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Estimation and Confidence Intervals

Dr. Rick Jerz

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Goals

• Define a point estimate• Define a level of confidence• Construct a confidence interval for the

population mean when the population standard deviation is known

• Construct a confidence interval for a population mean when the population standard deviation is unknown

• Apply a finite-population correction factor (PCF)• Calculate the sample size for sampling• Define “relative standard error,” (RSE)

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Parameters of Interest

• Population characteristics• Mean, 𝜇• Standard deviation, 𝜎

• Sample characteristics• Mean, 𝑋• Sample standard deviation, 𝑠

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Point and Interval Estimates

• A point estimate is the statistic, computed from sample information, which is used to estimate the population parameter

• A confidence interval estimate is a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence.

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Factors Affecting Confidence Interval Estimates

The factors that determine the width of a confidence interval are:1. The sample size, n2. The variability in the population,

usually 𝝈, or estimated by s3. The desired level of confidence4. Relative size of sample

𝑋 ± (𝑧 𝑜𝑟 𝑡)(𝜎 𝑜𝑟 𝑠)

𝑛∗

𝑁 − 𝑛𝑁 − 1

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Using the SamplingDistribution of the Sample Mean

(Sigma Known) • If a population follows the normal

distribution, the sampling distribution of the sample mean will also follow the normal distribution

• To determine the confidence interval using:

nXzs

µ-=

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Example: Confidence Interval for a Mean, 𝝈 Known

The American Management Association wishes to have information on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following question:

What is a reasonable range of values for the population mean?

Suppose the association decides to use the 95 percent level of confidence:

The confidence limits are $45,169 and $45,671The ±$251 is referred to as the margin of error

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Confidence Interval Interpretation

• For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated

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Simulating Confidence Intervals

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Using the SamplingDistribution of the Sample Mean

(Sigma Unknown) • If the sample is used with the normal

distribution, it is less accurate • William Gosset created the student t

distribution (~1900)• To determine the probability a sample

mean falls within a particular region, use:

nsXt µ-

=

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Characteristics of thet distribution

• It is, like the z distribution, a continuous distribution.

• It is, like the z distribution, bell-shaped and symmetrical.

• There is not one t distribution, but rather a family of t distributions. All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n.

• The t distribution is more spread out and flatter at the center than the standard normal distribution. As the sample size increases, however, the t distribution approaches the standard normal distribution.

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Confidence Interval Estimates for the Mean

Use Z distribution• If the population

standard deviation is known, or the sample is greater than 30.

Use t distribution• If the population

standard deviation is unknown.

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Comparing the z and tDistributions when n is small

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When to Use the z or tDistribution for Confidence

Interval Computation

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Example using the t distribution

• A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?

nstX

sx

n

n 1,2/

unknown) is (since dist.-t theusing C.I. theCompute

09.032.0

10:problem in theGiven

-±

==

=

a

s

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Using the t distribution Table

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Confidence Interval Estimates Excel’s “Data Analysis ToolPak”

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Finite Population Correction Factor

• When the population sampled is not infinite!

𝑁 − 𝑛𝑁 − 1

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Example of FPC

• Sample 15 students from a course containing 96 students

96−1596−1 = .923

• Reduce the confidence interval by 92.3%

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Selecting a Sample Size

There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population. They are:

• The degree of confidence selected• The maximum allowable error• The variation in the population

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Sample Size Determination for a Variable

• To find the sample size, 𝒏 , for a variable:

𝒏 =𝒛 ∗ 𝒔𝑬

𝟐

Where:E = The allowable errorz = the z value corresponding to the selected level of confidences = the sample deviation (from sample)

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Example: Sample Size Determination

A student in public administration wants to determine the mean amount members of city councils in large cities earn per month as remuneration for being a council member. The error in estimating the mean is to be less than $100 with a 95 percent level of confidence. The student found a report by the Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size?

Given in the problem:• E, the maximum allowable error, is $100• The value of z for a 95 percent level of

confidence is 1.96, • The estimate of the standard deviation is

$1,000

38516.384)6.19(100$000,1$96.1

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Sample Size Determination Example 2

• A consumer group would like to estimate the mean monthly electricity charge for a single-family house in July within $5 using a 99 percent confidence level. Based on similar studies the standard deviation is estimated to be $20. How large a sample is required?

1075

)20)(58.2( 2

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çèæ=n

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Sample Size, σ Unknown

• Use Excel’s “Goal Seek” utility

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Variability of the Spread

• Standard error= ⁄𝜎 𝑛, or = ⁄𝑠 𝑛

• Relative standard error (RSE)

=⁄(𝜎 𝑜𝑟 𝑠) 𝑛

𝑋• Typically, not stated, but the smaller the

better

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Summary

• Simple concepts, difficult calculations• Confidence interval• A new table, t distribution, and new logic• Variability precision• Finite-correction factor

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