Probability, Chi Square,  and Pop Beads

Purpose:  The purpose of this lab exercise is to carry out an experiment in order to test a given hypothesis and to use the Chi-Square test to determine if the results of the experiment  support the hypothesis.  Using the Chi Square test, you will determine the probability that the results of the experiment are due to chance alone.

Materials and Methods:  This investigation requires a bowl containing 100 red pop beads and 100 yellow pop beads thoroughly mixed.

If pop beads are removed randomly two at a time from a bowl containing 100 red and 100 yellow beads until all 200 are removed, then the beads will be removed in the following combinations:

 red and red
 red and yellow
yellow and red
yellow and yellow

Since red-yellow and yellow-red combinations are the same, then the expected ratio of bead removal is 1(RR):2(RY):1(YY) or 50RR:100RY:50YY.  Hypothesis:  If the beads are removed randomly, the actual (observed) ratio is due to chance alone and should not deviate from the expected ratio significantly.

This experiment will compare this expected ratio with the ratio that you actually observe and determine the probability of obtaining such a ratio (of deviating from the expected ratio) by the random removal of these beads (two at a time).

Experiment:  Two participants are required to conduct this investigation.  One student will randomly (no looking) remove two pop beads at a time until the bowl is empty.  The other student will record whether a red and red, red and yellow, or a yellow and yellow pair of beads were removed at each withdrawal.

Once all the beads are removed from the bowl, the data is entered in the following Chi Square formula:

o = observed
e = expected

To solve the formula, enter your data in the table below:

Once you have obtained the chi square value, you must determine the probability of obtaining your ratio of beads.  For this you must use the table below.  The degree of freedom is one less than the pop bead combinations that are possible.  In this experiment, 3 combinations are observed, RR, RY, and YY;  therefore, 3 - 1 = 2.   Locate 2 under the "Degrees of Freedom" and then find a value to the right that is close to your chi square value.  When you have done that, look at the top of the chart at the values under the heading, "Probability." 

With this Chi Square table, you are determining the probability that the deviation from the 1:2:1 ratio (25RR:50RY:25YY) is due to chance.  If the probability is 0.10 or greater, then the deviation is not considered significant, and the proposed hypothesis is a valid one.  On the other hand, a probability of less that 0.10 is significant because chance alone is not likely to result in such observed ratios.

Conclusion:  In a brief paragraph below, describe the outcome of your experiment and state the probability of such an outcome by chance alone.  Based on the probability, is the proposed hypothesis a valid one?

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