A consumer products company is formulating a new shampoo and is interested in fo

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 H_0: mu =175 mm versus H1: mu > 175 mm, using the results of n = 10 samples. Find the P-value sample average is x^bar = 185. What is the probability of type II error if the true mean foam height is 200 mm and we assume that alpha = 0.05? What is the power of the test? If the sample size is increased to n = 16. Where would the boundary of the critical be placed the type I error probability is 0.05?

Explanation / Answer

here std error of mean =std deviation/(nj)1/2 =6.325

hence test stat z=(X-mean)/std error =(185-175)/6.325 =1.5810

for above p value =0.0569

b) for 0.05 level corresponding value of X =175+6.325*1.64485 =185.4037

hence probabilty of type 2 error =0.5-P(X<185.4037)=0.5-P(Z<(185.4037-200)/6.325)=0.5-0.0105=0.4895

power of test =1-0.4895 =0.5105

c) for n=16; std error =20/(16)1/2=5

for 0.05 level; z=1.96

hence critical boundary =mean +/- z*std error =165.2002 ; 184.7998

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