Question

Suppose a coin is tossed 100 times and the number of heads are recorded. We want to test whether the coin is fair. Again, a coin is called fair if there is a fifty-fifty chance that the outcome is a head or a tail. We reject the null hypothesis if the number of heads is larger than 55 or smaller than 45.

Write your H_0 and H_A in terms of the probability of heads, say p.

Find the Type I error rate if the null hypothesis is true.

Calculate the power of the test if the true chance of head is 0.4

Find the p-value if the observed number of heads is 65.

Answer 1

Null Hypothesis H₀: p = 0.5

Alternative hypothesis H_A: p 0.5

Standard error of sample proportion = = 0.05

We reject the null hypothesis if the number of heads is larger than 55 or smaller than 45.

Type I error rate = Probability to reject the null hypothesis when it is true

= P(p < 0.45 | p = 0.5) + P(p > 0.55 | p = 0.5)

= P[Z < (0.45 - 0.5) / 0.05] + P[Z > (0.55 - 0.5) / 0.05]

= P[Z < -1] + P[Z > 1]

= 0.1587 + 0.1587

= 0.3174

Standard error of sample proportion for p =0.4 = = 0.049

Power of the test if the true chance of head is 0.4 = Probability to reject the null hypothesis when p = 0.4

= P(p < 0.45 | p = 0.4) + P(p > 0.55 | p = 0.4)

= P[Z < (0.45 - 0.4) / 0.049] + P[Z > (0.55 - 0.4) / 0.049]

= P[Z < 1.02] + P[Z > 3.06]

= 0.8461 + 0.0011

= 0.8472

If the observed number of heads is 65, sample proportion, = 65/00 = 0.65

Test statistic, z = ( - p) / Std error = (0.65 - 0.5) / 0.05 = 3

For two tail test, p-value = 2 * P(z > 3) = 0.0027