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ECO 321 Intermediate Business Statistics

Lesson 6 Sampling Distributions

And Confidence Interval Estimation

Lesson Topics

Estimation Process Point Estimates Interval Estimates

Sampling Distribution of the Mean

Properties of as an Estimate for Confidence Interval Estimation for the Mean ( Known)

X μ

Lesson Topics

Confidence Interval Estimation for the Mean ( Unknown) Confidence Interval Estimation for the Proportion Determining Sample Size Confidence Interval Estimation and Ethical Issues

(continued)

Population and Sample

Population Sample

Use parameters to summarize features

Use statistics to summarize features

Inference on the population from the sample

Estimation Process

Mean, μ, is unknown

Population Random Sample I conjecture

that the population

mean, μ, is 50

Sample

50X =

Point Estimates

Estimate Population Parameters …

with Sample Statistics

Mean

Proportion

Variance

Difference

μ π

2σ

1 2μ μ−

X p

1 2X X−

Another Point Estimate

Here is a link to some of the most recent poll results

Drawback of Point Estimates

Q. What is the probability that a point estimate will equal to the true parameter that is being estimated? A. Zero. Theoretically, you will never obtain a point estimate that equals the unknown parameter.

Interval Estimation Process

Mean, μ, is unknown

Population Random Sample I am 95% confident that μ is between 40 &

60.

Sample

50X =

Interval Estimates

Provides Range of Values Take into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Give Information about Closeness to Unknown Population Parameters Stated in terms of level of confidence

Never 100% sure

Why Study Sampling Distributions

Sample Statistics are Used to Estimate Population Parameters

E.g. estimates the population mean

Problems: Different Sample Provides Different Estimate

Large sample gives better estimate; large sample costs more How good is the estimate?

Approach to Solution: Theoretical Basis is Sampling Distribution

50X = Xμ

Sampling Distribution

Theoretical Probability Distribution of a Sample Statistic Sample Statistic is a Random Variable

Sample mean, sample proportion

Results from Taking All Possible Samples of the Same Size

When the Population is Normal

Central Tendency

Variation (Standard Error)

Sampling with Replacement

Population Distribution

Sampling Distributions

XXμ μ=

X X n

σ σ =

X16Xμ =

5 2.236X

n σ = =

25 1X

n σ = =

16Xμ =

5Xσ =

When the Population is Not Normal

Central Tendency

Sampling with Replacement

Population Distribution

Sampling Distributions

Xμ μ=

X n σ

σ =

X8.08Xμ =

5 2.782X

n σ = =

25 1.244X

n σ = =

8.08Xμ =

6.22Xσ =

Variation (Standard Error)

Central Limit Theorem

As Sample Size Gets Large Enough

Sampling Distribution Becomes Almost Normal Regardless of Shape of Population

Applet to Illustrate the CLT

Click here to access the applet that will illustrate the Central Limit Theorem in action.

How Large is Large Enough?

For Most Distributions, n>30

For Fairly Symmetric Distributions, n>15

For Normal Distribution, the Sampling Distribution of the Mean is Always Normally Distributed

This is a property of sampling from a normal population distribution and is NOT a result of the central limit theorem

The average of all possible estimates of equals

Unbiased Property of ( )

Xμ

BiasedUnbiased

XμXμ X

XXμ μ= X

( )P X

Xμ

Effect of Large Sample (Consistency )

Larger sample sizeSmaller

sample size

XXμ

For Sampling with Replacement: As n increases, decreasesXσ

X X n

σ σ =

( )P X

As the sample size increases, the variation of the estimator around the true parameter resulting from taking repeated samples decreases

Less Variability (Efficiency)

Sampling Distribution of Median Sampling

Distribution of Mean

Xμ X

Standard Error (Standard Deviation) of the Sampling Distribution of the sample mean is Less than the Standard Error of Other Unbiased Estimators

Xσ

( )P X

The sample mean has the smallest variation than any other unbiased estimators in repeated sampling

Confidence Interval Estimates

Mean

σ Unknown

Confidence Intervals

Proportion

σ Known

Confidence Interval for ( Known)

Assumptions Population standard deviation is known Population is normally distributed If population is not normal, need large sample

Confidence Interval Estimate

is called the sampling error or

margin of error

μ σ

/ 2 / 2X Z X Z n n

α α σ σ

μ− ≤ ≤ +

/ 2e Z n

α σ

Standard error

critical value

Obtaining Confidence Interval in PHStat

PHStat | Confidence Interval | Estimates for the Mean, sigma known See PHStat Animation #15

Example

A random sample of 15 stocks traded on the NASDAQ market showed an average shares traded to be 215000. From past experience, it is believed that the population standard deviation of shares traded is 195000 and the shares traded are very closed to a normal distribution. Construct a 99% confidence interval for the average shares traded on the NASDAQ market. Interpret your result.

Confidence Interval Estimate for the Mean

Population Standard Deviation 195000 Sample Mean 215000 Sample Size 15 Confidence Level 99% Standard Error of the Mean 50348.7835 Z Value -2.57583451 Interval Half W idth 129690.1343 Interval Lower Limit 85309.86569 Interval Upper Limit 344690.1343

PHStat output

85309 344690μ< < The 99% CI for the population mean:

Sampling Distribution of the Mean

Interval and Level of Confidence

Confidence Intervals

Intervals extend from

of the intervals constructed contain ;

do not.

XX Zσ−

Xσ / 2α / 2α

X Xμ μ=

1 α−

XX Zσ+

( )1 100%α−

100 %α

/ 2 XZαμ σ+ / 2 XZαμ σ−

Z/ 2 . .u c Zα=/ 2. .l c Zα= −

Applet to Illustrate the Confidence Interval

Click here to access the applet that will illustrate the Confidence Interval in action.

Example: Interpretation (continued)

If all possible samples of size 25 are taken and the corresponding 95% confidence intervals are constructed, 95% of the confidence intervals that are constructed will contain the true unknown population mean.

Using the confidence interval method on repeated sampling, the probability that we will have constructed a confidence interval that will have contained the unknown population mean is 95%.

Repeated Samples

Example: Interpretation (continued)

For this particular confidence interval [46.67, 53.50], the unknown population mean can either be in the interval or not in the interval. It is, therefore, incorrect to state that the probability is 95% that the unknown population mean will be in the interval [46.67, 53.50].

We are 95% confidence that the population average is between 46.67 and 53.50.

Single Sample

Assumptions Population standard deviation is unknown If population is normal, small sample can be used If population is not normal, use large sample

Use Student’s t Distribution Confidence Interval Estimate

Confidence Interval for ( Unknown)

μ σ

/ 2, 1 / 2, 1n n S S

X t X t n n

α αμ− −− ≤ ≤ +

Margin of Error Standard Error

Student’s t Distribution

Z t

t (df = 5)

t (df = 13)Bell-Shaped Symmetric

‘Fatter’ Tails

Standard Normal

Example

/ 2, 1 / 2, 1

8 8 50 2.0639 50 2.0639

25 25 46.69 53.30

n n S S

X t X t n n

α αμ

− −− ≤ ≤ +

− ≤ ≤ +

≤ ≤

A random sample of 25 has 50 and 8. Set up a 95% confidence interval estimate for

n X S μ

= = =

We are 95% confident that the unknown true population mean is somewhere between 46.69 and 53.30.

PHStat | Confidence Interval | Estimate for the Mean, sigma unknown Example in Excel Spreadsheet

Confidence Interval for ( Unknown) in PHStat

μ σ

Microsoft Excel Worksheet

Confidence Interval Estimate for the Mean

Data Sample Standard Deviation 8 Sample Mean 50 Sample Size 25 Confidence Level 95%

Standard Error of the Mean 1.6 Degrees of Freedom 24 t Value 2.063898137 Interval Half Width 3.302237019

Interval Lower Limit 46.70 Interval Upper Limit 53.30

Intermediate Calculations

Confidence Interval

Another Example

Data on the “Average Quiz Scores before Mid- term 1” and “Mid-term 1 Scores”

Microsoft Excel Worksheet

Confidence Interval Estimate for Proportion

Assumptions Two categorical outcomes Population follows Binomial distribution Normal approximation can be used if

and Confidence Interval Estimate

5nπ ≥ ( )1 5n π− ≥

( ) ( ) / 2 / 2

1 1p p p p p Z p Z

n nα α π

− − − ≤ ≤ +

Margin of Error Standard Error

Example

A random sample of 400 voters showed 32 preferred candidate A. Set up a 95% confidence interval estimate for p.

( ) ( )

/ /

1 1

.08 1 .08 .08 1 .08 .08 1.96 .08 1.96

400 400 .053 .107

p p p p p Z p Z

n nα α π

2 2

− − − ≤ ≤ +

≤ ≤ We are 95% confident that the proportion of voters who prefer candicate A is somewhere between 0.053 and 0.107.

Confidence Interval Estimate for Proportion in PHStat

PHStat | Confidence Interval | Estimate for the Proportion … Example in Excel Spreadsheet

Microsoft Excel Worksheet

Confidence Interval Estimate for the Mean

Data Sample Size 400 Number of Successes 32 Confidence Level 95%

Sample Proportion 0.08 Z Value -1.95996108 Standard Error of the Proportion 0.01356466 Interval Half Width 0.026586206

Interval Lower Limit 0.053413794 Interval Upper Limit 0.106586206

Intermediate Calculations

Confidence Interval

Elements of Confidence Interval Estimation

Level of Confidence Confidence in which the interval will contain the unknown population parameter

Precision (Sampling Error) Closeness to the unknown parameter

Cost (Sample Size) Cost required to obtain a sample of size n

( )1 100%α−

Level of Confidence

Denoted by A Relative Frequency Interpretation

In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter

A Specific Interval Will Either Contain or Not Contain the Parameter

No probability involved in a specific interval

( )100 1 %α−

Factors Affecting Interval Width (Precision)

Data Variation Measured by

Sample Size

Level of Confidence

Intervals Extend from

X n σ

σ =

( )100 1 %α−

/ 2 / 2 to X XX Z X Zα ασ σ− + Sampling Error = / 2 XZα σ

Determining Sample Size (Cost)

Too Big:

• Requires too many resources

Too small:

• Won’t do the job

Determining Sample Size for Mean

What sample size will be needed for the confident interval estimate for the mean to be correct within ± 5 with a 90% level of confidence? A pilot study suggested that the standard deviation is 45.

Round Up

( )2 22 2 2 2

1.645 45 219.2 220

Error 5 Z

n σ

= = = ≅

Determining Sample Size for Proportion

Out of a population of 1,000, we randomly selected 100 of which 30 were defective. What sample size is needed for the confident interval estimate for proportion to be within ± 5% with 90% confidence?

Round Up

( ) ( )( )2 2 2 2

1 1.645 0.3 0.7 Error 0.05

227.3 228

Z n

π π− = =

= ≅

Obtaining Sample Size in PHStat

PHStat | Sample Size | Determination for the mean… PHStat | Sample Size | Determination for the proportion… Here’s an Excel spreadsheet that contains the result of the previous examples:

Microsoft Excel Worksheet

Ethical Issues

Confidence Interval (Reflects Sampling Error) Should Always be Reported Along with the Point Estimate The Level of Confidence Should Always be Reported The Sample Size Should be Reported An Interpretation of the Confidence Interval Estimate Should Also be Provided

Some Examples of Good and Bad Reporting

Here is a link to a really terrible reporting Here is a link to a really bad practice Here is a link to some not so bad practices Here is a link to a better practice

Lesson Summary

Illustrated Estimation Process Discussed Point Estimates Addressed Interval Estimates Discussed Sampling Distribution of the Sample Mean Addressed Properties of as an Estimate for Discussed Confidence Interval Estimation for the Mean ( Known)

X μ

Lesson Summary

Discussed Confidence Interval Estimation for the Mean ( Known) Discussed Confidence Interval Estimation for the Mean ( Unknown) Addressed Determining Sample Size Discussed Confidence Interval Estimation and Ethical Issues