Question

I have a coin of unknown fairness. Devise a test procedure to test its fairness; that is, to decide whether or not the coin is fair. Take the null hypothesis to be "the coin is fair". Pick a level of significance, and number of tosses to perform.

(a) What is the probability that I will conclude the coin is unfair, whereas in fact it is fair?

(b) Assuming in reality, the probability that the coin will produce heads is 0.55. What is theprobability that I will conclude that the coin is fair?

Answer 1

We generally test for level of significance 0.10, 0.05, 0.01 or something like these.

Let, level of significance be

Let, number of tosses to perform be

We have to test for null hypothesis

against the alternative hypothesis

(a)

Probability to conclude that the coin is unfair, whereas in reality it is fair is same as probability of rejecting our null hypothesis when null hypothesis is true.

For rejection region we have,

Thus the given situation corresponds to type I error whose probability is same as level of significance of the test and so its probability is 0.05.

Hence, required probability is 0.05.

Note- This value entirely depends upon value of level of significance that we have chosen. So, taking another value of level of significance will produce different probability.

(b)

We know,

Thus the given situation corresponds to type II error.

To calculate required probability we first obtain rejection region using probability of type I error.

Under null hypothesis we have,

Corresponding test statistic is given by

Number of observations

We know,

  [Using R-code 'qnorm(1-(1-0.95)/2)']

Hence, our 95% acceptance region is given by .

Corresponding to the given situation, required probability is given by

Under alternative hypothesis ,

We observe that,

• 
• 

So, both conditions for normal approximation are satisfied.

In such instance, acceptance region is given by

.

[Using R-code 'pnorm(0.9648001)-pnorm(-2.974876)']

Hence, required probability is 0.831212.

Note- This value as well as our rejection/acceptance region entirely depends upon value of level of significance and number of tosses to perform that we have chosen. So, taking another value of level of significance or the number of tosses to perform will produce different probability.