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Section 7.3Estimating a Population mean µ

(σ known)

Objective

Find the confidence interval for a population mean µ when σ is known

Determine the sample size needed to estimate a population mean µ when σ is known

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Best Point Estimation

The best point estimate for a population mean µ (σ known) is the sample mean x

Best point estimate : x

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= population mean

= population standard deviation

= 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

x

Notation

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(1) The population standard deviation σ is known

(2) One or both of the following:

The population is normally distributed or

n > 30

Requirements

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Margin of Error

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Confidence Interval

( x – E, x + E )

where

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Definition

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

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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.

Round-Off Rules for Confidence Intervals Used to Estimate µ

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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

Example

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Using StatCrunch

Stat → Z statistics → One Sample → with Summary

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

Example

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Using StatCrunch

Enter Parameters

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

Example

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Using StatCrunch

Click Next

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

Example

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Using StatCrunch

Select ‘Confidence Interval’

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

Example

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Using StatCrunch

Enter Confidence Level, then click ‘Calculate’

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

Example

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Using StatCrunch

From the output, we find the Confidence interval is

CI = (35.862, 40.938)

Lower Limit

Upper Limit

Standard Error

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

Example

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Sample Size for Estimating a Population Mean

(z/2) n =

E

2

= population mean

σ = population standard deviation

= sample mean

E = desired margin of error

zα/2 = z score separating an area of /2 in the right tail of the standard normal distribution

x

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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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/2 = 0.025

z / 2 = 1.96

(using StatCrunch)

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?

Example

n = 1.96 • 15 = 96.04 = 97 3

2

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 .

What we know: = 0.05 E = 3 = 15

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SummaryConfidence Interval of a Mean µ

(σ known)

( x – E, x + E )

σ = population standard deviation

x = sample mean

n = number sample values

1 – α = Confidence Level

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(z/2) n =

E

2

E = desired margin of error

σ = population standard deviation

x = sample mean

1 – α = Confidence Level

SummarySample Size for Estimating a Mean µ

(σ known)