Question

In order to test whether a certain coin is fair, it is tossed ten times and the number of heads (X) is counted. Let p be the "head probability". We wish to test the null hypothesis: p = 0.5 against the alternative hypothesis: p > 0.5 at a significance level of 5%.

(a) Suppose we will reject the null hypothesis when X is smaller than h. Find the value of h.

(b) What is the probability of committing a type I error?

(c) What is the probability of committing a type II error if the true value of p is 0.75?

Answer 1

Under the null hypothesis, the random variables p = 0.5 is binomially distributed with parameters: number of trials p = 0.5 and probability of success p > 0.5 ("head probability"). Hence, the probability distribution function (PDF) of 5% is given by less-than, and greater-than cumulative probabilities for a random variable that has a binomial distribution with parameters

1. We would reject the null hypothesis for values of p = 0.5 that are significantly greater than p > 0.5 So we want to determine h such that if the null hypothesis is true, i.e., p = 0.5 then p > 0.5

Now,

p > 0.5

Hence, we would choose p = 0.5

However, if we allow the alpha error to exceed 0.5 by a small amount then we may opt to choose p<0.5

2. A Type I (alpha) error is committed when the a true null hypothesis is rejected.
3. A Type II (beta) error is committed when we fail to reject a false null hypothesis.

In this case, when p= 0.75 or p = 0.5 whichever was opted.