The Pepsi/Coke Challenge

Student Online Guide

Supplies needed for this activity:

• 2 chilled cans of Pepsi
• 2 chilled cans of Coke

• 20 3-oz paper cups
• A permanent marker (to number the cups)
• A small packet of crackers (to clean the palate)

Purpose

The purpose of this activity is to use the binomial distribution and to introduce the concept of hypothesis testing in an informal manner.

Procedure

1. Find a Subject for the test.
   Ask a friend if there is a difference between Coke and Pepsi. Choose one friend who is absolutely certain that he/she can tell the difference.
2. Establish probabilities
   1. State that you believe that the friend cannot tell the difference. That if you give the friend a cup to taste that the friend has a 50% chance of guessing correctly. State "my hypothesis is that you will be correct 50% of the time. Your job is to prove me wrong". This usually gets an enthusiastic response.
   2. Now the friend must declare how often they can tell. Settle on a value of about 90% (Can you tell the difference if you have a cold? How about after you have eaten something? )
3. Determine the Sample Size (This is the most difficult part of the experiment) These are binomial probabilities based on getting x successes (getting the answer correct) in N trials. You need to determine the number of trials necessary. There are two binomial distributions involved: One with the p = .50 and one with the p = .90
   1. If the friend tastes ONE cup then there is a 50% chance that the friend could guess correctly. If the friend does guess correctly and you declare the friend can tell the difference, there is a 50% chance you are wrong.
      You should not be willing to live with a 50% chance of error. The most you are willing to live with is 5%. Not only that, the friend (who can tell the difference 90% of the time) has a 10% chance of making an error on one cup. This is not fair to the friend!"
   2. What if the friend tries two cups? Draw two trees. One with the 50% probability and the other with 90% probability. If the friend is required to get both cups correct the probabilities are:
      1. There is a 25% chance that the friend can do that by guessing, that is that I will make an error if I declare the friend can tell the difference.
      2. There is a 19% chance that the friend will make an error and identify one or more cups incorrectly.

         If N = 2 Number correct	With p = .50 Probability of getting this number correct	With p = .9 Probability of getting this number correct
         0	0.25	0.01
         1	0.50	0.18
         2	0.25	0.81

         ( Error is highlighted in Yellow!)
   3. Increase the sample size to three cups:
      1. Now there is a 12.5% chance of your error
      2. The friend has a 27.1% chance of an error by missing one or more of the cups.
         Notice that my probability of error is decreasing but the friend’s probability of error is increasing.

         If N = 3 Number correct	With p = .50 Probability of getting this number correct	With p = .9 Probability of getting this number correct
         0	0.125	0.001
         1	0.375	0.027
         2	0.375	0.243
         3	0.125	0.729

   4. If the number of cups is increased to five then your probability of error will finally be below 5%. At 5 cups
      1. Your probability of error is 3.125%.
      2. However, the friend’s probability of guessing all five correct is only 59% so he has a 41% chance of not being able to do this even though he can tell 90% of the time.

         If N = 5 Number correct	With p = .50 Probability of getting this number correct	With p = .9 Probability of getting this number correct
         0	0.0313	0.0000
         1	0.1563	0.0005
         2	0.375	0.0081
         3	0.125	0.0729
         4	0.1563	0.3281
         5	0.0313	0.5905

         
         BUT WAIT!
   5. Suppose we increase the sample size to 8 cups. Now the friend only needs to get 7 out of the 8 correct and does not need to get all 8 correct for your error to be less than 5%.
      1. The probability of getting 7 out of 8, or 8 out of 8 by guessing is .031 + .004 = .035
      2. The probability that the friend is able to get 7 or 8 out of 8 is .383 + .430 = .813, so the error is .187
         Note: Both error rates have decreased! Keep going till both are under 5%

         If N = 8 Number correct	With p = .50 Probability of getting this number correct	With p = .9 Probability of getting this number correct
         0	0.0039	0.0000
         1	0.0313	0.0000
         2	0.1094	0.0000
         3	0.2188	0.0004
         4	0.2734	0.0046
         5	0.2188	0.0331
         6	0.1094	0.1488
         7	0.0313	0.3826
         8	0.0039	0.4305

   6. At 11 cups the friend only has to correctly identify 9 or more of the cups
      1. Your error rate is .027 + .005 + .000 = .032
      2. Friend’s error probability is : 1 – ( .213 + .384 .314) = .089
         This last is close but we can get BOTH error probabilities below 5% by increasing the sample size further.
   7. At n = 14 both error rates drop below 5% if the friend is required to get 11 or more correct
      1. The probability of guessing and getting 11, 12, 13, or 14 out of 14 correct is .022 + .006 + .001 +.000 = .027
      2. The probability that a friend makes an error is: 1 – (.114 + .257 + .356 + .229) = .044

         If N = 14 Number correct	With p = .50 Probability of getting this number correct	With p = .9 Probability of getting this number correct
         0	0.0000	0.0000
         1	0.0008	0.0000
         2	0.0056	0.0000
         3	0.0222	0.0000
         4	0.0611	0.0000
         5	0.1222	0.0000
         6	0.1833	0.0000
         7	0.2095	0.0002
         8	0.1833	0.0013
         9	0.1222	0.0078
         10	0.0611	0.0350
         11	0.0222	0.1142
         12	0.0056	0.2570
         13	0.0008	0.3559
         14	0.0000	0.2288

         The test is now set. The friend will need to get 11 or more out of the 14 correct to demonstrate that the friend can tell the difference.
4. Prepare the cups
   1. Number the cups from 1 to 14.
   2. Determine how the cups will be filled
   3. Explain that this will not be a "double blind" test so you not know what is in the cups.
   4. The cups should be filled randomly . To ensure that they are a coin will be flipped. If the coin comes ups heads the cup will get about an once of Pepsi, and if the coin comes up tails the cup will get an ounce of Coke. Be sure to keep a record of what is in which cup. Your friend should not be present. The friend may got get some water.
5. Run the test.
   1. The friend tests one cup at a time, must declare Coke or Pepsi and go on to the next. Between trials the friend may use a cracker or water to clean the palate.
6. Results
   1. Post the results to the BB. Comment on at least one result (could be yours)
      The following questions might be the basis for your comments:
   2. "Suppose the friend had gotten only 10 of the 14, what do you believe?"
   3. "Suppose the friend got 12 of the 14, what would you believe?"
   4. "Could we have set the standard 5% lower or higher?
   5. "How did increasing the sample size affect the error percentage?"
      
7. Relevance
   1. You have just completed a hypothesis test. You will be studying hypothesis testing
   2. The null hypothesis was that the friend was guessing or p = .5 and the alternative was that they are not guessing. If the friend guessed correctly on 11 or more of the cups then you rejected the null hypothesis (that he was guessing) and decided that the friend could tell the difference (the alternative hypothesis was true).
   3. If you concluded that the friend could tell the difference then the probability of a (Type I) error was 0.027
   4. If you concluded that the friend was guessing then the probability of a (Type II) error was the probability that the friend would miss it given that they could correctly identify the soda 90% of the time or 0.044

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