A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. For the Wald-Wolfowitz runs test, the value of the total number of runs R is ________.
A) 8
B) 11
C) 16
D) 19

Question

Quizplus · Accepted Answer

The Wald-Wolfowitz runs test involves counting the number of runs in a sequence of flips. A run is defined as consecutive flips with the same result. For example, in the sequence H H T H T H H T, there are four runs: H, T, H, and T. To calculate the expected number of runs for a sequence of n flips, we use the formula:

E(R) = 1 + (2n - 1)/3

For n = 30, we have E(R) = 1 + (2*30 - 1)/3 = 20. Therefore, if the coin is fair, we would expect to see 20 runs in the sequence.

To calculate the actual number of runs, we count the number of times the sequence changes from H to T or from T to H. In the given sequence, we have 16 runs: H H T H T H H T H H T H H T H | T | H H H T T T T T | H H H H | T. The vertical lines separate the runs. Therefore, the value of the total number of runs R is 16.

We can compare the observed value of R to the expected value of E(R) using a normal approximation. Under the null hypothesis that the coin is fair, the distribution of R is approximately normal with mean E(R) and variance:

Var(R) = (2n - 1)(2n - 3)/9

For n = 30, we have Var(R) = (2*30 - 1)(2*30 - 3)/9 = 80.44. Therefore, the standard deviation of R is approximately sqrt(Var(R)) = 8.97.

To test whether the observed value of R is significantly different from the expected value under the null hypothesis, we calculate the z-score:

z = (R - E(R))/sqrt(Var(R))

For R = 16 and E(R) = 20, we have z = (16 - 20)/8.97 = -0.45.

The p-value for this test is the probability of getting a z-score less than or equal to -0.45 or greater than or equal to 0.45 (since this is a two-tailed test). Using a standard normal distribution table or calculator, we find that the p-value is approximately 0.654. This is greater than the typical significance level of 0.05, so we fail to reject the null hypothesis.

Therefore, we cannot conclude that the coin is unfair based on the Wald-Wolfowitz runs test.

A Magician Has a Coin That May or May Not