McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.

Confidence IntervalsConfidence Intervals

Chapter 7Chapter 7

7-2

Confidence IntervalsConfidence Intervals

7.1z-Based Confidence Intervals for a Population Mean: s Known

7.2t-Based Confidence Intervals for a Population Mean: s Unknown

7.3 Sample Size Determination7.4 Confidence Intervals for a Population Proportion7.5

Confidence Intervals for Parameters of Finite Populations (optional)

7.6A Comparison of Confidence Intervals and Tolerance Intervals (optional)

7-3

z-Based Confidence Intervalsfor a Mean: s Known

z-Based Confidence Intervalsfor a Mean: s Known

• The starting point is the sampling distribution of the sample mean• Recall from Chapter 6 that if a population is

normally distributed with mean and standard deviation s, then the sampling distribution of is normal with mean = and standard deviation

• Use a normal curve as a model of the sampling distribution of the sample mean

• Exactly, because the population is normal• Approximately, by the Central Limit Theorem for

large samples

nx ss

7-4

The Empirical RuleThe Empirical Rule

• Recall the empirical rule, so…• 68.26% of all possible sample means are

within one standard deviation of the population mean

• 95.44% of all possible observed values of x are within two standard deviations of the population mean

• 99.73% of all possible observed values of x are within three standard deviations of the population mean

7-5

GeneralizingGeneralizing

• In the example, we found the probability that is contained in an interval of integer multiples of s

• More usual to specify the (integer) probability and find the corresponding number of s

• The probability that the confidence interval will not contain the population mean is denoted by • In the example, = 0.0456

7-6

Generalizing ContinuedGeneralizing Continued

• The probability that the confidence interval will contain the population mean is denoted by • 1 – is referred to as the confidence

coefficient• (1 – ) 100% is called the confidence level

• Usual to use two decimal point probabilities for 1 – • Here, focus on 1 – = 0.95 or 0.99

7-7

General Confidence IntervalGeneral Confidence Interval

• In general, the probability is 1 – that the population mean is contained in the interval

• The normal point z/2 gives a right hand tail area under the standard normal curve equal to /2

• The normal point - z/2 gives a left hand tail area under the standard normal curve equal to /2

• The area under the standard normal curve between -z/2 and z/2 is 1 –

ss

nzxzx x 22

7-8

Sampling Distribution Of AllPossible Sample Means

7-9

z-Based Confidence Intervals for aMean with s Known

• If a population has standard deviation s (known),

• and if the population is normal or if sample size is large (n 30), then …

• … a )100% confidence interval for is

s

s

s

nzx,

nzx

nzx 222

7-10

The Effect of on ConfidenceInterval Width

z/2 = z0.025 = 1.96 z/2 = z0.005 = 2.575

7-11

t-Based Confidence Intervals for aMean: s Unknown

• If s is unknown (which is usually the case), we can construct a confidence interval for based on the sampling distribution of

• If the population is normal, then for any sample size n, this sampling distribution is called the t distribution

ns

xt

7-12

The t DistributionThe t Distribution

• The curve of the t distribution is similar to that of the standard normal curve• Symmetrical and bell-shaped• The t distribution is more spread out than

the standard normal distribution• The spread of the t is given by the number

of degrees of freedom• Denoted by df• For a sample of size n, there are one fewer

degrees of freedom, that is,

df = n – 1

7-13

The t Distribution and Degrees of Freedom

The t Distribution and Degrees of Freedom

• For a t distribution with n – 1 degrees of freedom, • As the sample size n increases, the

degrees of freedom also increases• As the degrees of freedom increase, the

spread of the t curve decreases• As the degrees of freedom increases

indefinitely, the t curve approaches the standard normal curve

• If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very similar to the standard normal curve

7-14

t and Right Hand Tail Areast and Right Hand Tail Areas

• Use a t point denoted by t• t is the point on the horizontal axis under

the t curve that gives a right hand tail equal to

• So the value of t in a particular situation depends on the right hand tail area and the number of degrees of freedom• df = n – 1 = 1 – , where 1 – is the specified

confidence coefficient

7-15

Using the t Distribution TableUsing the t Distribution Table

• Rows correspond to the different values of df• Columns correspond to different values of • See Table 7.3, Tables A.4 and A.20 in

Appendix A and the table on the inside cover• Table 7.3 and A.4 gives t points for df 1 to 30,

then for df = 40, 60, 120, and ∞• On the row for ∞, the t points are the z points

• Table A.20 gives t points for df from 1 to 100• For df greater than 100, t points can be

approximated by the corresponding z points on the bottom row for df = ∞

• Always look at the accompanying figure for guidance on how to use the table

7-16

t-Based Confidence Intervals for aMean: s Unknown

If the sampled population is normally distributed with mean , then a )100% confidence interval for is

t/2 is the t point giving a right-hand tail area of /2 under the t curve having n – 1 degrees of freedom

n

stx 2

7-17

Sample Size Determination (z)

22

s

B

zn

If s is known, then a sample of size

so that is within B units of , with 100(1-)% confidence

7-18

Sample Size Determination (t)

2

2

B

stn

If s is unknown and is estimated from s, then a sample of size

so that is within B units of , with 100(1-)% confidence. The number of degrees of freedom for the t/2 point is the size of the preliminary sample minus 1

7-19

Confidence Intervals for aPopulation Proportion

n

p̂p̂zp̂

12

If the sample size n is large*, then a )100% confidence interval for p is

* Here n should be considered large if both

5ˆ15ˆ pnandpn

7-20

Determining Sample Size forConfidence Interval for p

2

21

B

zppn

A sample size

will yield an estimate , precisely within B units of p, with 100(1-)% confidence

Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p

p̂

7-21

Confidence Intervals for PopulationMean and Total for a Finite Population

For a large (n 30) random sample of measurements selected without replacement from a population of size N, a )100% confidence interval for is

A )100% confidence interval for the population total is found by multiplying the lower and upper limits of the corresponding interval for by N

N

nN

n

szx

2

7-22

Confidence Intervals for Proportionand Total for a Finite Population

For a large random sample of measurements selected without replacement from a population of size N, a )100% confidence interval for p is

A )100% confidence interval for the total number of units in a category is found by multiplying the lower and upper limits of the corresponding interval for p by N

N

nN

n

p̂p̂p̂

1

1

7-23

A Comparison of ConfidenceIntervals and Tolerance Intervals

A tolerance interval contains a specified percentage of individual population measurements

• Often 68.26%, 95.44%, 99.73%

A confidence interval is an interval that contains the population mean , and the confidence level expresses how sure we are that this interval contains • Often confidence level is set high (e.g., 95% or 99%)

– Because such a level is considered high enough to provide convincing evidence about the value of