Question

1. Stacy and Leslie are playing a very simple gambling game. They toss a coin and Stacy wins if it comes up “heads” while Leslie wins if it comes up “tails.” After 12 hours of gambling, Leslie begins to suspect that Stacy has been cheating because Stacy has won more games. Leslie accuses Stacy, but Stacy pleads innocent and proposes to test Leslie’s claim by doing an experiment in which the coin is tossed 14 times.

1. State the null and alternative hypotheses for this experiment.
2. If α = .01, what is the rejection region?
3. They do the experiment and “heads” occurs 8 times. What can they conclude? What if heads comes up 12 times?
4. Let’s assume that Stacy actually is cheating because she is using a coin that is biased to come up “heads” 55% of the time. What was the power of the experiment they did?
5. Given the power of the experiment, was Stacy clever or stupid to propose it as a test to Stacy?

Answer 1

This is a case of binomial testing for population proportion.

We are testing for the probability of getting heads, The claim is that Stacy is cheating that means the probability of getting heads is more than getting tails.

• State the null and alternative hypotheses for this experiment.

  The coin is unbiased and that Stacy is not cheating. (P(heads) = 0.5)

VS

  The coin is biased and that Stacy is cheating. (P(heads) > 0.5)

• If α = .01, what is the rejection region?

Since this a binomial experiment we take the normal approximation and also because it is being tested in only one direction (right )  we will use a 1-tailed z-test.

Critical value at 0.01 level of significance

= 2.3264 ............................using normal percentage tables.

• They do the experiment and “heads” occurs 8 times. What can they conclude? What if heads comes up 12 times?

The experiment is conducted 14 times. Therefore n = 14. heads appear 8 times. Therefore x = 8

Test Statistic: ........................ Null proportion

=

= 0.5400

We reject the null hypothesis if |Test Stat| > Critical value

Since 0.5400 < 2.3264

Decision: We do not reject the null hypothesis at 1% level of significance and conclude that P(heads) = 0.5 and Stacy is not cheating.

  If heads appears to comes up 12 times then our

Test Stat : = 3.8188

Since Test Stat > Critical

We would reject the null hypothesis and conclude that Stacy was cheating.

• Let’s assume that Stacy actually is cheating because she is using a coin that is biased to come up “heads” 55% of the time. What was the power of the experiment they did?

  Since we assume that Stacy is cheating, we conclude that P(heads) = 55% is assumed to be true.

Power of test =

where is the probability of making type 2 error. Type 2 error is not rejecting the null hypothesis when it is false.

The test states that a biased coin has 55% chance of getting heads. Therefore out of 14 trials  if the heads appears 8 or more than 8 () times we would reject it. Means Stacy is cheating.

Not rejecting the null hypothesis means Heads will appear less than 8 times.

Type 2 error is not reject null hypothesis if it is false. At false hypothesis P(Heads) = 0.55

= P( X < 8) | p = 0.55)

= P(X = 0) + P (X = 1 ) + P(X = 2)....P(X = 7) | p = 0.55

=

= 0.21

Power of test = 1 - = 1 - 0.21

Power of test = 0.79

• Given the power of the experiment, was Stacy clever or stupid to propose it as a test to Stacy?

Power of test means the probability of making a correct decision by rejecting the null hypothesis when it is false. A good power is 0.8 (80%).

In our case the power is 0.79 which is good. The decision of rejecting null hypothesis means that Stacy is cheating. A good power of the test is working against Stacy so Stacy was stupid to propose the test.