Transcript Ch8-6

Section 8.6
Testing a claim about a
standard deviation
Objective
For a population with standard deviation σ, use
a sample too test a claim about the standard
deviation.
Tests of a standard deviation use the
c2-distribution
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Notation
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Notation
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Requirements
(1) The sample is a simple random sample
(2) The population is normally distributed
Very strict condition!!!
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Test Statistic
Denoted c2 (as in c2-score) since
the test uses the c2 -distribution.
n
Sample size
s
Sample standard deviation
σ0
Claimed standard deviation
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Critical Values
Right-tailed test “>“
Needs one critical value (right tail)
Use StatCrunch: Chi-Squared Calculator
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Critical Values
Left-tailed test
“<”
Needs one critical value (left tail)
Use StatCrunch: Chi-Squared Calculator
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Critical Values
Two-tailed test
“≠“
Needs two critical values (right and left tail)
Use StatCrunch: Chi-Squared Calculator
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Example 1
Problem 14, pg 449
Statisics Test Scores
Tests scores in the author’s previous statistic classes have
followed a normal distribution with a standard deviation
equal to 14.1. His current class has 27 tests scores with a
standard deviation of 9.3.
Use a 0.01 significance level to test the claim that this
class has less variation than the past classes.
What we know:
σ0 = 14.1
n = 27
Claim: σ < 14.1
s = 9.3
using
α = 0.01
Note: Test conditions are satisfied since population is normally distributed
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Example 1 Using Critical Regions
What we know:
σ0 = 14.1 n = 27 s = 9.3
Claim: σ < 14.1 using α = 0.01
H0 : σ = 14.1
H1 : σ < 14.1 Left-tailed
Test statistic:
c2L = 12.20
Critical value:
(df = 26)
c2 = 11.31
c2 in critical region
Initial Conclusion: Since c2 in critical region, Reject H0
Final Conclusion: Accept the claim that the new class has
less variance than the past classes
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Calculating P-value for a Variance
Stat → Variance → One sample → with summary
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Calculating P-value for a Variance
Enter the
Sample variance (s2)
Sample size (n)
NOTE: Must use Variance
s2 = 9.32 = 86.49
Then hit Next
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Calculating P-value for a Variance
Select
Enter the
Select
Hypothesis Test
Null:variance (σ02)
Alternative (“<“, “>”, or “≠”)
σ02 = 14.12 = 198.81
Then hit Calculate
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Calculating P-value for a Variance
The resulting table shows both the
test statistic (c2) and the P-value
Test statistic (c2)
P-value
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Example 1 Using Critical Regions
What we know:
H0 : σ = 14.1
H1 : σ < 14.1
Using
StatCrunch
s2 = 86.49
σ02 = 198.81
σ0 = 14.1 n = 27 s = 9.3
Claim: σ < 14.1 using α = 0.01
Stat → Variance→ One sample → With summary
Sample variance:
Sample size:
86.49
27
● Hypothesis Test
Null: proportion=
Alternative
198.81
<
P-value = 0.0056
Initial Conclusion: Since P-value < α (α = 0.01), Reject H0
Final Conclusion: Accept the claim that the new class has
less variance than the past classes
We are 99.44% confident the claim holds
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Example 2
Problem 17, pg 449
BMI for Miss America
Listed below are body mass indexes (BMI) for recent
Miss America winners. In the 1920s and 1930s,
distribution of the BMIs formed a normal distribution
with a standard deviation of 1.34.
Use a 0.01 significance level to test the claim that recent
Miss America winners appear to have variation that is
different from that of the 1920s and 1930s.
19.5 20.3 19.6 20.2 17.8 17.9 19.1 18.8 17.6 16.8
Using StatCrunch: s = 1.1862172
What we know:
σ0 = 1.34
n = 10
Claim: σ ≠ 1.34
s = 1.186
using
α = 0.01
Note: Test conditions are satisfied since population is normally distributed
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Example 2 Using Critical Regions
What we know:
σ0 = 1.34
n = 10
Claim: σ ≠ 1.34
s = 1.186
using
α = 0.01
H0 : σ = 1.34
H1 : σ ≠ 1.34 two-tailed
0.005
Test statistic:
c2L = 2.088
Critical values:
(df = 26)
c2R = 26.67
c2 = 7.053
c2 not in critical region
Initial Conclusion: Since c2 not in critical region, Accept H0
Final Conclusion: Reject the claim recent winners have a
different variations than in the 20s and 30s
Since H0 accepted, the observed significance isn’t useful.
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Example 2 Using P-value
What we know:
σ0 = 1.34
n = 10
Claim: σ ≠ 1.34
H0 : σ2 = 1.796
H1 :
σ2
< 1.796
Using
StatCrunch
s2 = 1.407
σ02 = 1.796
s = 1.186
using
α = 0.01
Stat → Variation → One sample → With summary
Sample variance:
Sample size:
1.407
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● Hypothesis Test
Null: proportion=
1.796
Alternative
<
P-value = 0.509
Initial Conclusion: Since P-value ≥ α (α = 0.01), Accept H0
Final Conclusion: Reject the claim recent winners have a
different variations than in the 20s and 30s
Since H0 accepted, the observed significance isn’t useful.
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