# Statistics: What is the probability of type I error?

1. Mar 5, 2014

### sanctifier

1. The problem statement, all variables and given/known data

X is a random variable of binomial distribution of parameter n=10 and unknown parameter p. Hypotheses are given as follows:

$H_0 \;\; : \;\; p=0.6$

$H_1 \;\; : \;\; p \neq 0.6$

Suppose rejection region for $H_0$ is $\{X \leq 1\} \cup \{X \geq 9\}$

Question 1: What is the probability of type I error?

Question 2: If $H_1$ is changed to "$H_1 \;\; : \;\; p =0.3$", then what is the probability of type II error?

2. Relevant equations

Binomial Distribution of parameters n and p: $f(x) = \binom{n}x p^x(1-p)^{n-x}$

3. The attempt at a solution

$P(X \leq 1,\;\; X \geq 9|p=0.6)=1-P(2\leq X \leq 8|p=0.6)= 1-\sum_{k=2}^8 \binom{10}k 0.6^k(1-0.6)^{10-k} \approx 0.1689$

$P(2 \leq X \leq 8|p=0.3) = \sum_{k=2}^8 \binom{10}k 0.3^k(1-0.3)^{10-k} \approx 0.617$

Last edited: Mar 5, 2014
2. Mar 29, 2014

### sanctifier

Help!

Does anyone know the correct solution?

3. Mar 30, 2014

### Ray Vickson

I get 0.04803512320 ≈ 0.0480 for question 1 and .8505479682 ≈ 0.8585 for question 2.