Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the population mean µ?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the population mean µ?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the population mean µ?

Confidence Intervals for Normal Distribution

TheoremLet X1,X2, . . . ,Xn be a random sample from a normal distributionwith mean µ and variance σ2, where µ and σ are unknown. Therandom variable

T =X − µS/√

n

has a probability distribution called a t distribution with

n − 1 degrees of freedom (df). Here X is the sample meanand S is the sample standard deviation.

Confidence Intervals for Normal Distribution

TheoremLet X1,X2, . . . ,Xn be a random sample from a normal distributionwith mean µ and variance σ2, where µ and σ are unknown. Therandom variable

T =X − µS/√

n

has a probability distribution called a t distribution with

n − 1 degrees of freedom (df). Here X is the sample meanand S is the sample standard deviation.

Confidence Intervals for Normal Distribution

Confidence Intervals for Normal Distribution

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

Properties of t Distributions:

Let tν denote the density function curve for ν df.

1. tν is governed by only one parameter ν, the number ofdegrees of freedom.

2. Each tν curve is bell-shaped and centered at 0.

3. Each tν curve is more spread out than the standard normal(z) curve.

4. As ν increases, the spread of the corresponding tν curvedecreases.

5. As ν →∞, the sequence of tν curves approaches the standardnormal curve (so the z curve is often called the t curve withdf=∞).

Confidence Intervals for Normal Distribution

NotationLet tα,ν = the number on the measurement axis for which the areaunder the t curve with ν df to the right of tα,ν is α; tα,ν is called at critical value.

Confidence Intervals for Normal Distribution

NotationLet tα,ν = the number on the measurement axis for which the areaunder the t curve with ν df to the right of tα,ν is α; tα,ν is called at critical value.

Confidence Intervals for Normal Distribution

NotationLet tα,ν = the number on the measurement axis for which the areaunder the t curve with ν df to the right of tα,ν is α; tα,ν is called at critical value.

Confidence Intervals for Normal Distribution

Proposition

Let x̄ and s be the sample mean and sample standard deviationcomputed from the results of a random sample from a normalpopulation with mean µ. Then a 100(1− α)% confidenceinterval for µ is(

x̄ − t α2,n−1 ·

s√n, x̄ + t α

2,n−1 ·

s√n

)or, more compactly, x̄ ± t α

2,n−1 · s√

n.

An upper confidence bound for µ is

x̄ + tα,n−1 ·s√n

and replacing + by − in this latter expression gives a lowerconfidence bound for µ, both with confidence level 100(1− α)%.

Confidence Intervals for Normal Distribution

Proposition

Let x̄ and s be the sample mean and sample standard deviationcomputed from the results of a random sample from a normalpopulation with mean µ. Then a 100(1− α)% confidenceinterval for µ is(

x̄ − t α2,n−1 ·

s√n, x̄ + t α

2,n−1 ·

s√n

)or, more compactly, x̄ ± t α

2,n−1 · s√

n.

An upper confidence bound for µ is

x̄ + tα,n−1 ·s√n

and replacing + by − in this latter expression gives a lowerconfidence bound for µ, both with confidence level 100(1− α)%.

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the 11th component?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the 11th component?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval for the 11th component?

Confidence Intervals for Normal Distribution

Proposition

A prediction interval (PI) for a single observation to be selectedfrom a normal population distribution is

x̄ ± t α2,n−1 · s

√1 +

1

n

The prediction level is 100(1− α)%.

Confidence Intervals for Normal Distribution

Proposition

A prediction interval (PI) for a single observation to be selectedfrom a normal population distribution is

x̄ ± t α2,n−1 · s

√1 +

1

n

The prediction level is 100(1− α)%.

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval such that at least 90% of thevalues in the population are inside this interval?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval such that at least 90% of thevalues in the population are inside this interval?

Confidence Intervals for Normal Distribution

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is the 95% confidence interval such that at least 90% of thevalues in the population are inside this interval?

Confidence Intervals for Normal Distribution

Proposition

A tolerance interval for capturing at least k% of the values in anormal population distribution with a confidence level 95%has theform

x̄ ± (tolerance critical value) · s

The tolerance critical values for k = 90, 95, and 99 in combinationwith various sample sizes are given in Appendix Table A.6.

Confidence Intervals for Normal Distribution

Proposition

A tolerance interval for capturing at least k% of the values in anormal population distribution with a confidence level 95%has theform

x̄ ± (tolerance critical value) · s

The tolerance critical values for k = 90, 95, and 99 in combinationwith various sample sizes are given in Appendix Table A.6.

Confidence Intervals for Normal Distribution

Proposition

A tolerance interval for capturing at least k% of the values in anormal population distribution with a confidence level 95%has theform

x̄ ± (tolerance critical value) · s

The tolerance critical values for k = 90, 95, and 99 in combinationwith various sample sizes are given in Appendix Table A.6.

Confidence Intervals for the Variance of a NormalPopulation

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is a 95% confidence for the population variance σ2?

Confidence Intervals for the Variance of a NormalPopulation

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is a 95% confidence for the population variance σ2?

Confidence Intervals for the Variance of a NormalPopulation

Example (a variant of Problem 62, Ch5)The total time for manufacturing a certain component is known tohave a normal distribution. However, the mean µ and variance σ2

for the normal distribution are unknown. After an experiment inwhich we manufactured 10 components, we recorded the sampletime which is given as follows:

1 2 3 4 5time 63.8 60.5 65.3 65.7 61.9

6 7 8 9 10time 68.2 68.1 64.8 65.8 65.4

with

X = 64.95, s = 2.42

What is a 95% confidence for the population variance σ2?

Confidence Intervals for the Variance of a NormalPopulation

TheoremLet X1,X2, . . . ,Xn be a random sample from a distribution withmean µ and variance σ2. Then the random variable

(n − 1)S2

σ2=

∑(Xi − X )2

σ2

has s chi-squared (χ2) probability distribution with n − 1 degreesof freedom (df).

Confidence Intervals for the Variance of a NormalPopulation

TheoremLet X1,X2, . . . ,Xn be a random sample from a distribution withmean µ and variance σ2. Then the random variable

(n − 1)S2

σ2=

∑(Xi − X )2

σ2

has s chi-squared (χ2) probability distribution with n − 1 degreesof freedom (df).

Confidence Intervals for the Variance of a NormalPopulation

Confidence Intervals for the Variance of a NormalPopulation

Confidence Intervals for the Variance of a NormalPopulation

NotationLet χ2

α,ν , called a chi-squared critical value, denote the numberon the measurement axis such that α of the area under thechi-squared curve with ν df lies to the right of χ2

α,ν .

Confidence Intervals for the Variance of a NormalPopulation

NotationLet χ2

α,ν , called a chi-squared critical value, denote the numberon the measurement axis such that α of the area under thechi-squared curve with ν df lies to the right of χ2

α,ν .

Confidence Intervals for the Variance of a NormalPopulation

NotationLet χ2

α,ν , called a chi-squared critical value, denote the numberon the measurement axis such that α of the area under thechi-squared curve with ν df lies to the right of χ2

α,ν .

Confidence Intervals for the Variance of a NormalPopulation

Proposition

A 100(1− α)% confidence interval for the variance σ2 of anormal population has lower limit

(n − 1)s2/χ2α2,n−1

and upper limit(n − 1)s2/χ2

1−α2,n−1

A confidence interval for σ has lower and upper limits that arethe square roots of the corresponding limits in the interval for σ2.

Confidence Intervals for the Variance of a NormalPopulation

Proposition

A 100(1− α)% confidence interval for the variance σ2 of anormal population has lower limit

(n − 1)s2/χ2α2,n−1

and upper limit(n − 1)s2/χ2

1−α2,n−1

A confidence interval for σ has lower and upper limits that arethe square roots of the corresponding limits in the interval for σ2.