Answer:
a) the probability of committing a type I error if the true proportion is p = 0.6 is 0.0548
b)
- the probability of committing a type II error for the alternative hypotheses p = 0.3 is 0.3504
- the probability of committing a type II error for the alternative hypotheses p = 0.4 is 0.6177
- the probability of committing a type II error for the alternative hypotheses p = 0.5 is 0.8281
Step-by-step explanation:
Given the data in the question;
proportion p = 0.6
sample size n = 10
binomial distribution
let x rep number of orders for raw materials arriving late in the sample.
(a) probability of committing a type I error if the true proportion is p = 0.6;
∝ = P( type I error )
= P( reject null hypothesis when p = 0.6 )
= ³∑
b( x, n, p )
= ³∑
b( x, 10, 0.6 )
= ³∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.6)^x](https://tex.z-dn.net/?f=%280.6%29%5Ex)
![( 1 - 0.6 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.6%20%29%5E%7B10-x)
∝ = 0.0548
Therefore, the probability of committing a type I error if the true proportion is p = 0.6 is 0.0548
b)
the probability of committing a type II error for the alternative hypotheses p = 0.3
β = P( type II error )
= P( accept the null hypothesis when p = 0.3 )
= ¹⁰∑
b( x, n, p )
= ¹⁰∑
b( x, 10, 0.3 )
= ¹⁰∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.3)^x](https://tex.z-dn.net/?f=%280.3%29%5Ex)
![( 1 - 0.3 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.3%20%29%5E%7B10-x)
= 1 - ³∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.3)^x](https://tex.z-dn.net/?f=%280.3%29%5Ex)
![( 1 - 0.3 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.3%20%29%5E%7B10-x)
= 1 - 0.6496
= 0.3504
Therefore, the probability of committing a type II error for the alternative hypotheses p = 0.3 is 0.3504
the probability of committing a type II error for the alternative hypotheses p = 0.4
β = P( type II error )
= P( accept the null hypothesis when p = 0.4 )
= ¹⁰∑
b( x, n, p )
= ¹⁰∑
b( x, 10, 0.4 )
= ¹⁰∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.4)^x](https://tex.z-dn.net/?f=%280.4%29%5Ex)
![( 1 - 0.4 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.4%20%29%5E%7B10-x)
= 1 - ³∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.4)^x](https://tex.z-dn.net/?f=%280.4%29%5Ex)
![( 1 - 0.4 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.4%20%29%5E%7B10-x)
= 1 - 0.3823
= 0.6177
Therefore, the probability of committing a type II error for the alternative hypotheses p = 0.4 is 0.6177
the probability of committing a type II error for the alternative hypotheses p = 0.5
β = P( type II error )
= P( accept the null hypothesis when p = 0.5 )
= ¹⁰∑
b( x, n, p )
= ¹⁰∑
b( x, 10, 0.5 )
= ¹⁰∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.5)^x](https://tex.z-dn.net/?f=%280.5%29%5Ex)
![( 1 - 0.5 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.5%20%29%5E%7B10-x)
= 1 - ³∑
![\left[\begin{array}{ccc}10\\x\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%5C%5Cx%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![(0.5)^x](https://tex.z-dn.net/?f=%280.5%29%5Ex)
![( 1 - 0.5 )^{10-x](https://tex.z-dn.net/?f=%28%201%20-%200.5%20%29%5E%7B10-x)
= 1 - 0.1719
= 0.8281
Therefore, the probability of committing a type II error for the alternative hypotheses p = 0.5 is 0.8281