Section 7.3

Estimating a Population mean µ ( σ known)

Objective

Find the

confidence interval

mean

µ

when

σ

is known for a population Determine the

sample size

needed to estimate a population mean

µ

when

σ

is known Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

1

Best Point Estimation

The

best point estimate mean

µ

(

σ

known) is the for a

population sample mean x Best point estimate : x

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2

Notation

 =

population mean

 =

population standard deviation

x

=

sample mean

n

=

number of sample values

E

=

margin of error

z



/

2 =

z-score

separating an area of

α

/2 in the right tail of the standard normal distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

3

Requirements

(1)

The population standard deviation

σ is known (2)

One or both of the following: The population is

normally distributed n

or

> 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

4

Margin of Error

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5

Confidence Interval

( x – E, x + E )

where

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6

Definition

The two values x – E and x + E are called confidence interval limits .

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7

Round-Off Rules for Confidence Intervals Used to Estimate

µ

1. When using the

original set of data

, round the confidence interval limits to

one more decimal place

than used in original set of data. 2. When the original set of data is unknown and only the

summary statistics (n, x, s)

are used, round the confidence interval limits to the

same number of decimal places

used for the sample mean.

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8

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Direct Computation

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9

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch

Stat → Z statistics → One Sample → with Summary

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10

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch Enter Parameters

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11

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch Click Next

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12

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch Select ‘Confidence Interval’

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13

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch Enter Confidence Level, then click ‘Calculate’

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14

Example

Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4

Using StatCrunch Standard Error Lower Limit Upper Limit From the output, we find the Confidence interval is CI = (35.862, 40.938)

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15

Sample Size for Estimating a Population Mean



= population mean σ = population standard deviation

x

= sample mean E = desired margin of error

z

α/2 = z score separating an area of



/2 in the right tail of the standard normal distribution

n

= (

z



/

2

)



E

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

2 16

Round-Off Rule for Determining Sample Size

If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

Examples: n = 310.67 round up to 311 n = 295.23 round up to 296 n = 113.01 round up to 114

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17

Example

We want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want

95% confidence

that the sample mean is

within 3 IQ points

of the population mean?

What we know:

 = 0.05

E

= 3  = 15 

/ 2

= 0.025

n = 1.96 • 15 = 96.04 = 97 3

z



/ 2

= 1.96

(using StatCrunch) With a simple random sample of only

97 statistics students

, we will be 95% confident that the sample mean is within 3 IQ points of the true population mean  .

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18

Summary

Confidence Interval of a Mean

µ

(

σ

known)

σ

=

population standard deviation

x

=

sample mean

n

=

number sample values

1 –

α

=

Confidence Level

( x – E, x + E )

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19

Summary

Sample Size for Estimating a Mean

µ

(

σ

known)

E

=

desired margin of error

σ

=

population standard deviation

x

=

sample mean

1 –

α

=

Confidence Level

n

= (

z



/

2

)



E



2

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20