Binomial Distribution Example_Hypothesis test

Example) Mathematical Statistics and Data Analysis 3ED, Chapter 9. Q1. 

A coin is thrown independently 10 times to test the hypothesis that the probability of head is $\frac {1}{2}$ . $H_{1}: p\neq \frac {1}{2}$. The test rejects if either 0 or 10 heads are observed. 

a) What's the significance level of test?

b) If in fact, the probability of head is 0.1. What's the power of the test? 

$\triangleright$ Think First!

This is a binomial example. X~Bin(10, 0.5) as a coin is thrown 10 times and the probability of head is 0.5. 

$\triangleright$  Solution (a)

$\alpha =$ P(reject $H_{0}$ |$H_{0}$ is true) = P(X=0 | $H_{0}$) + P(X=10 | $H_{0}$)

Under $H_{0}$, $\alpha = \binom{10}{0}(0.5)^0(0.5)^{10}+ \binom{10}{10}(0.5)^{10}(0.5)^0 = \frac{1}{1024}+\frac{1}{1024}=$ 0.0020   

$\triangleright$  Solution (b)

$1-\beta$ = P(reject $H_{0}$ when $H_{1}$ is true) =  P(X=0|$H_{1}$)+P(X=10|$H_{1}$)

         = $\binom{10}{0}(0.1)^0(0.9)^{10}+ \binom{10}{10}(0.1)^{10}(0.9)^0 =$ 0.3487