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Business Statistics: A Decision-Making Approach

8th Edition

Chapter 8Estimating Single

Population Parameters

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Chapter Goals

After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence

interval estimate Construct and interpret a confidence interval estimate for a

single population mean using both the z and t distributions Determine the required sample size to estimate a single

population mean or a proportion within a specified margin of error

Form and interpret a confidence interval estimate for a single population proportion

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Overview of the Chapter

Builds upon the material from Chapter 1 and 7 Introduces using sample statistics to estimate population

parameters Because gaining access to population parameters can be

expensive, time consuming and sometimes not feasible Confidence Intervals for the Population Mean, μ

when Population Standard Deviation σ is Known when Population Standard Deviation σ is Unknown

Confidence Intervals for the Population Proportion, p Determining the Required Sample Size for means and

proportions

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Estimation Process

(mean, μ, is unknown)

Population

Random Sample Mean

(point estimate)

Mean x = 50

Sample

I am 95% confident that μ is between 40 & 60.

confidenceinterval

Confidence Level

Point Estimate

Suppose a poll indicate that 62% (sample mean) of the people favor limiting property taxes to 1% of the market value of the property.

The 62% is the point estimate of the true population of people who favor the property-tax limitation.

EPA tested average Automobile Mileage (point estimate)

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Confidence Interval

The point estimate (sample mean) is not likely to exactly equal the population parameter because of sampling error.

Probability of “sample mean = population mean” is zero Problem: how far the sample mean is from the

population mean. To overcome this problem, “confidence interval” can be

used as the most common procedure.

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Point and Interval Estimates

So, a confidence interval provides additional information about variability within a range.

The interval incorporates the sampling error

Point Estimate

Lower

Confidence

Limit

Upper

Confidence

Limit

Width of confidence interval

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Confidence Level

Confidence level = 95% Meaning: In the long run, 95% of all the confidence

intervals will contain the true parameter Describes how strongly we believe that a particular

sampling method will produce a confidence interval that includes the true population parameter.

A percentage (less than 100%) Most common: 90% (α = 0.1), 95% (α = 0.05), 99%

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Common Levels of Confidence

Commonly used confidence levels are 90%, 95%, and 99%

Confidence Level

Critical value, z

1.28

1.645

1.96

2.33

2.58

3.08

3.27

80%

90%

95%

98%

99%

99.8%

99.9%

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General Formula

The general formula for the confidence interval is:

Point Estimate Margin of Error

Margin of Error: Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval

(z or t - Value) * (Standard Error)

n

σzx

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

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General Formula

According to our textbook

Point Estimate (Critical Value)(Standard Error)

z or t -value based on the level of

confidence desired

Conditions for finding the Critical Value

The Central Limit Theorem states that the sampling distribution of a statistic will be normally (nearly normally) distributed if at least n > 30.

The sampling distribution of the mean is normally distributed when the population distribution is normally distributed.

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How to Find the z Value

When one of these conditions is satisfied, the critical value can be expressed as a z score.

To find the z value, see the next slide.

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Finding the z Value

Consider a 95% confidence interval:

n

σzx

-z = -1.96 z = 1.96

.0252

α .025

2

α

Point EstimateLower Confidence Limit

UpperConfidence Limit

z units:

x units: Point Estimate

0

1.96z

n

σzx x

0.95/2 (because of lower and upper limit) = 0.475

find on z table = 1.96

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When the population distribution type is unknown or when the sample size is small (less than 30), the t value is preferred.

The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after

sample mean has been calculated

d.f. = n – 1

Only n-1 independent pieces of data information left in the sample because the sample mean has already been obtained

How to Find the t value

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t Distribution:Try the t simulation one the website

t0

t (df = 5)

t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal

Standard Normal

(t with df = )

Note: t compared to z as n increases

As n the estimate of s becomes better so t

converges to z

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t Table

Confidence Level

df 0.50 0.80 0.90

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0 2.920The body of the table contains t values, not probabilities

Let: n = 3 df = n - 1 = 2 confidence level: 90%

/2 = 0.05

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t Distribution Values

With comparison to the z value

Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____

0.80 1.372 1.325 1.310 1.28

0.90 1.812 1.725 1.697 1.64

0.95 2.228 2.086 2.042 1.96

0.99 3.169 2.845 2.750 2.58

Note: t compared to z as n increases

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Example

A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ

d.f. = n – 1 = 24, so

The confidence interval is

2.0639t0.025,241n,/2 t

25

8(2.0639)50

n

stx

46.698 53.302

Summary of When to use z or t

Use z value (or score) Population is normally distributed (very rare and even

though sample size is small) Sample size is n > 30

Use t value (or score) Do not know population distribution type (not normal)

Do not know Pop. Mean and Std Many real world situations

Sample size is less than 30

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Using Analysis ToolPak

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Download “Confidence Interval Example” Excel file

Using Analysis ToolPak

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Margin of Error

Confidence Interval

119.9 (mean) ± 2.59

117.31 ------ 122.49

small sample: apply “t” distribution automatically)

Don’t even worry about p* = 1 - α/2

Using Analysis ToolPak(large sample: use normal (z) distribution automatically )

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Determining Required Sample Size

Wishful situation High confidence level, low margin of error, and right sample size

In reality, conflict among three…. For a given sample size, a high confidence level will tend to

generate a large margin of error For a given confidence level, a small sample size will result in an

increased margin of error Reducing of margin of error requires either reducing the confidence

level or increasing the sample size, or both

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Determining Required Sample Size

How large a sample size do I really need? Sampling budget constraint Cost of selecting each item in the sample

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Determining Required Sample SizeWhen σ is known

The required sample size can be found to reach a desired margin of error (e) and

level of confidence (1 - ) Required sample size, σ known:

2

222

e

σz

e

σzn

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Required Sample Size Example

If = 45 (known), what sample size is needed to be 90% confident of being correct within ± 5?

(Always round up)

219.195

(45)1.645

e

σzn

2

22

2

22

So the required sample size is n = 220

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If σ is unknown (most real situation)

If σ is unknown, three possible approaches

1. Use a value for σ that is expected to be at least as large as the true σ

2. Select a pilot sample (smaller than anticipated sample size) and then estimate σ with the pilot sample standard deviation, s

3. Use the range of the population to estimate the population’s Std Dev. As we know, µ ± 3σ contains virtually all of the data. Range = max – min.

Thus, R = (µ + 3σ) – (µ - 3σ) = 6σ. Therefore, σ = R/6 (or R/4 for a more conservative estimate, producing a larger sample size)

Example when σ is unknown

Example 8-5: Jackson’s Convenience Stores Using a pilot sample approach Available on the class website

Make sure to review all the examples from page 306 to 327.

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Confidence Intervals for the Population Proportion, π

Try by yourself! An interval estimate for the population

proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ). For example, estimation of the proportion of

customers who are satisfied with the service provided by your company

Sample proportion, p = x/n

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Confidence Intervals for the Population Proportion, π

Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation

We will estimate this with sample data:

(continued)

n

p)p(1sp

n

π)π(1σπ

See Chpt. 7!!

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Confidence Interval Endpoints

Upper and lower confidence limits for the population proportion are calculated with the formula

where z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size

n

p)p(1zp

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Example

A random sample of 100 people

shows that 25 are left-handed.

Form a 95% confidence interval for

the true proportion of left-handers

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Example

A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.

1.

2.

3.

0.0433 /1000.25(0.75)p)/np(1S

0.2525/100 p

p

0.3349 0.1651

(0.0433) 1.96 0.25

(continued)

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Interpretation

We are 95% confident that the true percentage of left-handers in the population is between

16.51% and 33.49%

Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

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Changing the sample size

Increases in the sample size reduce the width of the confidence interval.

Example: If the sample size in the above example is

doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at 0.25, but the width shrinks to

0.19 0.31

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Finding the Required Sample Size

for Proportion Problems

n

π)π(1ze

Solve for n:

Define the margin of error:

2

2

e

π)(1πzn

π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.50 – worst possible variation thus the largest sample size)

Will be in % units

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What sample size...?

How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?

(Assume a pilot sample yields p = 0.12)

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What sample size...?

Solution:For 95% confidence, use Z = 1.96e = 0.03p = 0.12, so use this to estimate π

So use n = 451

450.74(0.03)

.12)0(0.12)(1(1.96)

e

π)(1πzn

2

2

2

2

(continued)

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Using PHStat

PHStat can be used for confidence intervals for the mean or proportion

Two options for the mean: known and unknown population standard deviation

Required sample size can also be found Download from the textbook website

The link available on the class website

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PHStat Interval Options

options

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PHStat Sample Size Options

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Using PHStat (for μ, σ unknown)

A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ

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Using PHStat (sample size for proportion)

How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?

(Assume a pilot sample yields p = 0.12)

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Chapter Summary

Discussed point estimates Introduced interval estimates Discussed confidence interval estimation for

the mean [σ known] Discussed confidence interval estimation for

the mean [σ unknown] Addressed determining sample size for mean

and proportion problems Discussed confidence interval estimation for

the proportion

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