Confidence Interval for the Variance of a Normal Population Bernd - - PowerPoint PPT Presentation

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Confidence Interval for the Variance of a Normal Population Bernd - - PowerPoint PPT Presentation

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2 -Distribution Confidence Interval for the Variance Confidence Interval for the Variance of a Normal Population Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Confidence Interval

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SLIDE 1

logo1 χ2-Distribution Confidence Interval for the Variance

Confidence Interval for the Variance of a Normal Population

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 2

logo1 χ2-Distribution Confidence Interval for the Variance

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 3

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 4

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 5

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ, then the random variable

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 6

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ, then the random variable (n−1)S2 σ2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 7

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ, then the random variable (n−1)S2 σ2 has a chi-squared distribution (χ2-distribution) with n−1 degrees of freedom.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 8

logo1 χ2-Distribution Confidence Interval for the Variance

  • Theorem. The distribution of the sample variances S2 of

samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ, then the random variable (n−1)S2 σ2 has a chi-squared distribution (χ2-distribution) with n−1 degrees of freedom. Remember that the χ2-distribution is a Gamma distribution with α = n 2 and β = 2.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 9

logo1 χ2-Distribution Confidence Interval for the Variance

Definition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 10

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. The function

fn(x) :=

  • 1

Γ( n

2)2 n 2 x n 2−1e− x 2;

for x > 0, 0;

  • therwise,

is the density of the χ2-distribution with n degrees of freedom.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 11

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. The function

fn(x) :=

  • 1

Γ( n

2)2 n 2 x n 2−1e− x 2;

for x > 0, 0;

  • therwise,

is the density of the χ2-distribution with n degrees of freedom. Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 12

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. The function

fn(x) :=

  • 1

Γ( n

2)2 n 2 x n 2−1e− x 2;

for x > 0, 0;

  • therwise,

is the density of the χ2-distribution with n degrees of freedom.

  • Theorem. Let Cn be a random variable with a χ2-distribution

with n degrees of freedom.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 13

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. The function

fn(x) :=

  • 1

Γ( n

2)2 n 2 x n 2−1e− x 2;

for x > 0, 0;

  • therwise,

is the density of the χ2-distribution with n degrees of freedom.

  • Theorem. Let Cn be a random variable with a χ2-distribution

with n degrees of freedom. Then E(Cn) = n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-14
SLIDE 14

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. The function

fn(x) :=

  • 1

Γ( n

2)2 n 2 x n 2−1e− x 2;

for x > 0, 0;

  • therwise,

is the density of the χ2-distribution with n degrees of freedom.

  • Theorem. Let Cn be a random variable with a χ2-distribution

with n degrees of freedom. Then E(Cn) = n and V(Cn) = 2n.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 15

logo1 χ2-Distribution Confidence Interval for the Variance

Definition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 16

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 17

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 18

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-19
SLIDE 19

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 20

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 21

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 22

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 23

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 24

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 25

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l)

  • l-α

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 26

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l)

  • l-α

α 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 27

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. For 0 < α < 1, the number χ2

α(n) is the unique real

number such that the area that is to the right of χ2

α(n) under the

density of the χ2-distribution with n degrees of freedom is α.

✲ ✻ α 2 χ2

α 2 (n-l)

  • l-α

α 2 χ2

l- α

2 (n-l) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 28

logo1 χ2-Distribution Confidence Interval for the Variance

1−α = P

  • χ2

1− α

2 (n−1) < (n−1)S2

σ2 < χ2

α 2 (n−1)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 29

logo1 χ2-Distribution Confidence Interval for the Variance

1−α = P

  • χ2

1− α

2 (n−1) < (n−1)S2

σ2 < χ2

α 2 (n−1)

  • =

P χ2

1− α

2 (n−1)

(n−1)S2 < 1 σ2 < χ2

α 2 (n−1)

(n−1)S2

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

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SLIDE 30

logo1 χ2-Distribution Confidence Interval for the Variance

1−α = P

  • χ2

1− α

2 (n−1) < (n−1)S2

σ2 < χ2

α 2 (n−1)

  • =

P χ2

1− α

2 (n−1)

(n−1)S2 < 1 σ2 < χ2

α 2 (n−1)

(n−1)S2

  • =

P

  • (n−1)S2

χ2

1− α

2 (n−1) > σ2 > (n−1)S2

χ2

α 2 (n−1)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-31
SLIDE 31

logo1 χ2-Distribution Confidence Interval for the Variance

1−α = P

  • χ2

1− α

2 (n−1) < (n−1)S2

σ2 < χ2

α 2 (n−1)

  • =

P χ2

1− α

2 (n−1)

(n−1)S2 < 1 σ2 < χ2

α 2 (n−1)

(n−1)S2

  • =

P

  • (n−1)S2

χ2

1− α

2 (n−1) > σ2 > (n−1)S2

χ2

α 2 (n−1)

  • =

P

  • (n−1)S2

χ2

α 2 (n−1) < σ2 <

(n−1)S2 χ2

1− α

2 (n−1)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-32
SLIDE 32

logo1 χ2-Distribution Confidence Interval for the Variance

Definition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-33
SLIDE 33

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. If a random sample x1,...,xn is taken from a

normal distribution

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-34
SLIDE 34

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. If a random sample x1,...,xn is taken from a

normal distribution, then the interval

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-35
SLIDE 35

logo1 χ2-Distribution Confidence Interval for the Variance

  • Definition. If a random sample x1,...,xn is taken from a

normal distribution, then the interval

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • is a confidence interval for the variance σ2 with confidence

coefficient 1−α.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-36
SLIDE 36

logo1 χ2-Distribution Confidence Interval for the Variance

Example.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-37
SLIDE 37

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-38
SLIDE 38

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-39
SLIDE 39

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-40
SLIDE 40

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-41
SLIDE 41

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-42
SLIDE 42

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-43
SLIDE 43

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-44
SLIDE 44

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-45
SLIDE 45

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-46
SLIDE 46

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-47
SLIDE 47

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • ≈

19·0.01 floz2 32.852 , 19·0.01 floz2 8.906

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-48
SLIDE 48

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • ≈

19·0.01 floz2 32.852 , 19·0.01 floz2 8.906

  • ≈
  • 0.0058 floz2,0.0213 floz2

.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-49
SLIDE 49

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • ≈

19·0.01 floz2 32.852 , 19·0.01 floz2 8.906

  • ≈
  • 0.0058 floz2,0.0213 floz2

.

Confidence interval for the standard deviation:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-50
SLIDE 50

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • ≈

19·0.01 floz2 32.852 , 19·0.01 floz2 8.906

  • ≈
  • 0.0058 floz2,0.0213 floz2

.

Confidence interval for the standard deviation:

  • 0.0058 floz2,
  • 0.0213 floz2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

slide-51
SLIDE 51

logo1 χ2-Distribution Confidence Interval for the Variance

  • Example. The bottlers of the new soft drink “Guzzle” are

experiencing problems with the filling mechanism for their 16 floz bottles. To estimate the standard deviation of the fill volume, the filled volume for 20 bottles was measured, yielding a sample standard deviation of 0.1 floz. Compute a 95% confidence interval for the standard deviation σ. Assuming normality, n = 20, s2 = 0.01 floz2, α = 0.05, χ2

0.025(19) ≈ 32.852, χ2 0.975(19) ≈ 8.906, confidence interval

for the variance:

  • (n−1)s2

χ2

α 2 (n−1),

(n−1)s2 χ2

1− α

2 (n−1)

  • ≈

19·0.01 floz2 32.852 , 19·0.01 floz2 8.906

  • ≈
  • 0.0058 floz2,0.0213 floz2

.

Confidence interval for the standard deviation:

  • 0.0058 floz2,
  • 0.0213 floz2

≈ (0.076 floz,0.146 floz).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population