Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet
specifications, a random sample of 10 welds is selected, and tests are
conducted on each weld in the sample. The sample data is recorded as
follows
101.9 100.4 101.2 100.9 101.7
with X = 101.10.
101.5 100.9 100.1 101.6 100.8
It is known that the weld strength is normally distributed with mean µ and
standard deviation σ = 2. If the specifications state that the mean
strength should be equal to 100 lb/in2 , shall we accept that the pipe
welds meet the specifications with significance level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
1 / 11
Test about a Population Mean
1.
2.
3.
4.
Parameter of interest:
µ = population average strength.
Null hypothesis:
H0 : µ = µ0 = 100.
Alternative hypothesis:
Ha : µ 6= 100.
Test statistic value:
z=
x̄ − µ0
x̄ − 100
√ =
√
σ/ n
2/ n
5. Rejection region: z ≥ z.025 or z ≤ −z.025 , where z.025 = 1.96.
6. Substituting n = 10 and x̄ = 101.10,
z=
101.10 − 100
√
= 1.74
2/ 10
7. Since −1.96 < 1.74 < 1.96, i.e., the value of the test statistic does
not fall in the rejection region (−∞, 1.96) ∪ (1.96, ∞), we can not
reject H0 at significance level .05.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
2 / 11
Test about a Population Mean
Test for Population Mean of A Normal Population with Known σ
Null hypothesis:
Test statistic value
H0 : µ = µ 0
x̄−µ
√0
z = σ/
n
Alternative Hypothesis
Ha : µ > µ0
Ha : µ < µ0
Ha : µ 6= µ0
Liang Zhang (UofU)
Rejection Region for Level α Test
z ≥ zα (upper-tailed test)
z ≤ −zα (lower-tailed test)
z ≥ zα/2 or z ≤ −zα/2 (two-tailed test)
Applied Statistics I
July 21, 2008
3 / 11
Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet
specifications, a random sample of 10 welds is selected, and tests are
conducted on each weld in the sample. The sample data is recorded as
follows
101.9 100.4 101.2 100.9 101.7
with X = 101.10.
101.5 100.9 100.1 101.6 100.8
It is known that the weld strength is normally distributed with mean µ and
standard deviation σ = 2. If the specifications state that the mean
strength should be equal to 100 lb/in2 , shall we accept that the pipe
welds meet the specifications with significance level .05? What is the
probability of making type II error then?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
4 / 11
Test about a Population Mean
H0 : µ = µ0 v.s. Ha : µ > µ0
Then the rejection region for level α test is z ≥ zα , or equivalently
√
x̄ ≥ µ0 + zα · σ/ n.
Let µ0 denote a particular value of µ that is less than the null value µ0 ,
then
β(µ0 ) = P(H0 is not rejected | µ = µ0 )
√
= P(X < µ0 + zα · σ/ n | µ = µ0 )
X − µ0
µ − µ0
√ < z α + 0 √ | µ = µ0
=P
σ/ n
σ/ n
0
µ0 − µ
√
= Φ zα +
σ/ n
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
5 / 11
Test about a Population Mean
Alternative Hypothesis
Ha : µ > µ0
Ha : µ < µ0
Ha : µ 6= µ0
Liang Zhang (UofU)
Type II Error Probability β(µ0 ) for Level α Test
0
√
Φ zα + µσ/0 −µ
n
0
µ0 −µ
1 − Φ −zα + σ/√n
0
√
Φ zα/2 + µσ/0 −µ
−
Φ
−zα/2 +
n
Applied Statistics I
0
µ0 −µ
√
σ/ n
July 21, 2008
6 / 11
Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet
specifications, a random sample of n welds is selected, and tests are
conducted on each weld in the sample.
It is known that the weld strength is normally distributed with mean µ and
standard deviation σ = 2. And the specifications state that the mean
strength should be equal to 100 lb/in2 . To construct a hypothesis with
α = .05 and β = .1, how large should n be?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
7 / 11
Test about a Population Mean
The sample size n for which a level α test also has β(µ0 ) = β at the
alternative value µ0 is
h
i
σ(zα +z ) 2


for a one-tailed (upper or lower) test
 µ0 −µ0β
n = σ(z +z ) 2

β
α/2

for a two-tailed test (an approximate solution)

µ0 −µ0
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
8 / 11
Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet
specifications, a random sample of 50 welds is selected, and tests are
conducted on each weld in the sample. The average strength of this
sample is X = 101.10, and the standard deviation is s = 2. If the
specifications state that the mean strength should exceed 100 lb/in2 , shall
we accept that the pipe welds meet the specifications with significance
level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
9 / 11
Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet
specifications, a random sample of 10 welds is selected, and tests are
conducted on each weld in the sample. The sample data is recorded as
follows
101.9 100.4 101.2 100.9 101.7
with X = 101.10 and s = .585.
101.5 100.9 100.1 101.6 100.8
It is known that the weld strength is normally distributed with mean µ. If
the specifications state that the mean strength should exceed 100 lb/in2 ,
shall we accept that the pipe welds meet the specifications with
significance level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
10 / 11
Test about a Population Mean
Test for Population Mean of A Normal Population with Unknown σ
Null hypothesis:
Test statistic value
H0 : µ = µ 0
√0
t = x̄−µ
s/ n
Alternative Hypothesis
Ha : µ > µ0
Ha : µ < µ0
Ha : µ 6= µ0
Liang Zhang (UofU)
Rejection Region for Level α Test
t ≥ tα,n−1 (upper-tailed test)
t ≤ −tα,n−1 (lower-tailed test)
t ≥ tα/2,n−1 or t ≤ −tα/2,n−1 (two-tailed test)
Applied Statistics I
July 21, 2008
11 / 11