Chapter 8

Confidence Intervals

Statistics for Business(ENV)

1

Confidence Intervals8.1 z-Based Confidence Intervals for a

Population Mean: σ Known8.2 t-Based Confidence Intervals for a

Population Mean: σ Unknown8.3 Sample Size Determination8.4 Confidence Intervals for a Population

Proportion

Reminder: Sampling distribution

– If a population is normally distributed with mean and standard deviation σ, then the sampling distribution of is normal with mean M = and standard deviation

– Use a normal curve as a model of the sampling distribution of the sample mean• Exactly, if the population is normal• Approximately, by the Central Limit Theorem

for large samples

nM

Example: SAT scores• The population of scores (X) on the SAT forms a

normal distribution with mean = 500 and = 100.

• In a random sample of n = 25 students, – the distribution of the sample mean is a normal

distribution with a mean=500– and standard deviation is

4

2025

100

nx

5

xx

In a random sampling of students of sample size=25, we are confident that 80% of the samples will have a mean between - 1.28 and + 1.28 , orwithin the interval [474.4, 525.6]

6

The probability that will be within ±25.6 of µ is 80%, ORIf we know then there is 80% probability that µ will be within ±25.6 away from OR we are 80% confident that µ will be within an interval ±25.6 away from

In other words:

A point estimate is a single value (sample statistic) used to estimate a population parameter. s

Point and Interval EstimatesA confidence interval is a range of values in which the population parameter (say ) is expected to be there.

Usually, people consider the 95% and 99% confidence intervals.

is between ? and ?

Example 3

The value of the population mean is unknown. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.

The Dean of the Business School wants to estimate the mean number of hours students studied per week. A sample of 49 students showed a mean of 24 hours with a sd of 4 hours. What is the population mean?

%95)12.2588.22(

%9549

)4(96.14.2

49

)4(96.14.2

%9549

)4(96.14.2

49

)4(96.14.2

%9596.196.1

%9596.196.1

%9596.1/

96.1

%5.4796.1/

0

P

P

P

nX

nXP

nX

nP

n

XP

n

XP

Constructing General Confidence Intervals for µ

n96.1X

nzX

n58.2X

95% CI for the (population) mean

99% CI for the (population) mean

12.100.24

49

496.100.24

n96.1X

The 95% CI for the mean is from 22.88 to 25.12.

95 percent confidence interval for the population mean

What if we don’t know ?

However, normally, we don’t know the population (sd) . So, normally, people just replace (estimate) the by s.

It doesn't matter too much since normally we are considering a large sample(n30).

Factors that determine the width of a Factors that determine the width of a confidence intervalconfidence interval

The sample size, nThe level of confidence, 1-The s.d. of the population, (usually estimated by the sample s.d., or s)

Constructing General Confidence Intervals for µ

n

stX

Confidence interval for the mean (n < 30 and the underlying distribution is normal)

The value of t depends on the confidence level as well as the degrees of freedom (df=n-1).

There is a family of t distributions, There is a family of t distributions, determined by its degrees of freedom (n-1). The t-distribution approaches N(0, 1) as n approaches infinity.

Characteristics of the t distributionCharacteristics of the t distributionIt is a continuous, bell-shaped and symmetrical distribution, which is flatter than a normal distribution.

16

Distributions of the t statistic for different values of degrees of freedom are compared to a normal distribution.

Confidence Interval for a Population Proportion

n

PPzP

)1(

NNP

)1(,~

Let X ~ Bin(N, ), then P =X/N is called a population proportion.The distribution for a population proportion. Both n and n(1- ) > 5

A point estimate of the population proportion is given by the sample proportion . Confidence Interval for a Population Proportion, obtained from the sample proportion P

P

A sample of 500 executives who own their own home revealed 175 planned to sell their house after they retire. Develop a 98% CI for the proportion of executives that plan to sell their house.

0497.35. 500

)65)(.35(.326.235.0

EXAMPLE 4

Here, the sample proportion is p=175/500=0.35

Selecting a Sample Size

nzE

Let E be the error term appear in the CI Let E be the error term appear in the CI

E is also known as the width of the C.I E is also known as the width of the C.I divided by 2.divided by 2.

2

E

zn

where n is the size of the sampleE is the allowable error z the z- value corresponding to the selected level of confidence the population s.d..

Selecting a Sample Size 2

Example 6

1075

)20)(58.2(2

n

A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies the s.d. is estimated to be $20.00. How large a sample is required?

Sample Size for Proportions

n p pZ

E

( )1

2

wherep is the estimated proportion, based on past experience

or a pilot surveyz is the z value associated with the degree of confidence

selectedE is the maximum allowable error the researcher will

tolerate

The formula for determining the sample size in the case of a proportion is

Example 7

89703.

96.1)70)(.30(.

2

n

The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.

24

“The nationwide telephone survey was conducted Friday through Wednesday with 1,224 adults and has a margin of sampling error of plus or minus three percentage points.”

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Chapter 8Estimation and Confidence Estimation and Confidence IntervalsIntervals When you have completed this chapter, you will be able to:

ONEDefine what is meant by a point estimate.TWO Construct a confidence interval for the mean when the population standard deviation is known and the sample size is large enough or underlying distribution is normal.THREEConstruct a confidence interval for the mean when the population standard deviation is unknown and sample size is large enough or underlying distribution is normal.

Chapter 8 continued

FOURConstruct a confidence interval for the population proportion. FIVE Construct a confidence interval for the mean when the population size is finite.SIXDetermine the sample size for attribute and variable sampling.