point_and_interval_estimation_for_population_mean.pdf

7/30/2019 Point_and_Interval_Estimation_for_Population_Mean.pdf

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Point and Interval

Estimation for PopulationMean

Ravindra S. Gokhale

IIM Indore

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Case Leasing

An auto manufacturer leases cars to small businesses for use in

visiting clients and other business travel. The contracted lease does

not specify a mileage limit and instead includes a depreciation fee of

$0.30 per mile. The contract includes other origination, maintenance,

and damage fees in addition to the fee that covers the mileage.

These leases run for a year. A sample of 150 cars (all were a

particular model of four-door sedan) returned to their dealers

approximately towards the end of this program averaged 21,000

miles, with standard deviation s = 2,352 miles. Currently this

manufacturer has leased approximately 10,000 of these vehicles.

When the program was launched, the planning budget projected that

the company would earn (in depreciation fees) $6,500 on average

per car.

2Source: Statistics for Business Decision Making and Analysis by Stine and Foster


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Case Leasing

Motivation:

Should the manufacturer assume that if it were to check every leased car,

the average would be 21,000 miles driven?

Can the manufacturer use a confidence interval to check on the claim of

$6,500 earnings in depreciation fees?

Method:

Are the conditions using a 95% confidence interval for the mean number

of miles driven per year satisfied?

Does the method of sampling raise any concerns?

Can the manufacturer estimate, with a range, the amount it can expect to

earn in depreciation fees per leased vehicle, on average?



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Case Leasing

Mechanics:

Construct the 95% confidence interval for the number of miles driven per

year on average for leased cars of this type.

Construct the 95% confidence interval for earnings over the one-year

period of the lease, in a form suitable for presentation.

Message:

Interpret the 95% confidence interval for the number of miles driven over

the one year period of the lease.

Interpret the 95% confidence interval for the average amount earned per

vehicle. What is the implication fee for the budget claim?

Communicate a range for the total earnings of this program, assuming

10,000 vehicles.



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Case Leasing

Summary:

Auto manufacturer leases cars to small businesses.

For each mile car has travelled, manufacturer gets $0.30.

Lease runs for one year.

Manufacturer is expecting to earn $6500 on an average per car.

Currently the manufacturer has leased approximately 10000 cars

of a particular model.

Business Question: Are we going to make profits as expected?

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Case Leasing

Data in hand:

A sample of 150 cars returned to the dealers approximately at the

end of this program averaged 21,000 miles with standard

deviation 2352 miles.

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Case Leasing

Statistical Problem:

What will be the average mileage for all the 10000 cars leased?

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Estimating Population Mean UsingSample Data

Point Estimation:

Point estimator is a sample statistic that best describes the

population parameter.

e.g. Sample mean is an estimate of population mean

Notation: An estimator of a parameter is denoted by ^ over the parameter

symbol

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Case Leasing

An estimate of the mileage?

An estimate of the profit?

Based on the point estimate of the average profit, what do you

conclude about the average profit from all 10000 cars?

Is it good to have just one number as the estimate?

If yes, how? If no, what can be a solution?

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Interval Estimation

Instead of using one number (point estimate) to describe the

parameter, we want to use an interval to describe the parameter.

Such intervals are popularly known as confidence intervals.

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Confidence Interval

For a given sample, if x-bar is the sample mean of a random

sample of size n from a normal population with a known

variance 2, then a 100(1 )% confidence interval on is

given by:

where, z/2 is the upper 100(/2) percentage point of the

standard normal distribution

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nz+x

nz-x /2/2


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Confidence Interval

For example, if x-bar is the sample mean of a random sample

of size n from a normal population with a known variance 2,then a 95% confidence interval on is given by:

How do you get 1.96?

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n

1.96+x

n

1.96-x


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Confidence Interval (cont)

Development of a confidence interval

X-bar ~ Normal(, 2/n)

Converting it to standard normal distribution,

Z = [X-bar ] / [ / square-root(n)]

We write as:

Re-arranging the terms:

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)(1}zn

XzP{ /2/2

)(1}n

zX

n

zXP{ /2/2


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Case Leasing

Can we assume that X150 follows a Normal Distribution?

Yes, because of Central Limit Theorem

Calculate a 95% confidence interval for in the case using X150

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Case Leasing

How to interpret the Confidence Interval that you have calculated?

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Confidence Interval

Example of interpreting a 95% Confidence Interval:

Under repeated sampling, on an average out of 100 confidence

intervals, 95 will contain the true population parameter.

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Interpretation of the confidence interval:

A Confidence Interval is a random interval (depends on the

content of the sample used to calculate it)

Wrong interpretation (example):

The probability that true value of lies within the 95%

confidence interval is 0.95

Correct interpretation (example):

If a large number of random samples are collected and 95%

confidence interval for is computed from each sample, then

95% of these intervals will contain the true value of

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In real life, do you have the luxury to collect infinite number of

random samples?

Then how will you interpret it?

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It is the faith (or the confidence) that you can put in the interval

that you have calculate.

Practical interpretation:

You have a 95% confidence that the interval calculated by

you contains the true value of the population parameter.

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Popular choices of the confidence intervals:

90%, 95%, 99%

Implications of various percentages?

Which is better, a wider confidence interval or a narrower

confidence interval?

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Case Leasing

What can you conclude?

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Case Leasing

While calculating the 95% confidence interval for the average

mileage of the 10000 cars, we used s instead of . Is that

correct?

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Case Leasing



correct?

No

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The t-distribution

When calculating confidence interval using s instead of, we have

to use percentiles from t-distribution instead of z-distribution.

The t-distribution:

follows a t-distribution with n-1 degrees of freedom.

What is meaning ofdegrees offreedom?

Understanding the t-distribution table.

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1-nt~ns-X

ns

-X


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Standard Error

is the standarderror of the estimate

In general,

Standard Error in reporting a point estimate:

The point estimate is a statistic

The statistic is a random variable and hence subject to variation

Standard error is the standard deviation of the point estimator

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ns


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The t-distribution

For a larger sample size (say n > 40), t distribution can be

approximated well by a normal distribution. Hence for a larger

sample size, a z-percentile will do a good job irrespective of

whether sample standard deviation is used for calculating

confidence interval or population standard deviation is used for

calculating confidence interval.

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Case Leasing



correct?

Since the sample size is 150 which is quite large, the t-distribution can be approximated by a Normal Distribution.

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point_and_interval_estimation_for_population_mean.pdf

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