Question

The desired percentage of SiO₂ in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO₂ in a sample is normally distributed with σ = 0.32 and that x = 5.21. (Use α = 0.05.)

(a) Does this indicate conclusively that the true average percentage differs from 5.5?
State the appropriate null and alternative hypotheses.

H₀: μ = 5.5
H_a: μ > 5.5

H₀: μ = 5.5
H_a: μ < 5.5     

H₀: μ = 5.5
H_a: μ ≠ 5.5

H₀: μ = 5.5
H_a: μ ≥ 5.5

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

(b) If the true average percentage is μ = 5.6 and a level α = 0.01 test based on n = 16 is used, what is the probability of detecting this departure from H₀? (Round your answer to four decimal places.)


(c) What value of n is required to satisfy α = 0.01 and β(5.6) = 0.01? (Round your answer up to the next whole number.)
n = _____ samples

Answer 1

a)H₀: μ = 5.5
H_a: μ ≠ 5.5

z=(5.21-5.5)*sqrt(16)/0.32 = -3.63

p value =0.0002 ( please try 0.0003 if this comes wrong)

b)

probability of detecting this departure from H₀ =0.0919

c)

required sample size             =n						=(Zα/2+Zβ)2σ2/(μo-μa)2
						=	247.00