CHEN 3010
Applied Data Analysis
Fall 2004
Homework 07 Solutions
Problem 1
443.
a) In order to use
χ
2
statistic in hypothesis testing and confidence interval construction, we need to assume
that the underlying distribution is normal.
1)The parameter of interest is the true standard deviation of the diameter,
σ
. However, the answer can be
found by performing a hypothesis test on
σ
2
.
2) H
0
:
σ
2
= 0.0004
3) H
1
:
σ
2
> 0.0004
4)
α
= 0.05
5)
χ
0
2
=
()
ns
−
1
2
2
σ
6) Reject H
0
if
χχ
α
0
2
1
2
>
−
,n
where
2
14
,
05
.
0
χ
= 23.685
7) n = 15, s = 0.016
χ
0
2
=
96
.
8
0004
.
0
)
016
.
0
(
14
s
)
1
n
(
2
2
2
=
=
−
σ
8) Since 8.96 < 23.685 do not reject H
0
and conclude there is insufficient evidence to indicate the true
standard deviation of the diameter exceeds 0.02 at
α
= 0.05.
b) Pvalue = P(
χ
2
> 8.96) for 14 degrees of freedom:
0.5 < Pvalue < 0.9
Using the EXCEL function CHIDIST(8.96,14), the Pvalue can be more accurately determined as 0.833.
c) 95% lower confidence interval on
σ
2
:
For
α
= 0.05 and n = 15,
χ
α
−
=
1
2
=
2
14
,
05
.
0
χ
23.68
68
.
23
)
016
.
0
(
14
2
<
σ
2
0.00015 <
σ
2
With 95% confidence, we believe the true variance of the hole diameter is greater than 0.00015 mm
2
. With
95% confidence, we believe the true standard deviation of the hole diameter is greater than 0.012 mm
d) Based on the lower confidence bound, we cannot reject the null hypothesis.
Problem 2
444.
a) In order to use
χ
2
statistic in hypothesis testing and confidence interval construction, we need to assume
that the underlying distribution is normal.
1) The parameter of interest is the true variance of the sugar content,
σ
2
.
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 Spring '08
 KOMPALA,DH
 Statistics, Normal Distribution, Null hypothesis, Statistical hypothesis testing

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