Chapter 7 Confidence Intervals and Sample Sizes

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Chapter 7Confidence Intervals and

Sample Sizes7.2 Estimating a Proportion p

7.3 Estimating a Mean µ (σ known)

7.4 Estimating a Mean µ (σ unknown)

7.5 Estimating a Standard Deviation σ

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Example 1

TRICK QUESTION!We only know the sample proportion s, We do not know the population proportion σ.

BUT…The proportion of the sample (0.7) is our best point estimate (i.e. best guess).

In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming.Q: What is the proportion of the adult population that believe in global warming?

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Definition

PopulationParameter

Best Point Estimate

Proportion

Mean

Std. Dev.

p

µ

σ

p

x

s

≈

≈

≈

Point EstimateA single value (or point) used to approximate a population parameter

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Definition

Confidence Interval : CIThe range (or interval) of values to estimate the true value of a population parameter.

It is abbreviated as CI

In Example 1, the 95% confidence interval for the population proportion p is CI = (0.677, 0.723)

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Definition

Confidence Level : 1 – α The probability that the confidence interval actually contains the population parameter.

The most common confidence levels used are 90%, 95%, 99%

90% : α = 0.1 95% : α = 0.05 99% : α = 0.01

In Example 1, the Confidence level is 95%

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Definition

Margin of Error : EThe maximum likely difference between the observed value and true value of the population parameter (with probability is 1–α)

The margin of error is used to determine a confidence interval (of a proportion or mean)

In Example 1, the 95% margin of error for the population proportion p is E = 0.023

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A: 0.7 is the best point estimate of the proportion of all adults who believe in global warming.

The 95% confidence interval of the population proportion p is:

CI = (0.677, 0.723)

( with a margin of error E = 0.023 )

What does it mean, exactly?

In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming.

Q: What is the proportion of the adult population that believe in global warming?

Example 1 Continued…

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For the 95% confidence interval CI = (0.677, 0.723)we say:

We are 95% confident that the interval from 0.677 to 0.723 actually does contain the true value of the population proportion p.

This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals, then 95% of them would actually contain the value of the population proportion p.

Interpreting a Confidence Interval

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Know the correct interpretation of a confidence interval

It is incorrect to say “ the probability that the population parameter belongs to the confidence interval is 95% ”because the population parameter is not a random variable, its value cannot change

The population is “set in stone”

!!! Caution !!!

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Do not confuse the two percentages

The proportion can be represented by percents (like 70% in Example 1)

The confidence level may be represented by percents (like 95% in Example 1)

Proportions can be any value from 0% to 100%

Confidence levels are usually 90%, 95%, or 99%

!!! Caution !!!

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( y – E, y + E )y = Best point estimate

E = Margin of Error

Confidence Interval Formula

• Centered at the best point estimate

• Width is determined by E

The value of E depends the critical value of the CI

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Finding the Point Estimate and E from a Confidence Interval

Margin of Error : E

E = (upper confidence limit) — (lower confidence limit)

2

Point estimate : y

y = (upper confidence limit) + (lower confidence limit)

2

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Definition

Critical Value The number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.

A critical value is dependent on a probability distribution the parameter follows and the confidence level (1 – α) .

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Normal Dist. Critical ValuesFor a population proportion p and mean µ (σ known), the critical values are found using z-scores on a standard normal distribution

The standard normal distribution is divided into three regions: middle part has area 1 – α and two tails (left and right) have area α/2 each:

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The z-scores z/2 and –z/2 separate the values:

Likely values ( middle interval )

Unlikely values ( tails )

Use StatCrunch to calculate z-scores (see Ch. 6)

z/2–z/2

Normal Dist. Critical Values

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The value z/2 separates an area of /2 in the right tail of the z-dist.

The value –z/2 separates an area of /2 in the left tail of the z-dist.

The subscript /2 is simply a reminder that the z-score separates an area of /2 in the tail.

Normal Dist. Critical Values

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Section 7.2Estimating a Population Proportion

Objective

Find the confidence interval for a population proportion p

Determine the sample size needed to estimate a population proportion p

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Definitions

The best point estimate for a population proportion p is the sample proportion p

Best point estimate : p

The margin of error E is the maximum likely difference between the observed value and true value of the population proportion p (with probability is 1–α)

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Margin of Error for Proportions

2

ˆ ˆpqE z

n

E = margin of error

p = sample proportion

q = 1 – p

n = number sample values

1 – α = Confidence Level

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Confidence Interval for a Population Proportion p

( p – E, p + E )ˆ ˆ

where

2

ˆ ˆpqE z

n

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Finding the Point Estimate and E from a Confidence Interval

Margin of Error:

E = (upper confidence limit) — (lower confidence limit)

2

Point estimate of p:

p = (upper confidence limit) + (lower confidence limit)

2

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Round-Off Rule for Confidence Interval Estimates of p

Round the confidence interval limits for p to

three significant digits.

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Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

Direct Computation

Example 1

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Using StatCrunch

Stat → Proportions → One Sample → with Summary

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Enter Values

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Click ‘Next’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Select ‘Confidence Interval’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Enter Confidence Level, then click ‘Calculate’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

From the output, we find the Confidence interval is

CI = (0.578, 0.762)

Lower Limit

Upper Limit

Standard Error

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Sample Size

Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

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Determining Sample Size

(solve for n by algebra)

( )2 ˆp qZ n =

ˆE 2

zE =

p qˆ ˆn

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Sample Size for Estimating Proportion p

When an estimate of p is known: ˆˆ( )2 p qn =

ˆE 2

z

When no estimate of p is known:use p = q = 0.5

( )2 0.25n =

E 2

z

ˆˆ ˆ

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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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A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet.

How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points when…

(a) In 2006, 73% of adults used the Internet.

(b) No known possible value of the proportion.

Example 2

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(a) Given:

Given a sample has proportion of 0.73, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 842 adults.

Example 2

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(b) Given:

For any sample, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 1068 adults.

Example 2

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SummaryConfidence Interval of a Proportion

2

ˆ ˆpqE z

n

( p – E, p + E )

E = margin of error

p = sample proportion

n = number sample values

1 – α = Confidence Level

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When an estimate of p is known:

ˆ( )2 p qn =

ˆE 2

z

When no estimate of p is known (use p = q = 0.5)

( )2 0.25n =

E 2

z

SummarySample Size for Estimating a Proportion

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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)