Question

Bottles of popular soda is supposed to contain 270 millilitres of soda. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. The distribution of contents is normal with standard deviation σ = 3ml. The hypothesis one is trying to test is

H0 : µ = 270 H1 : µ < 270

Find the power of the test against the alternative µ = 269. (Note: You are not asked to do hypothesis testing. You are being asked to find power of test.)

Answer 1

Assume sample size =15 because not given in the data
Given that,
Standard deviation, σ =3
Sample Mean, X =269
Null, H0: μ=270
Alternate, H1: μ<270
Level of significance, α = 0.05
From Standard normal table, Z α/2 =1.6449
Since our test is left-tailed
Reject Ho, if Zo < -1.6449 OR if Zo > 1.6449
Reject Ho if (x-270)/3/√(n) < -1.6449 OR if (x-270)/3/√(n) > 1.6449
Reject Ho if x < 270-4.9347/√(n) OR if x > 270-4.9347/√(n)
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Suppose the size of the sample is n = 15 then the critical region
becomes,
Reject Ho if x < 270-4.9347/√(15) OR if x > 270+4.9347/√(15)
Reject Ho if x < 268.7259 OR if x > 271.2741
Implies, don't reject Ho if 268.7259≤ x ≤ 271.2741
Suppose the true mean is 269
Probability of Type II error,
P(Type II error) = P(Don't Reject Ho | H1 is true )
= P(268.7259 ≤ x ≤ 271.2741 | μ1 = 269)
= P(268.7259-269/3/√(15) ≤ x - μ / σ/√n ≤ 271.2741-269/3/√(15)
= P(-0.3539 ≤ Z ≤2.9359 )
= P( Z ≤2.9359) - P( Z ≤-0.3539)
= 0.9983 - 0.3617 [ Using Z Table ]
= 0.6366
For n =15 the probability of Type II error is 0.6366
power of the test = 1-type 2 error
power of the test = 1-0.6366
power of the test = 0.3634
Answer:
power of the test = 0.3634