Answer to Question #205022 in Statistics and Probability for marwa

Question

To test the hypothesis that a coin is fair, the following decision rules are adopted: (1) Accept the hypothesis if the number of heads in a single sample of 100 tosses is between 40 and 60 inclusive, (2) reject the hypothesis otherwise. a. Find the probability of rejecting the hypothesis when it is actually correct. b. Interpret graphically the decision rule and the result of part (a). c. What conclusions would you draw if the sample of 100 tosses yielded 53 heads? 60 heads?

Answer 1

If a coin is fair p=0.5

$H_0: p=0.5 \\ H_1: p≠0.5$

a) If the number of heads in a single sample of 100 tosses is between 40 and 60 inclusive we will not reject H0. We will reject H0 if the number of heads is less than 40 or the number of heads are greater than 60.

Let X denotes the number of heads in 100 tosses of coin.

X~Binomial (n=100,p)

We have to find P of rejecting the hypothesis when it is actually correct.

When hypothesis is correct:

X~Binomial (x=100,p=0.5)

Here we will apply Central limit theorem we have large enough random sample.

np>10

n(1-p)>10

We will find the probability using Central limit theorem.

$X~N(50, \sigma= 5)$

We will reject hypothesis when p(x<40)

Applying continuity correction

$p(x<39.5) = p(Z< \frac{39.5-50}{5}) \\ = p(Z< -2.1) \\ = 0.01786$

Also, we will reject the hypothesis when p(x>60)

Applying continuity correction

$p(x>60.5) = p(Z> \frac{60.5-50}{5}) \\ = p(Z>2.1) \\ = 0.01786$

The probability of rejecting the hypothesis when it is a actually true is

=0.01786 + 0.01786 = 0.03572

Let’s check the actual probability using binomial distribution

p(x<40)=0.0176

p(x>60) = 0.0176

Required probability is 0.0352

These values are close enough.

b)

Z=±2.1 are the critical values and of the value of test statistics lie in critical region. Reject the null hypothesis.

c) If the toss yielded 53 heads the test statistics will not lie in critical region: we will not reject the null hypothesis and the coin is fair.

If the toss yielded 60heads the test statistics will not lie in critical region: we will not reject the null hypothesis and the coin if fair.