L8 Estimate 2014

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Chapter 8

Confidence Interval

Estimation

David Chow

Oct 2014

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Learning Objectives

To construct and interpret confidence interval estimates for the mean and the proportion.

To determine the necessary sample size for a confidence interval.

Section 8.5: Applications in Auditing (NOT covered)

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Basic Concepts

A point estimate is a single number

Eg: For the population mean (), a point estimate is ____

A confidence interval is an interval estimate. It provides additional information about variability.

Eg: Giant pandas mean age = 22 yrs old

Point Estimate

Lower Confidence

Limit

Upper Confidence

Limit

Width of confidence interval

Eg: According to a survey, the 95% confidence interval mean wage of private tutoring is between $110 to $150 per hour

I.e., = $130 20

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Basic Concepts

The general formula for all confidence intervals (C.I.) is:

Critical values (Z) are related to the level of confidence (1- , also called confidence level).

Eg: 95% confidence: (1 - ) = 0.95, or = 5%.

With a given , critical values can be obtained from the Z-table.

Then, a C.I. can be computed:

Eg: 95% confidence: (1 - ) = 0.95, or = 5%.

Point Estimate Margin of Error, where

Margin of Error (e) = (Critical Value) x (Standard Error)

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Remarks

This chapter focuses on two parameters, and

Lets start with the easiest case: estimating with a known population standard deviation ()

A more realistic case ( unknown) follows

Concepts versus Computation:

As always, statistic concepts can be a bit abstract at first, but computations have standard steps to follow

We will work out a few examples, master the computations first, then go back to think about the rationale and interpretation behind your math

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Estimating ( Known)

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Confidence Interval for ( Known)

Assume population standard deviation is known.

Also assume n is large enough (n > 30), or the population

is normally distributed. Such assumptions ensure ____.

A two-tailed confidence interval estimate:

Z, also written as Z/2, is the standardized normal distribution

critical value for a probability of ____ in each tail.

n

ZX

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Critical Values of Z

Consider a 95% confidence interval:

Z= -1.96 Z= 1.96

.951

.0252

.025

2

Lower Confidence Limit

Upper Confidence Limit Point Estimate

0

Find the critical values

Z0.05 and Z0.005.

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Eg: Length of A4 Paper

A paper producer wants to check if the

paper produced has the correct mean length

of 11 inches

Find the 95% confidence interval of the

population mean paper length based on a

sample of 100, sample meanx = 10.998 in

is known to be 0.02 in

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Eg: Length of A4 Paper

The 95% confidence interval is given by:

=x Z/2 x

Step 1: Find Z0.025 = 1.96

Step 2: Z/2 x = 1.96 (0.02)/10 = 0.00392

The required confidence interval is:

= 10.998 0.00392 inches, or

10.99408 < < 11.00192

Find the 99% interval. What is the effect of raising the confidence level?

Eg: Mean Resistance

A sample of 11 circuits

from a large normal

population has a mean

resistance of 2.20 ohms.

Past testing shows that

the population standard

deviation is 0.35 ohms.

Determine a 95%

confidence interval for

the true mean resistance

of the population.

2.4068) , (1.9932

.2068 2.20

)11(.35/ 1.96 2.20

n

025.0

ZX

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We are 95% confident that the true

mean resistance is between 1.9932

and 2.4068 ohms

I.e., 95% of intervals formed in this

manner will contain the true

population mean.

Is it correct to use the Z-distribution?

ANSWER

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Recap: Choosing Confidence Level

A bigger confidence level raises

the confidence (of the interval

containing the true mean)

But a wider interval estimate also

means ____ precision

95% is the most common choice

It provides a good balance between

precision and confidence

Example: Body Temperature

n = 106,x = 98.20F, = 0.62F

1. Find the 95% confidence interval

2. How to obtain a narrower interval estimate?

1. Margin of error = ____ = 0.12

CI: 98.08 to 98.32

2. Smaller sigma, bigger n, or smaller (1-alpha)

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ANSWER

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, Confidence Intervals and Sampling Distribution

x

Confidence Intervals

Intervals:

to (1-) x 100% of intervals constructed

contain ;

() x 100% do not.

Sampling Distribution

n

ZX

n

ZX

x

x1

x2

/2 /21

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Interpreting Confidence Level

Suppose we select many different samples of

size n from a population.

A 95% confidence interval is constructed for

each sample.

Then 95% of those interval estimates would

actually contain the true value of .

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Estimating ( Unknown)

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Confidence Interval for ( Unknown)

Usually is unknown

Use sample standard deviation S instead

This will introduce extra uncertainty

because S varies from sample to sample

So another distribution (the t distribution) is used

It is flatter than the standard normal distribution

The t distribution requires that the original population is normally distributed

This is assumed in most cases

Strictly speaking, this assumption should be checked at first

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Confidence Interval for ( Unknown)

With an unknown , you need to be sure that

(1) the sample size is large enough (n 30), or

(2) the population is normal

Such assumptions enable the use of Students t dist:

Confidence Interval Estimate:

where t, also written as t/2,n-1, is the critical value of the t

distribution with n-1 degree of freedom, and an area of /2 in

each tail)

n

StX 1-n

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Critical Values of t

The critical value of t is characterized by two elements:

The confidence level (1- ), and

The degrees of freedom (df).

What is d.f.?

It is the number of observations that are free to vary after sample mean has been calculated.

In this section, df = n-1.

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Degrees of Freedom

Given a mean value of 8.0, X3 must be 9

(i.e., X3 is not free to vary) Here, n = 3, so degrees of freedom = n 1 = 3 1 = 2

You are free to choose 2 values (X1 and X2),

but the third is set for a given mean.

Eg: Suppose the mean of 3 numbers is 8.0

Let X1 = 7, X2 = 8

What is X3?

In this example d.f. = 2.

What does it mean?

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Degrees of Freedom

t 0

t (df = 5)

t (df = 13) t-distributions: bell-shaped, symmetric,

but fatter tails than Z

Standard Normal (t distribution with df = )

Note: t Z as n increases

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Critical Values of t

Upper Tail Area

df

.25 .10 .05

1

1.000

3.078

6.314

20 0.687 1.325 1.724

21

0.686

1.323

1.721

t 0 1.724

The body of the table contains

t values, not ____

Suppose n = 21, and = 0.10.

Then df = ____,

upper-tail area = ____

/2 = 0.05

d.f. = 20

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Eg: Mean Age of Retirement

A random sample of 25 retirees has mean age = 50 and std = 8. Find the 95% confidence interval for .

Must assume a normal population.

From t-table, t0.025, 24 = 2.0639

The confidence interval is

25

8(2.0639)50

n

S1-n /2, tX

(46.698 , 53.302)

Eg: Heating Oil Consumption

A random sample of 35 households has mean consumption of heating oilx = 1122.75 gallons, and S = 295.72 gallons.

Find the 95% confidence interval for .

ANSWER

Critical values are t0.025, 34 = 2.0322.

= 1122.75 101.58 gallons.

Based on the sample evidence, we are 95% confident that the interval 1122.75 101.58 gallons covers the population mean.

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ANSWER

NOTE: Z or t?

If n 30, it is commonly acceptable to use Z (instead of t) as an approximation.

But if you can find a more precise answer (using t-values), why not?

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Estimating

Population Proportion

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Confidence Intervals for the


Recall that the distribution of the sample proportion is

approximately normal if the sample size is large, with

standard deviation

We will estimate this with sample data:

n

p)p(1

n

)(1p

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Confidence Intervals for the


The confidence interval for the population proportion is given by:

where

Z = critical Z-value given the level of confidence

p = sample proportion

n = sample size

Such interval estimate for is based on a point estimate (p), plus an allowance for uncertainty arising from sampling

n

p)p(1Zp

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Example: Vegetarians

1. A random sample of 100 people shows that 25 of them are vegetarians.

Form a 95% confidence interval for the true proportion of vegetarians in the

population.

2. Compute the 95% confidence interval if n=1000.

00.25(.75)/196.125/100

p)/np(1p

Z

0.3349) , (0.1651

(.0433) 1.96 .25 Interpretation

95% of intervals formed from

samples of size 100 in this manner

will cover the true proportion

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Sample Size

Determination

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Sample Size Determination

Recall that sample size (n) affects the margin

of error (e, also called sampling error),

where

If e is set before conducting a survey, this

equation helps you determine the sample size

for a pre-set value of e (the acceptable level

of error): 2

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e

Zn

n

Ze

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If = 45, what sample size is needed to estimate

the mean within 5 with 90% confidence?

219.195

(45)(1.645)

e

Zn

2

22

2

22

Round up to the next integer to get the

required sample size n = 220

Eg: A4 Paper Again

In the paper manufacturer example, = 0.02, n =

100, and the 95% interval estimate is = 10.998

0.00392 inches.

Suppose the manufacturer wants to limit the error to

0.003 by choosing a larger sample. What is n?

ANSWER

The required sample size is n = 171. 7.1700.003

(0.02)(1.96)2

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2

22

e

Zn

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ANSWER

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To determine the required sample size for the proportion, you

must know:

The critical value Z (from a confidence level of 1-),

The acceptable sampling error (e), and

The true proportion .

If is unknown, use the sample value p, or set = 0.50.

2

2 )1(

e

Zn

Now solve

for n to get n

Ze)1(

Eg: Quality Control

Out of a population of 1,000 light bulbs, we randomly selected 100 of

which 30 were defective. What sample size is needed to be within

0.05 with 90% confidence?

(a) Since the true population proportion is unknown, use the sample

value here.

(b) Now, set = 0.50 and compare the result with (a).

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2 22 2

1 1.645 0.3 0.7

Error 0.05

227.3 228

Z p pn

(b) The required sample size

increases to 271.

NOTE: The product (1- ) ranges from 0 to 0.25. By assuming a value

of 0.25, we are in fact playing safe by

sampling more than necessary.

ANSWER

(a)

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More on the

t Distribution

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The t distribution is a family of probability distributions. It is bell-shaped, symmetric, & flatter than the Z distribution..

t Distribution

A specific t distribution depends on a parameter known as the degrees of freedom (d.f.).

Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.

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A t distribution with more degrees of freedom has ____ dispersion. As the number of d.f. increases, the difference between t distribution and Z distribution becomes smaller and smaller.

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Degrees Area in Upper Tail

of Freedom .20 .10 .05 .025 .01 .005

. . . . . . .

50 .849 1.299 1.676 2.009 2.403 2.678

60 .848 1.296 1.671 2.000 2.390 2.660

80 .846 1.292 1.664 1.990 2.374 2.639

100 .845 1.290 1.660 1.984 2.364 2.626

.842 1.282 1.645 1.960 2.326 2.576

Look familiar? They are ____.

t Distribution What is this 2.009?

Review Questions

A population has a standard deviation of 50. A random sample of 100 from this population is selected, and the sample mean is 600. At 95% confidence, the margin of error is ____

As the number of degrees of freedom for a t distribution ____, the difference between the t distribution and the standard normal distribution becomes smaller

For the interval estimation of when is known and the sample is large, the proper distribution to use is ____

1. 9.8

2. Increases

3. The normal distribution

ANSWER

Review Questions

4. The t value for a 95% confidence interval estimation with 24 degrees of freedom is ____

5. A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to 0.90, the interval for a. becomes narrower

b. becomes wider

c. does not change

d. becomes 0.1

6. In a random sample of 144 observations, sample proportion p = 0.6. The 95% confidence interval for is a. 0.52 to 0.68

b. 0.144 to 0.200

c. 0.60 to 0.70

d. 0.50 to 0.70

4. 2.064

5. A

6. A

ANSWER

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