y axis scale) and how the visualisation could be improved [data visualisation could be a whole talk in itself!!!]
There are common ways we analyse music – things like key signature, time signature, tempo (speed), genre/style, instrumentation etc. – but I used one that I thought would not be too hard to teach during the talk: whether a song is in the major or minor key. However, listening to music first was really just a fun “gateway” to learn more about how the Spotify API provides “audio features” about songs in its database, in particular supervised machine learning. According to Spotify, the Ed Sheeran song Beautiful people is in the minor key, but me and guitar chords published online think that it’s in the major key. What’s the lesson here? We can’t just take data that comes from a model as being the truth.
I also wanted to talk more about how songs make us feel, to extend thinking about the modality of the song (major = happy, minor = sad), to the lyrics used in the song as well. How can we take a set of lyrics for a song and analyse these in terms of overall sentiment – positive or negative? There’s lots of approaches, but a common one is to treat each word independently (“bag of words”) and to use a pre-existing lexicon. The slides show the different ways I introduce this type of analysis, but the important point is how common it is to transfer a model trained within one data context (for the bing lexicon, customer reviews online) and use it for a different data context (in this case, music lyrics). There might just be some issues with doing this though!
Overall, what I tried to do in this talk was not to showcase computer programming (coding) and mathematics, since often we make these things the “star attraction” in talks about data science education. The talk I gave was totally “powered by code” but do we need to start with code in our teaching? When I teach statistics, I don’t start with pulling out my calculator! We start with the data context. I wanted to give real examples of ways that I have engaged and supported all students to participate in learning data science: by focusing on what humans think, feel and see in the modern world first, then bringing in (new) ways of thinking statistically and computationally, and then teaching the new skills/knowledge needed to support this thinking.
We have an opportunity to introduce data science in a real and meaningful way at the school level, and we HAVE to do this in a way that allows ALL students to participate – not just those in enrichment/extension classes, coding clubs, and schools with access to flash technology and gadgets. While my focus is the senior levels (Years 11 to 13), the modern world of data gives so many opportunities for integrating statistical and computational thinking to learn from data across all levels. We need teachers who are confident with exploring and learning from modern data, and we need new pedagogical approaches that build on the effective ones crafted for statistics education. We need to introduce computational thinking and computer programming/coding (which are not the same things!) in ways that support and enrich statistical thinking.
I’m pretty excited about the talks and workshops I’m doing over the next month or so! Below are the summaries or abstracts for each talk/workshop and when I get a chance I’ll write up some of the ideas presented in separate posts.
Keynote: Searching for meaningful sampling in apple orchards, YouTube videos, and many other places! (AMA, Auckland, September 14, 2019)
In this talk, I shared some of my ideas and adventures with developing more meaningful learning tasks for sampling. Using the “Apple orchard” exemplar task, I presented some ideas for “renovating” existing tasks and then introduced some new opportunities for teaching sample-to-population inference in the context of modern data and associated technologies. I shared a simple online version of the apple orchard and also talked about how my binge watching of DIY YouTube videos led to my personal (and meaningful) reason to sample and compare YouTube videos.
Workshop: Expanding your toolkit for teaching statistics (AMA, September 14, Auckland, 2019)
In this workshop, we explored some tools and apps that I’ve developed to support student’s statistical understanding. Examples were: an interactive dot plot for building understanding of mean and standard deviation, a modelling tool for building understanding of distributional variation, tools for carrying out experiments online and some new tools for collecting data through sampling.
The slides for both the keynote and workshop are embedded below:
Talk: Introducing high school statistics teachers to code-driven tools for statistical modelling (VUW/NZCER, Wellington, September 30, Auckland, 2019)
Abstract: The advent of data science has led to statistics education researchers re-thinking and expanding their ideas about tools for teaching and learning statistical modelling. Algorithmic methods for statistical inference, such as the randomisation test, are typically taught within NZ high school classrooms using GUI-driven tools such as VIT. A teaching experiment was conducted over three five-hour workshops with six high school statistics teachers, using new tasks designed to blend the use of both GUI-driven and code-driven tools for learning statistical modelling. Our findings from this exploratory study indicate that teachers began to enrich and expand their ideas about statistical modelling through the complementary experiences of using both GUI-driven and code-driven tools.
Keynote: Follow the data (NZAMT, Wellington, October 3, 2019)
Abstract: Data science is transforming the statistics curriculum. The amount, availability, diversity and complexity of data that are now available in our modern world requires us to broaden our definitions and understandings of what data is, how we can get data, how data can be structured and what it means to teach students how to learn from data. In particular, students will need to integrate statistical and computational thinking and to develop a broader awareness of, and practical skills with, digital technologies. In this talk I will demonstrate how we can follow the data to develop new learning tasks for data science that are inclusive, engaging, effective, and build on existing statistics pedagogy.
Workshop: Just hit like! Data science for everyone, including cats (and maybe dogs) (NZAMT, Wellington, October 2, 2019)
Abstract: Data science is all about integrating statistical and computational thinking with data. In this hands-on workshop we will explore a collection of learning tasks I have designed to introduce students to the exciting world of image data, measures of popularity on the web, machine learning, algorithms, and APIs. We’ll explore questions such as “Are photos of cats or dogs more popular on the web?”, “What makes a good black and white photo?”, “How can we sort photos into a particular order?”, “How can I make a cat selfie?” and many more. We’ll use familiar statistics tools and approaches, such as data cards, collaborative group tasks and sampling activities, and also try out some new computational tools for learning from data. Statistical concepts covered include features of data distributions, informal inference, exploratory data analysis and predictive modelling. We’ll also discuss how each task can also be extended or adapted to focus on specific aspects and levels of the statistics curriculum. Please bring along a laptop to the workshop.
Recently I’ve been developing and trialling learning tasks where the learner is working with a provided data set but has to do something “human” that motivates using a random sample as part of the strategy to learn something from the data.
Since I already had a tool that creates data cards from the Quick, Draw! data set, I’ve created a prototype for the kind of tool that would support this approach using the same data set.
I’ve written about the Quick, Draw! data set already:
For this new tool, called different strokes, users sort drawings into two or more groups based on something visible in the drawing itself. Since you have the drag the drawings around to manually “classify” them, the larger the sample you take, the longer it will take you.
There’s also the novelty and creativity of being able to create your own rules for classifying drawings. I’ll use cats for the example below, but from a teaching and assessment perspective there are SO many drawings of so many things and so many variables with so many opportunities to compare and contrast what can be learned about how people draw in the Quick, Draw!
Here’s a precis of the kinds of questions I might ask myself to explore the general question What can we learn from the data about how people draw cats in the Quick, Draw! game?
Are drawings of cats more likely to be heads only or the whole body? [I can take a sample of cat drawings, and then sort the drawings into heads vs bodies. From here, I could bootstrap a confidence interval for the population proportion].
Is how someone draws a cat linked to the game time? [I can use the same data as above, but compare game times by the two groups I’ve created – head vs bodies. I could bootstrap a confidence interval for the difference of two population means/medians]
Is there a relationship between the number of strokes and the pause time for cat drawings? [And what do these two variables actually measure – I’ll need some contextual knowledge!]
Do people draw dogs similarly to cats in the Quick, Draw! game? [I could grab new samples of cat and dog drawings, sort all drawings into “heads” or “bodies”, and then bootstrap a confidence interval for the difference of two population proportions]
Here’s a scenario. You buy a jumbo bag of marshmallows that contains a mix of pink and white colours. Of the 120 in the bag, 51 are pink, which makes you unhappy because you prefer the taste of pink marshmallows.
Time to write a letter of complaint to the company manufacturing the marshmallows?
The thing we work so hard to get our statistics students to believe is that there’s this crazy little thing called chance, and it’s something we’d like them to consider for situations where random sampling (or something like that) is involved.
For example, let’s assume the manufacturing process overall puts equal proportions of pink and white marshmallows in each jumbo bag. This is not a perfect process, there will be variation, so we wouldn’t expect exactly half pink and half white for any one jumbo bag. But how much variation could we expect? We could get students to flip coins, with each flip representing a marshmallow, and heads representing white and tails representing pink. We then can collate the results for 120 marshmallows/flips – maybe the first time we get 55 pink – and discuss the need to do this process again to build up a collection of results. Often we move to a computer-based tool to get more results, faster. Then we compare what we observed – 51 pink – to what we have simulated.
In particular, you can show that models other than 50% (for the proportion of pink marshmallows) can also generate data (simulated proportions) consistent with the observed proportion. So, not being able to reject the model used for the test (50% pink) doesn’t mean the 50% model is the one true thing. There are others. Like I told my class – just because my husband and I are compatible (and I didn’t reject him), doesn’t mean I couldn’t find another husband similarly compatible.
Note: The app is in terms of percentages, because that aligns to our approach with NZ high school students when using and interpreting survey/poll results. However, I first use counts for any introductory activities before moving to percentages, as demonstrated with this marshmallow example. The app rounds percentages to the closest 1% to keep the focus on key concepts rather than focusing on (misleading) notions of precision. I didn’t design it to be a tool for conducting formal tests or constructing confidence intervals, more to support the reasoning that goes with those approaches.