Shockwheat. Statistics 1, Activity 1

Transcription

1 Statistics 1, Activity 1 Shockwheat Students require real experiences with situations involving data and with situations involving chance. They will best learn about these concepts on an intuitive or informal level without the rules and algorithms traditionally associated with them. Students need an opportunity to reflect on and interpret situations and work at describing them in order to answer questions about the population from which the data is gathered. A secondary purpose is that of communicating information about the data to others. In this first activity, data is collected and analysed for the purpose of answering specific questions posed by the Board of an imaginary breakfast cereal company, Kooleg s, around the launch of a new product, Shockwheat. This problem assumes that our students have met many of the ideas involved in summarising data before (graphical representation, descriptive statistics and data distribution) but the emphasis here changes from How to draw a graph or How to calculate the mean to What does the data tell us? and How does this help us to put forward a convincing argument? Features of this activity A simulation of a real-world situation based on a realistic mathematical model A hands-on activity involving students as participants rather than observers Student input is encouraged at all stages of the activity. Learning occurs throughout the activity as different problems arise. Students contrast their gut reaction with the simulation data. Use of a computer (optional) Page 1 of 9

2 Materials and preparation Introductory video (to be supplied) Introductory PowerPoint (shockwheat_memos.ppt) Cereal box with model (possibly many) Several sets of monsters (possibly on cards) A class set of dice Optional Graphic calculators Random number tables Page 2 of 9

3 Introducing the problem Either introduce the problem using the introductory video or through the PowerPoint up to slide 6. Question: On average, how many packets will a customer have to buy to get a full set of monsters? Encourage all students to make a guess based on their gut feeling do not talk about which average yet. Record their guesses. Sorted data Stress to your students that this is unsorted data which, by its nature, is often confusing and difficult to summarise. Ask for suggestions as to how this could be sorted and recorded more effectively. Respond positively to their suggestions but try to steer them to a dot plot and then into a stem and leaf diagram. At each stage ask them to explain what they can see from the data. Notice that in the stem and leaf plot, we have included the totals in each group, recorded in brackets, for clarity, and a key. It is important to stress checking that we haven t lost any data. Page 3 of 9

4 The wisdom of the crowd Now ask your students how they could summarise the group s initial reactions or gut feeling. Explain to them that this is often referred to as the wisdom of the crowd. This could be linked to How many sweets in the jar? Return to the question, On average, how many packets will we have to buy to get a full set of monsters? Now ask your students what on average might mean. Try to get them to recall the terms and meaning of mean, mode and median as well as how to find them in this case. The mode and the median can be easily found from the stem and leaf plot. The mean clearly requires some calculation! At this point, it is a good idea to introduce some more formal notation: x (sigma x) stands for the sum of all of our guesses x (x bar) stands for the mean of our guesses n represents the number of guesses and x =!! Now ask your students how they intend to answer the original question: On average, how many packets will a customer have to buy to get a full set of monsters? This should lead to a discussion on the relative merits of the three averages. Page 4 of 9

5 But how can we find out if we are right? Throw this question out to your students. Lead your students if necessary into actually suggesting buying packets or opening boxes. Explain to the students that in your demonstration box there are actually envelopes with pictures of monsters inside and they can in effect each buy a packet by drawing out an envelope and identifying their monster. At this stage it doesn t matter if they put the envelopes back or not. Go round the room until you have collected a set. Now ask if we can confidently answer the original question On average, how many packets will a customer have to buy to get a full set of monsters? Hopefully, they will say they need more evidence (that is nearly always the answer when analysing data!). Okay. Any suggestions as to what we should do next? Try to steer the conversation towards repeating the experiment over and over again. Okay, has anyone got any ideas as to how we might be able to do that? Again lead if necessary to a simulation using a dice. Page 5 of 9

6 Repeating the experiment It is probably easier to work in pairs so, after you have explained to your students how to record their data, give each pair a six-sided die. We suggest that every pair works through at least three cycles as shown below. You will now have lots of data that needs to be collected and analysed. Once again, it is a good idea to record all of the raw data on the board as it arises. Students need to get used to working with untidy data. We now follow the same path as before, reinforcing both the wording and understanding of some basic statistical tests. Again, represent this as a dot plot and on a stem and leaf diagram. As before, at each stage ask your students to share what they see from the diagrams. We now return to the question, On average, how many packets will we have to buy to get a full set of monsters? Once again we find the three averages and comment on the differences. Page 6 of 9

7 We now return to the question On average, how many packets will a customer have to buy to get a full set? Now encourage your students to share their ideas. Page 7 of 9

8 Meanwhile back in the board room Now show your students slide 22 or the next video clip. Throw this open to the group for discussion in pairs. They might come up with an intuitive approach, suggest a cumulative frequency graph, or they might suggest a box and whisker. Give them time to discuss their ideas with their partner or within their group. Introduce the notation lower quartile (Q1) and upper quartile (Q3), and confirm median as (Q2). This will obviously need some clarification depending on the number of results. Also bring in minimum value (Q0) and maximum (Q4). At this point it is worth asking the group to find the five key points from their original estimates and compare the two sets of results using box plots. Page 8 of 9

9 This then leads into a discussion as to what the two stem and leaf plots actually show. Make sure that you introduce the terms spread, range and interquartile range and that your students understand the significance of these terms. The board also asked for the 90th percentile. As you probably know, there are different ways of interpreting this, but we have gone for probably the simplest way, which is to divide the data into 10 equal groups and take the last result in the 9th group. Finding outliers The board have one final request Throw the question back to the group and ask for their opinions. They may well think that anyone who has bought over 19 packets should be offered a free set of monsters, which would be a sensible recommendation. But eventually lead your students to a generally agreed definition that an outlier is more than 1.5 x IQR beyond the lower and upper quartiles as shown above. Now ask your students to make their full recommendation to the board. Page 9 of 9