Chi Square Test (GOF and Independence) Name Per. Chi Square Modeling Using M & M's Candies Introduction: Have you ever wondered why the package of M&Ms you just bought never seems to have enough of your favorite color? Or, why is it that you always seem to get the package of mostly brown M&Ms? What's going on at the Mars Company? Is the number of the different colors of M&Ms in a package really different from one package to the next, or does the Mars Company do something to insure that each package gets the correct number of each color of M&M? You've probably stayed up nights pondering this! & S One way that we could determine if the Mars Co. is true to its word is to sample a package of M&Ms and do a type of statistical test known as a "goodness of fit" test. These type of statistical tests allow us to determine if any differences between our observed measurements (counts of colors from our M&M sample) and our expected (what the Mars Co. claims) are simply due to chance sample error or some other reason (i.e. the Mars Co.'s sorters aren't really doing a very good job of putting the correct number of M&M's in each package). The goodness of fit test we will be doing today is called a Chi Square Analysis. This test is generally used when we are dealing with discrete data (i.e. count data, or non continuous data). We will be calculating a statistic called a Chi square or X We will be using a table to determine a probability of getting a particular X" value. Remember, our probability values tell us what the chances are that the differences in our data are due simply to chance alone (sample error). The Chi Square test (X?) is often used in science to test if data you observe from an experiment is the same as the data that you would predict from the experiment. This investigation will help you to use the Chi Square test by allowing you to practice it with a population of familiar objects, M&M candies. Objectives: After this investigation you should be able to: write a null hypothesis that pertains to the investigation; determine the degrees of freedom (df) for an investigation; calculate the X' value for a given set of data; use the critical values table to determine if the calculated value is equal to or less than the critical value; determine if the Chi Square value exceeds the critical value and if the null hypothesis is accepted or rejected.Chi Square Test (GOF and Independence) CHI-SQUARE DISTRIBUTION TABLE Accept Hypothesis Reject Hypothesis Probability (p) Degrees of 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.01 0.001 Freedom 1 0.004 0.02 0.06 0.15 0.46 1.07 1.64 2.71 3.84 6.64 10.83 0.10 0.21 0.45 0.71 1.39 2.41 3.22 4.60 5.99 9.21 13.82 0.35 0.58 1.01 1.42 2.37 3.66 4.64 6.25 7.82 11.34 16.27 0.71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.38 18.47 1.14 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09 20.52 10 CO V O UT A W N 1.63 2.20 3.07 3.83 5.35 7.23 8.56 10.64 12.59 16.81 22.46 2.17 2.83 3.82 4.67 6.35 8.38 9.80 12.02 14.07 18.48 24.32 2.73 3.49 4.59 5.53 7.34 9.52 11.03 13.36 15.51 20.09 26.12 3.32 4.17 5.38 6.39 8.34 10.66 12.24 14.68 16.92 21.67 27.88 10 3.94 4.86 6.18 7.27 9.34 11.78 13.44 15.99 18.31 23.21 29.59 2, Draw your Chi Square Curve and put in the critical value (p = 0.05). 3. What is the 12 value for your data? 4. Is your null hypothesis accepted or rejected? Explain why or why not.\fChi Square Test (GOF and Independence) M&M DATA (Individual) What Colors Come In Your Bag? Here are the percentages given by M&M on their website for each color. Brown = 12% . Green = 15% . Red = 12% Orange = 23% Yellow = 15% . Blue =23% 1) Open 2 bags of M&Ms. (If you do not have 2 bags of M&M's email me and I will send you a se of data.) 2) Separate the M&Ms into color categories and count the number of each color. 3) Record your M&M color totals in the data table. Table 1 Brown Red Yellow Green Orange Blue 28 Total Number of M&M's 4) Calculate the expected number of M&Ms in your package by multiplying the total number of M&Ms in the package by the color percent listed on page 1 of the activity. For example, if your package contains 500 M&Ms and you want to find the expected number of red M&Ms you will need to multiply 500 by 20% (500 x 0.20). Record your calculations in the data table. 5) Calculate the difference between the observed and expected numbers for each M&M color. Record your calculations in the data table. 6) Square the difference between the observed and expected. Record your calculations in the data table. 7) Divide the square of the difference by the expected. Record your calculations in the data table. 8) Total all the answers from step 7 to determine the chi-square (12) value. Record the chi-square (12) in the data table