Question

A consumer products company is formulating a new shampoo and is interested in foam height (in mm). Foam height is approximately normally distributed and has a standard deviation of 20 mm. The company wishes to test H₀: μ = 175 mm versus H₁: μ > 175 mm, using a random sample of n = 10 samples.

(a) Find P-value if the sample average is  = 185? Round your final answer to 3 decimal places.

(b) What is the probability of type II error if the true mean foam height is 200 mm and we assume that α = 0.05? Round your intermediate answer to 1 decimal place. Round the final answer to 4 decimal places.

(c) What is the power of the test from part (b)? Round your final answer to 4 decimal places.

Answer 1

a)

Ho :   µ =   175  
Ha :   µ >   175   (Right tail test)
          
Level of Significance ,    α =    0.050  
population std dev ,    σ =    20.0000  
Sample Size ,   n =    10  
Sample Mean,    x̅ =   185.0000  
          
'   '   '  
          
Standard Error , SE = σ/√n =   20/√10=   6.3246  
Z-test statistic= (x̅ - µ )/SE =    (185-175)/6.3246=   1.5811  
          

          
p-Value   =   0.0569   [ Excel formula =NORMSDIST(-z) ]

b)

true mean ,    µ =    200
      
hypothesis mean,   µo =    175
significance level,   α =    0.05
sample size,   n =   10
std dev,   σ =    20.0000
      
δ=   µ - µo =    25
      
std error of mean=σx = σ/√n =    20/√10=   6.3246

Zα =   1.6449   (right tailed test)
P(type II error) , ß =   P(Z < Zα - δ/σx)  
= P(Z <    1.6449-(25)/6.3246)  
=P(Z<   -2.308   ) =
type II error, ß =   0.0105  

c)

power =    1 - ß =   0.9895