M & M Statistics: A Chi Square Analysis AP BIOLOGY Mrs. Madalon

Objective Apply understanding of a statistical test known as a ”goodness of fit” test, specifically Chi Square Analysis, to determine differences between observed and expected values. Use chi-square to determine if the Mars Company insures that each package gets the correct number of each color of M&M.

Let’s start by stating our null hypothesis… Null Hypothesis: (in a statistical test) your expected results and observed results are the same! …any observed difference being due to sampling or experimental error. NULL HYPOTHESIS The number of M&M’s in our bag would be representative of the factory %’s and is due to chance, not selection!

Procedure Wash your hands Working with your partner… open up your bag of M&Ms DO NOT EAT ANY OF THE M&Ms ... Not yet anyway  Separate the M&Ms into color categories Count the # of each color of M&M you have You may now eat any purple M&Ms you find... Record your color counts under each category in DATA TABLE 1: Individual Data

Observed Record your number of observed M&Ms for each color in DATA TABLE 1: Individual Data NOW- also put this information on the board to calculate the class data in DATA TABLE 2: Class Data

How to Calculate Expected (e) Value When calculating Expected (e) data… you must refer to the factory %’s for each color in a plain bag of M&Ms. This information is provided on the front page! NOTE: You cannot use %... You must use numbers. Therefore, convert the factory expected %’s to a number. For example, Factory expected % for brown M&Ms is 13%... 13/100 = 0.13 To obtain (e)...(e) x Total M&Ms in bag .... For example: 0.13 x 16 = 2.08 expected value for brown M&Ms

How to Calculate Deviation Deviation (Difference between expected & observed)… To calculate this, (observed – expected). Say I only observed 2 brown M&Ms in my bag... Using the expected value from the previous slide... my deviation would then be: 2 – 2.08 = -0.08 Deviation Squared (d2): (-0.08)2 = 0.064 D2/e: 0.064/2.08 = 0.03

How to calculate X2 This is the sum of all of the d2/e values!

How to calculate degrees of freedom Degrees of Freedom: (n – 1) … this is the number of categories – 1 You need to determine degrees of freedom before you can accept or reject null hypothesis (interpret X2 on the provided table) You have 6 scenarios in this activity (brown, blue, orange, green, red, & yellow) Therefore, your degrees of freedom for this activity are (6 – 1) = 5 WHY MINUS 1? … This is how many other options you have other than the option you got... Therefore, there are 5 other options.

How to interpret X2 number… Looking at your degrees of freedom… find X2 value on the provided table... If X2 is > 0.05 then you ACCEPT your null hypothesis If X2 is < 0.06 then you REJECT your null hypothesis The lower your chi2 (X2 ) number is... The more accurate your experiment! The larger the sample size (ie: class data) the better/more accurate your data!

Now complete DATA TABLE 2: Class Data using the class observed data on the board!

Answer the analysis questions and enjoy your M&Ms! 