Toss The Coin Four Times If The Coin Lands All Heads

Toss The Coin Four Times If The Coin Lands Either All Heads Or All Ta

Toss the coin four times. If the coin lands either all heads or all tails, reject Ho: p=1/2 (the p denotes the chance for the coin to land on heads.) Complete parts a and b. a. what is the probability of a type 1 error for this procedure? b. if p=4/5, what is the probability of a type II error for this procedure?

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The exercise involves conducting a hypothesis test regarding the fairness of a coin based on the outcomes of four successive tosses. Specifically, the null hypothesis (Ho) asserts that the probability of landing heads (p) is 1/2, while the alternative hypothesis implies a deviation from this probability. The testing procedure involves tossing the coin four times and rejecting Ho if all four outcomes are either heads or tails, indicating an extreme result that suggests the coin might not be fair.

Understanding the Testing Procedure

The described test is based on observing the extreme outcomes—either all heads or all tails—acist that the coin might be biased. If the coin is fair, the probability of these extreme sequences should be small, and such outcomes are used as evidence against Ho. Conversely, if the coin is biased, especially in the direction where p = 4/5, there's a higher chance of observing such extreme sequences.

Part a: Probability of a Type I Error

A Type I error occurs when the null hypothesis is true, but the test incorrectly rejects it. In this case, Ho states p = 1/2; thus, the probability of a Type I error is the probability that, under this true condition, the test results in rejection—that is, observing all heads or tails in four tosses.

Since the decision rule is to reject Ho if all four coins are the same (either all heads or all tails), the probability of a Type I error, denoted as α, is the probability of observing these two events under the null hypothesis.

Calculations:

- Probability of getting four heads (HHHH): (1/2)^4 = 1/16

- Probability of getting four tails (TTTT): (1/2)^4 = 1/16

The total probability of observing either sequence when \( p = 1/2 \):

α = 2 × (1/16) = 2/16 = 1/8 = 0.125

Thus, the probability of a Type I error for this procedure is 0.125.

Part b: Probability of a Type II Error with p = 4/5

A Type II error occurs when the null hypothesis is false, but the test fails to reject it. Here, when the actual p = 4/5, and the testing procedure does not observe the extreme outcomes (all heads or all tails), it results in a Type II error.

The probability that the test fails to reject when p = 4/5 is:

- The probability of NOT observing all heads or all tails in four tosses, given p = 4/5.

Calculations:

- Probability of all heads:

P(all heads) = (4/5)^4 = (0.8)^4 = 0.4096

- Probability of all tails:

P(all tails) = (1 - p)^4 = (1/5)^4 = (0.2)^4 = 0.0016

The probability of the test failing to reject Ho when p = 4/5 (the complement of observing the rejection conditions) is:

1 - [P(all heads) + P(all tails)]

= 1 - (0.4096 + 0.0016)

= 1 - 0.4112

= 0.5888

Therefore, the probability of a Type II error when p = 4/5 is approximately 0.5888.

Conclusion

This testing procedure, based on extreme outcomes—either all heads or all tails—has a Type I error probability of 12.5%, which is the chance of incorrectly rejecting the fairness hypothesis when it is true. Conversely, if the coin is biased with p = 4/5, there is approximately a 58.88% chance that the test will fail to detect this bias (Type II error).

Such analysis underscores the importance of understanding the power and limitations of simple hypothesis tests based on small sample outcomes. The test is relatively conservative, with the likelihood of Type II errors high when the bias is moderate, necessitating either larger sample sizes or alternative testing procedures for more rigorous analysis.

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