Question

The weight (kg) of chocolate is normally distributed with population mean ? and population standard deviation 1.2 kg. The manager claimed that the weight of this chocolate is 9.0 kg. We are now doing a hypothesis testing: ?0: ? =
10.1 ?? ?1: ? > 10.1 at 5% significance level and the sample mean is 11.4 kg.

(a) (10) Find the least sample size such that the null hypothesis will be rejected.

(b) (5) Find the rejection region in term of sample mean if sample size is 25.

(c) (10) State the Type I error of this testing and what is the probability of Type I

error.

(d) (10) State the Type II error of this testing

(e) (15) Assume the sample size of the test is “105”, find the probability of Type II error if the TRUE population mean is 10.5kg.

Answer 1

a)

to reject null hypothesis,

Z stat > critical value

α=0.05, critical value=1.645

Z-test statistic= (x̅ - µ )/(s/√n) =    (11.4-10.1)/(1.2/√n)= 1.645

n=((1.645*1.2)/(11.4-10.1) )² = 2.3 ≈ 3

answer: 3

b)

hypothesis mean,   µo =    10.1
significance level,   α =    0.05
sample size,   n =   25
std dev,   σ =    1.2000
      
δ=   µ - µo =    -10.1
      
std error of mean=σx = σ/√n =    1.2/√25=   0.2400

Zα =   1.6449   (right tailed test)
      
We will reject the nullif we get a Z statistic ≥ 1.645  
this Z-critical value corresponds to X critical value( X critical), such that      
      
(x̄ - µo)/σx ≥ Zα      
x̄ ≥ Zα*σx + µo      
x̄ ≥ 1.6449*0.24+10.1  
x̄ ≥ 10.4948   (rejection region)

c) Type I error is concluding that true mean is greater than 10.1, while in actual it is not greater than 10.1

P(type I error) = 0.05

d) Type II error is concluding that true mean is not greater than 10.1, while in actual it is greater than 10.1

e)

true mean ,    µ =    10.5
      
hypothesis mean,   µo =    10.1
significance level,   α =    0.05
sample size,   n =   105
std dev,   σ =    1.2000
      
δ=   µ - µo =    0.4
      
std error of mean=σx = σ/√n =    1.2/√105=   0.1171

Zα =   1.6449   (right tailed test)
P(type II error) , ß =   P(Z < Zα - δ/σx)  
= P(Z <    1.6449-(0.4)/0.1171)  
=P(Z<   -1.771   ) =
type II error, ß =   0.0383