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Statistics: What is the probability of type I error?

  1. Mar 5, 2014 #1
    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:

    [itex] H_0 \;\; : \;\; p=0.6 [/itex]

    [itex] H_1 \;\; : \;\; p \neq 0.6 [/itex]

    Suppose rejection region for [itex] H_0 [/itex] is [itex] \{X \leq 1\} \cup \{X \geq 9\} [/itex]

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

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

    2. Relevant equations

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

    3. The attempt at a solution

    Answer 1: Desired probability is

    [itex] 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 [/itex]

    Answer 2:

    [itex] 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 [/itex]

    Are these answers correct? Thank you in advance!
     
    Last edited: Mar 5, 2014
  2. jcsd
  3. Mar 29, 2014 #2
    Help!

    Does anyone know the correct solution?
     
  4. Mar 30, 2014 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

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