Confidence Interval Module - faculty.nps.edufaculty.nps.edu/rdfricke/OA3102/Interval Estimation -...

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Module 5: Interval Estimation Statistics (OA3102)

Professor Ron Fricker Naval Postgraduate School

Monterey, California

Reading assignment:

WM&S chapter 8.5-8.9

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Goals for this Module

• Interval estimation – i.e., confidence intervals

– Terminology

– Pivotal method for creating confidence intervals

• Types of intervals

– Large-sample confidence intervals

– One-sided vs. two-sided intervals

– Small-sample confidence intervals for the mean,

differences in two means

– Confidence interval for the variance

• Sample size calculations

Interval Estimation

• Instead of estimating a parameter with a

single number, estimate it with an interval

• Ideally, interval will have two properties:

– It will contain the target parameter q

– It will be relatively narrow

• But, as we will see, since interval endpoints

are a function of the data,

– They will be variable

– So we cannot be sure q will fall in the interval

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Objective for Interval Estimation

• So, we can’t be sure that the interval

contains q, but we will be able to

calculate the probability the interval

contains q

• Interval estimation objective: Find an

interval estimator capable of generating

narrow intervals with a high probability

of enclosing q

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Why Interval Estimation?

• As before, we want to use a sample to infer

something about a larger population

• However, samples are variable

– We’d get different values with each new sample

– So our point estimates are variable

• Point estimates do not give any information about

how far off we might be (precision)

• Interval estimation helps us do inference in such a

way that:

– We can know how precise our estimates are, and

– We can define the probability we are right

Terminology

• Interval estimators are commonly called

confidence intervals

• Interval endpoints are called the upper

and lower confidence limits

• The probability the interval will enclose

q is called the confidence coefficient or

confidence level – Notation: 1-a or 100(1-a)%

– Usually referred to as “100(1-a)” percent CIs Revision: 1-12 6

• Via the CLT, we know that is within 2 std

errors ( ) of m 95% of the time

(Unobserved) population distribution (pdf of Y)

(Unobserved) sampling distribution of the mean

(Unobserved) mY

7

Y

Confidence Intervals: The Main Idea

Y n

y

2Y Y nm

95% confidence

interval for mY

• So, m must be within 2 SEs of 95% of the time Y

• A two-sided confidence interval:

• A lower one-sided confidence interval:

• An upper one-sided confidence interval:

Upper confidence

limit

In General

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ˆ ˆPr 1L Uq q q a

Confidence

coefficient

Target

parameter

Lower confidence

limit

ˆPr 1Lq q a

ˆPr 1Uq q a

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Pivotal Method: A Strategy

for Constructing CIs

• Pivotal method approach

– Find a “pivotal quantity” that has following two

characteristics:

• It is a function of the sample data and q, where

q is the only unknown quantity

• Probability distribution of pivotal quantity does

not depend on q (and you know what it is)

• Now, write down an appropriate probability

statement for the pivotal quantity and then

rearrange terms…

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Example: Constructing a

95% CI for m, known (1)

• Let Y1, Y2, …, Yn be a random sample from a

normal population with unknown mean mY and

known standard deviation Y

• Create a CI for mY based on the sampling

distribution of the mean:

• To start, we know that (via standardizing):

~ (0,1)

/

Y

Y

YN

n

m

2~ , /Y YY N nm

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• Now for Z ~ N(0,1) we know

– That is, there is a 95% probability that the random

variable Z lies in this fixed interval

• Thus

• So, let’s derive a 95% confidence interval…

Pr( 1.96 1.96) 0.95Z

-Pr -1.96 1.96 0.95

/

Y

Y

Y

n

m

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-Pr -1.96 1.96 0.95

/

Y

Y

Y

n

m

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• So, If Y1 = y1, Y2 = y2, …, Yn = yn are observed

values of a random sample from a

with known, then

• We can be 95% confident that the interval

covers the population mean

– Interpretation: In the long run, 19 times out of 20

the interval will cover the true mean and 1 time out

of 20 it will not

1.96 YYy

n

mis a 95% confidence interval for

2,mN

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Calculating a Specific CI

• Consider an experiment with sample size

n=40, and Y=0.1

• Calculate a 95% confidence interval for mY

5.426y

Example 8.4

• Suppose we obtain a single observation Y

from an exponential distribution with mean q.

Use Y to form a confidence interval for q with

confidence level 0.9.

• Solution:

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Example 8.4 (continued)

Example 8.5

• Suppose we take a sample of size n=1 from a

uniform distribution on [0,q ], were q is

unknown. Find a 95% lower confidence

bound for q.

• Solution:

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Large-Sample Confidence Intervals

• If is an unbiased statistic, then via the CLT

has an approximate standard normal

distribution for large samples

• So, use it as an (approximate) pivotal quantity

to develop (approximate) confidence intervals

for q

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ˆ

ˆZ

q

q q

q̂

Example 8.6

• Let . Find a confidence interval

for q with confidence level 1-a.

• Solution:

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ˆˆ ~ ( , )N

qq q

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One-Sided Limits

• Similarly, we can determine the 100(1-a)%

one-sided confidence limits (aka confidence

bounds):

–

–

• What if you use both bounds to construct a

two-sided confidence interval?

– Each bound has confidence level 1-a, so resulting

interval has a 1-2a confidence level

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ˆˆ100(1 )% za q

a q q lower bound for

ˆˆ100(1 )% za q

a q q upper bound for

Example 8.7

• The shopping times of n=64 randomly

selected customers were recorded with

minutes and . Estimate m, the true

average shopping time per customer with

confidence level 0.9.

• Solution:

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33y 2 256ys

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Example 8.8

• Two brands of refrigerators, A and B, are

each guaranteed for a year. Out of a random

sample of nA=50 refrigerators, 12 failed before

one year. And out of an independent random

sample of nB=60 refrigerators, 12 failed before

one year. Give a 98% CI for pA-pB.

• Solution

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What is a Confidence Interval?

• Before collecting data and calculating it, a confidence

interval is a random interval

– Random because it is a function of a random variable (e.g., )

• The confidence level is the long-run percentage of

intervals that will “cover” the population parameter

– It is not the probability a particular interval contains the

parameter!

• This statement implies that the parameter is random

• After collecting the data and calculating the CI

the interval is fixed

– It then contains the parameter with probability 0 or 1

Y

A CI Simulation

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• Simulated 20 95%


with samples of size

n=10 drawn from

N(40,1) distribution

• One failed to cover

the true (unknown)

parameter, which is

what is expected on

average

Another CI Simulation

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• Simulated 100 95%


with samples of size

n=10 drawn from

N(40,1) distribution

• 6 failed to cover the

true (unknown)

parameter

– Close to the

expected number: 5

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Illustrating Confidence Intervals

This is a demonstration showing confidence

intervals for a proportion.

Applets created by Prof Gary McClelland, University of Colorado, Boulder

You can access them at

www.thomsonedu.com/statistics/book_content/0495110817_wackerly/applets/seeingstats/index.html

TO DEMO

C:/Users/RonFricker/Desktop/My Documents/Teaching-Courses/NPS/Resources/Wackerly/Media/Applets/chapter17/seeingstats/Chpt8/confIntervalP.html

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Summary: Constructing a Two-sided

Large-Sample Confidence Interval

• For an unbiased statistic , determine

• Choose the confidence level: 1-a

• Find

– E.g., for a = 0.05,

• Given data, calculate and

• Then the 100(1-a)% confidence interval for q is

ˆ ˆ/2 /2

ˆ ˆ,z za aq qq q

0.025 1.96z /2za

q̂q̂

q̂q̂

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E.g., Constructing a Two-sided

Large-Sample 95% CI for m

• is an unbiased estimator for m, and we

know

The confidence level is 1-a = 0.95

• So

• Given data, calculate and the 95% CI for m

is

Y

/2 0.025 1.96z za

YYn

y

1.96 , 1.96Y Yy n y n

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E.g., Constructing a Two-sided

Large-Sample 95% CI for p

• For Y, the number of successes out of n trials,

an unbiased estimator for p is

• Then note that

– Follows from:

– And, since we don’t know p,

• As before, for a confidence level of 1-a =

0.95,

• So, the 95% CI for m is

/2 0.025 1.96z za

ˆ ˆ ˆ ˆ ˆ ˆ1.96 1 , 1.96 1p p p n p p p n

ˆ /p Y n

ˆ (1 ) /p p p n

ˆˆ ˆ ˆ(1 ) /p p p n

2 2Var( / ) Var( ) / (1 ) /Y n Y n np p n

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How Confidence Intervals Behave

• Width of CI’s:

• Margin of error:

– Bigger s.d. bigger s.e. wider intervals

– Bigger sample size smaller s.e. narrower

intervals

– Higher confidence bigger z-values wider

intervals

/22 Yw zn

a

/2YE zn

a

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Sample Size Calculations

• Often desire to determine necessary sample

size to achieve a particular error of estimation

– Must specify the estimation error B and know or

well estimate the population standard deviation

• Then for a 100(1-a)% two-sided CI solve

for n:

/2B zn

a

2

/2zn

w

a

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Example

• We want to estimate the average daily yield m

of a chemical, where we know =21 tons

• Find the sample size (n) so that a 95% CI for

m has an error of estimation to be less than

B=5 tons

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Example 8.9

• A stimulus reaction may take two forms: A or

B. If we want to estimate the probability the

reaction will be A, what sample size do we

need if

– We want the error of estimation less than 0.04

– The probability p is likely to be near 0.6

– And we plan to use a confidence level of 90%

• Solution:

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Example 8.10

• We’re going to compare the effectiveness of

two types of training (for an assembly op)

– Subjects to be divided into 2 equally sized groups

– Measurement range expected to be about 8 mins

– Estimate mean difference in assembly time to

within 1 minute with 95% confidence

• Solution:

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Small-Sample Confidence

Interval for m ( Unknown)

• For small n and unknown, standardized statistic no longer normally distributed

• But, if is the mean of a random sample of size n from a distribution with mean m,

has a t distribution with n-1 degrees of freedom – Precisely if population has normal distribution

• See Theorems 7.1 & 7.3 and Definition 7.2

– Approximately for sample mean via CLT

1/

n

YT

s n

m

Y

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Very Similar to Confidence

Interval for m with Known

• So, we can use the t distribution to build a CI!

• Deriving using T as the pivotal quantity:

/2, 1 1 /2, 1 /2, 1 /2, 1

/2, 1 /2, 1

/2, 1 /2, 1

Pr Pr/

Pr / /

Pr / /

n n n n n

n n

n n

Yt T t t t

s n

t s n Y t s n

Y t s n Y t s n

a a a a

a a

a a

m

m

m

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So, Constructing a 95% Confidence

Interval for m (with Unknown)

• Choose the confidence level: 1-a

• Remember the degrees of freedom () = n -1

• Find

– Example: if a = 0.05, df=7 then = 2.365

• Calculate and

• Then the 95% confidence interval for m is

y ns /

2.365 , 2.365s s

y yn n

1,2/ nta

7,025.0t

Remember, this value also depends on the dfs

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Example 8.11

• A manufacturer of gunpowder has developed

a new powder. Eight tests gave the following

muzzle velocities in feet per second:

3,005 2,925 2,935 2,965

2,995 3,005 2,937 2,905

Find a 95% CI for the true average velocity m

• Solution:

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Interval for m1-m2

• Suppose we want to compare the means of two normally distributed populations

– Population 1:

– Population 2:

• Then

• Can use this as a pivotal quantity

1 2 1 2

2 2

1 2

1 2

~ (0,1)Y Y

Z N

n n

m m

2

1 1mean , variance m 2

2 2mean , variance m

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Interval for m1-m2 , continued

• If we can further assume that , then

• But if is unknown, then need to appropriately estimate it

• To do so, first estimate the two sample means

1 2 1 2

1 2

~ (0,1)1 1

Y YZ N

n n

m m

2 2 2

1 2

1

1 1

11

1 n

i

i

Y Yn

2

2 2

12

1 n

i

i

Y Yn

• Then, the pooled estimate of variance:

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1 22 2

1 1 2 22 1 1

1 2

( ) ( )

2

n n

i ii ip

y y y ys

n n

Sample mean for population Y1

Sample mean for population Y2

• Can also express as a weighted average of

and :

2

1s2

2s

Average squared deviation from different means

Pooled Estimate of the Variance

2 22 1 1 2 2

1 2

( 1) ( 1)

2p

n s n ss

n n

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Interval for m1-m2 , continued

• So, assuming , we have

2

1 2 1 2 1 2

2

1 21 2

1 2 1 2

1

1 2

2

2/ 1 1

~1 1

p

n

p

Y Y n n SZ

n nW n n

Y YT

Sn n

m m

m m

2 2 2

1 2

Example 8.12

• Lengths of time for two groups of employees

to assemble a device:

– Standard: Employees received standard training

– New: Employees received a new type of training

• Estimate the true mean difference in training

(m1-m2) with 95% confidence

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Training

Type

Time to Assemble

Measurements

Standard 32 37 35 28 41 44 35 31 34

New 35 31 29 25 34 40 27 32 31

Example 8.12 Solution

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CI for the Variance

• Let X1, X2, …, Xn be a random sample from a

normal population with mean m and standard

deviation

• Consider the the pivotal quantity

• Then a confidence interval for the variance is:

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22 2

1 /2, 1 /2, 12

( 1)Pr 1n n

n Sa a a

2 22

2 2

/2, 1 1 /2, 1

( 1) ( 1)Pr 1

n n

n S n S

a a

a

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Example: 95% CI for Variance

• After observing s2 = 25.4 for n=20 obs, calculate a

95% CI for 2

– For =19, chi-squared critical values are 8.906 and 32.852

– So:

• Remember, the distribution is not symmetric, so be

careful with a and a

– Lower limit divides by the bigger critical value

2 22

2 2

/2, 1 1 /2, 1

2

( 1) ( 1)Pr 1

19 25.4 19 25.4or, 0.95

32.852 8.906

Thu s, the 95% CI [14.69, 54.19

n n

n s n s

a a

a

Example 8.13

• We want to assess the variability of a

measuring methodology. Three independent

measurements are taken: 4.1, 5.2, and 10.2.

Estimate 2 with confidence level 90%.

• Solution:

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Why Calculate CIs for ?

• Just like with m, is a population parameter – Sometimes need to know how well it is estimated

by s

• E.g., the precision of a weapon is inversely proportional to its standard deviation – if the standard deviation is large, the weapon is not precise – Confidence intervals for provide information

about the likely range of the impact error

– Big difference between a of 3 meters and a of 300 meters with implications for both collateral damage and friendly troops

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Bootstrap Confidence Intervals

• Can use the bootstrap method to estimate


• Basic idea:

– Use bootstrap methodology to create an empirical

sampling distribution for statistic of interest

– Then take the appropriate quantiles of the

empirical distribution for upper and lower end-

points of confidence interval

• As with point estimation, useful when it’s hard

to analytically specify sampling distribution

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Caution! Confidence Intervals

are Not for Prediction

• CI is an interval estimate for the population parameter

• CIs do not predict the likely range of the next observation - common pitfall!

• Interval for next observation is called a prediction interval

• Prediction interval has variability of original random variable plus the uncertainty about the population parameter

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• Interval estimation – i.e., confidence intervals

– Terminology

– Pivotal method for creating confidence intervals

• Types of intervals

– Large-sample confidence intervals

– One-sided vs. two-sided intervals

– Small-sample confidence intervals for the mean,

differences in two means

– Confidence interval for the variance

• Sample size calculations

What We Covered in this Module

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Homework

• WM&S chapter 8.5-8.9 – Required exercises: 40, 41, 42, 60, 63, 64, 71,

82, 91, 96

– Extra credit: 94

• Useful hints: Problems 8.91 and 8.96: Here’s you’re given the

raw data and must calculate the necessary

statistics first

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