Percent+of+Red+M&Ms

toc [|Post your questions] here about this lesson

FOM11 students typically are headed to the Liberal Arts courses where statistics are needed.

This experiment (and its off-shoot) is to give students:
 * the experience of sampling and analyzing data
 * rate problems to solve
 * an example of a non-linear function - radioactive decay simulation

It is the backbone of the Stats unit and drives the learning.

When we need info to analyze the data we will refer to the appropriate section in the text or to online resources. ('just in time learning')

flat =M&Ms & confidence intervals= ref: Nelson's Foundations of Mathematics 11

Documents in order of appearance
 * [|M&M Spreadsheet - Nov 09] containing [[file:Student & group data.pdf|Day 1 handout]]
 * [|M&M data - Nov 30.xls] containing [[file:M&M Day 2 Spreadsheet practice & Graph of data.pdf]]
 * [|Statistics - selected exercises.xls] from Nelson's text for practice and discussion; relies heavily on use of spreadsheet features
 * [|selected z-score full solutions.pdf]: hand-written solutions from Nelson p264; z-score table on pp 592-93 or use
 * [|z-score table.pdf]
 * [|Practice test data.xls] [|Key Practice test data.xls]
 * [|Test data B.xls] [|Test data B Key.xls]
 * [|Data generator B for excel test.xls] used to generate Data Set A & B

Purpose
To predict the percent of red M&Ms in the original batch.

Materials & equipment - 30 students
5 packages of 200 gm M&M Milk Chocolate 30 portion cups with lids 1 sharpie 30 recording sheets 1 class data collection sheet spreadsheet software access to Internet [|calories per day calculator]

Method

 * 1) Prepare the samples. There will be about 40 M&Ms for each student.
 * 2) Get students to complete the colour tally for their sample.
 * 3) Get students to record results from other students in their group.
 * 4) Ask students to enter their data into the class data sheet.

Observations

 * 1) Questions on the handout are rate problems.
 * 2) Questions about the sample - individual portion
 * 3) Questions about the larger sample - group data for the bag
 * Data analysis
 * **Mean** & 10-question review: []
 * **Median** & 8-question review: []
 * **Mode** & 6-question review: []
 * draw box & whisker plot, [|video]
 * **Frequency tables & calculating means from the tables**, a 5-question review: []
 * frequency table, frequency polygon, histogram
 * **Standard deviation** lesson & 8-question review: []
 * Data interpretations:
 * Predict what was the **real** percent red M&Ms in the original batch;
 * Were the 30 samples distributed normally?
 * **Normal distribution** lesson & 8-question review: []


 * Other results
 * see links in spreadsheet for others who have posted results online


 * Sources of errors
 * randomness of packaging


 * Looking back: What was the z-score for your percent of red M&Ms compared to the class results
 * **z-score** lesson (text) & 6-question review: []
 * note that it is **NOT** appropriate to use the z-score if the distribution is not a normal distribution. Even so we will be assuming that our sample size is large enough that it approaches the population size and the mean of the population ('mu')

Conclusion
Do what do you think is the real percent of red M&Ms? How sure are you?

=Video help= is here:
 * [|sample mean vs population mean] There is an error in notation that Khan addresses in the next video
 * variance of a [|population] vs a [|sample]
 * because our sample size is small (30) the standard deviation used in the textbook is not valid
 * as sample sizes become larger the standard deviation, s, adjusted for the sample size approaches the standard deviation, sigma, for the population

flat

Presentation of the above
rubrics

flat =Work flow=
 * Day 0**
 * divide up the M&M into smaller samples (~40 per student)
 * prepare [|handout]; It's one of the sheets in [|FOM11 - M&M Math - Nov 09.xls].


 * Day 1**
 * Explain that this is a simulation where we can pretend that each M&M is a person, where the colour is the response to a question in your survey
 * collect individual & group data: [[file:Student & group data.pdf|Sample handout]]
 * collect class data in the spreadsheet. My sample is [|M&M Spreadsheet - Nov 09]
 * conduct the exponential decay when all the data is collected
 * solve rate problems on the handout

> **Example** > I have a student who wants to be involved in the fashion industry. The question might be asked in a survey "What colour of clothing do you think looks best on you?". The results would be used to determine how many coloured shirts to order, say.

,
 * Day 2 to 4 using a spreadsheet program**
 * teacher uses spreadsheet to collate class results & calculate "Percent red M&M in samples". Sample [|M&M spreadsheet - Nov 15.xls] updated Nov 30
 * [[file:M&M Day 2 Spreadsheet practice & Graph of data.pdf|Day 2 handout]] prepared from class data
 * Each student
 * opens a copy of the spreadsheet [|M&M spreadsheet - Nov 15.xls] updated Nov 30
 * uses spreadsheet formulas to calculate Sample size, Average, Highest value, 3rd quartile, 2nd quartile (median), 1st quartile, lowest value, and range. NB: the formulas are in Open Office syntax
 * transfers the data to the [[file:M&M Day 2 Spreadsheet practice & Graph of data.pdf|Day 2 handout]] to construct
 * box & whisker plot. See [|video lesson from Khan Academy]
 * frequency table counting by 2's
 * frequency table counting by 5's
 * histogram for the **//by 2's//** table
 * frequency polygon for the **//by 5's//** table
 * Direct students' attention to the **//Statistics Assessment//** tab of the spreadsheet to show students how they will be assessed: M&M experiment, Data Analysis on Data set B, Media use of data . The two extra columns //Selected exercises in textbook'// and //Data Set A statistical analysis// will be practice sessions leading up to the final assessments. [[file:Statistics Assessment.pdf]]
 * Discussion questions comparing M&M results to statistics on a math test
 * Why do you want to know the class average on a particular math test?
 * why do you want to know the highest score? the lowest?
 * Why won't teachers tell you these statistics on a math test?


 * Day 4, 5, & 6**
 * Complete the frequency table.
 * Draw the histograms

Review Practice using the spreadsheet of selected exercises


 * Day 4**

 A distribution is normal if
 * Normal distribution** aka Gaussian distribution, p250
 * its histogram is symmetric
 * the mean, median, and mode occur simultaneously at the axis of symmetry
 * the extremes taper to zero (are less likely to occur)

[|Interactive example]: vary the standard deviation, bin size, & sample size (trials) and note the changes to the distribution. The purple curve is **fitted** to the data, while the black curve is the theoretical (expected) normal curve.


 * Z-scores** standardizes a data item by comparing it to the average and the standard deviation for the data. p255

=Discussion threads=
 * critiques
 * adaptations
 * further exploration