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6-16-1

Chapter 6Chapter 6

Estimates and Sample Estimates and Sample Size with One SampleSize with One Sample

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6-26-2 OutlineOutline

6-1 Introduction 6-3 Estimating a Population

Mean with: known 6-4 Estimating a Population

Mean with: unknown

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6-46-4 ObjectivesObjectives

Find the confidence interval for the mean when is known or n 30.

Determine the minimum sample size for finding a confidence interval for the mean.

Find the confidence interval for the mean when is unknown and n 30.

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6-66-66-3 6-3 Confidence Intervals for the Mean Confidence Intervals for the Mean (( known or known or nn 30) and Sample Size 30) and Sample Size

X

A point estimate is a specific numerical value estimate of a parameter. The best estimate of the population mean is thesample mean .

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6-76-7

The estimator must be an unbiased unbiased estimatorestimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.

6-3 Three Properties of a Good 6-3 Three Properties of a Good Estimator Estimator

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6-86-8

The estimator must be consistent. For a consistent estimatorconsistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.

6-3 Three Properties of a Good 6-3 Three Properties of a Good Estimator Estimator

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6-96-9

The estimator must be a relatively relatively efficient estimatorefficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

6-3 Three Properties of a Good 6-3 Three Properties of a Good Estimator Estimator

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6-106-10 6-3 Confidence Intervals6-3 Confidence Intervals

An interval estimateinterval estimate of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.

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6-116-11 6-3 Confidence Intervals6-3 Confidence Intervals

A confidence intervalconfidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate.

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6-126-12 6-3 Confidence Intervals6-3 Confidence Intervals

The confidence levelconfidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.

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6-136-13

The confidence levelconfidence level is the percentage equivalent to the decimal value of 1 – .

6-3 Formula for the Confidence 6-3 Formula for the Confidence Interval of the Mean for a SpecificInterval of the Mean for a Specific

X zn

X zn

2 2

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6-146-146-3 Maximum Error of Estimate 6-3 Maximum Error of Estimate or Margin of Error (E)or Margin of Error (E)

The maximum error of estimatemaximum error of estimate or

margin of error (E)margin of error (E) is the maximum difference between the point estimate of a parameter and the actual value of the parameter.

nzE

2

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6-156-15

The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 50 students is selected, and the mean is found to be 23.2 years. Find the 95% confidence interval of the population mean.

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

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6-166-16

Since the confidence

is desired z Hence

substituting in the formula

X zn

X zn

one gets

, ,

– +

2

95%

196

2 2

interval

. .

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

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6-176-17

2322

5023.2

2

232 0 6 232 0 6

22 6 238

95%

22 6 238

50

. (1.96)( ) (1.96)( )

. . . .

. . or 23.2 0.6 years.

, ,

,

. .

, .

Hence the president can say with

confidence that the average age

of the students is between and

years based on students

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

50

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6-186-18

A certain medication is known to increase the pulse rate of its users. The standard deviation of the pulse rate is known to be 5 beats per minute. A sample of 30 users had an average pulse rate of 104 beats per minute. Find the 99% confidence interval of the true mean.

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

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6-196-19

Since the confidence

is desired z Hence

substituting in the formula

X zn

X zn

one gets

, ,

– +

2

99%

2 58

2

interval

. .

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

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6-206-20

104 (2.58)530

104 (2.58)530

104 2 4 104 2 4

1016 1064

99%

1016 106.4

. ( ) ( )

. .

. . .

, ,

,

.

Hence one can say with

confidence that the average pulse

rate is between andbeats per minute, based on 30 users.

6-3 Confidence Intervals -6-3 Confidence Intervals - Example

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6-216-216-36-3 Formula for the Minimum Sample Size Formula for the Minimum Sample Size Needed for an Interval Estimate of the Needed for an Interval Estimate of the Population MeanPopulation Mean

.

2

2

mbera whole nuto obtain

upthe answerry, round If necessa

.n of errore or margiof estimat

um error the maximwhere E is

E

zn

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6-226-22

The college president asks the statistics teacher to estimate the average age of the students at their college. How large a sample is necessary? The statistics teacher decides the estimate should be accurate within 1 year and be 99% confident. From a previous study, the standard deviation of the ages is known to be 3 years.

6-3 Minimum Sample Size Needed for an Interval 6-3 Minimum Sample Size Needed for an Interval

Estimate of the Population Mean -Estimate of the Population Mean - ExampleExample

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6-236-23

Since or

z and E substituting

in nz

Egives

n

= . ( – . ),

= . , = ,

= ( . )( )

0 01 1 0 99

2 58 1

2 58 3

159 9 60

2

2

2

2

. .

6-3 Minimum Sample Size Needed for an Interval 6-3 Minimum Sample Size Needed for an Interval

Estimate of the Population Mean -Estimate of the Population Mean - ExampleExample

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6-246-246-4 Characteristics of the 6-4 Characteristics of the t-Distributiont-Distribution

The t-distribution shares some characteristics of the normal distribution and differs from it in others. The t-distribution is similar to the standard normal distribution in the following ways:

It is bell-shaped. It is symmetrical about the mean.

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6-256-256-4 Characteristics of the6-4 Characteristics of thet-Distributiont-Distribution

The mean, median, and mode are equal to 0 and are located at the center of the distribution.

The curve never touches the x axis. The t distribution differs from the

standard normal distribution in the following ways:

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6-266-266-4 Characteristics of the6-4 Characteristics of thet-Distributiont-Distribution

The variance is greater than 1. The t distribution is actually a family of

curves based on the concept of degrees of freedomdegrees of freedom, which is related to the sample size.

As the sample size increases, the t distribution approaches the standard normal distribution.

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6-276-276-4 Standard Normal Curve and 6-4 Standard Normal Curve and

the the tt Distribution Distribution

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6-136-13

When n < 30 and is unknown use t-distribution with degrees of freedom = n – 1.

6-4 Formula for the Confidence 6-4 Formula for the Confidence Interval of the Mean for a SpecificInterval of the Mean for a Specific

n

stX

n

stX 22

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6-286-28

Ten randomly selected automobiles were stopped, and the tread depth of the right front tires were measured. The mean was 0.32 inches, and the standard deviation was 0.08 inches. Find the 95% confidence interval of the mean depth. Assume that the variable is approximately normally distributed.

6-4 Confidence Interval for the Mean 6-4 Confidence Interval for the Mean ( ( unknown and unknown and nn < 30) - < 30) - Example

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6-296-29

Since is unknown and s must replace it, the t distribution must be used with = 0.05. Hence, with 9 degrees of freedom, t/2 = 2.262 (see Table F in text).

From the next slide, we can be 95% confident that the population mean is between 0.26 and 0.38.

6-4 Confidence Interval for the Mean 6-4 Confidence Interval for the Mean ( ( unknown and unknown and nn < 30) - < 30) - Example

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6-306-30 6-4 Confidence Interval for the Mean 6-4 Confidence Interval for the Mean ( ( unknown and unknown and nn < 30) - < 30) - Example

Thus the confidence

of the population mean is found by

substituting in

X ts

X tsn

0.32–(2.262)0.0810

(2.262)0.0810

95%

0 32

0 26 0 38

2 2

interval

.

. .

n