Point Estimation and Confidence Interval

Point and Interval Estimation

Arun Kumar, Ravindra Gokhale, and NagarajanKrishnamurthy

Quantitative Techniques-I, Term I, 2012Indian Institute of Management Indore

Leasing

An auto manufacturer leases cars to small businesses for use invisiting clients and other business travel. The contracted leasedoes not specify a mileage limit and instead includes adepreciation fee of $0.30 per mile. The contract includes otherorigination, maintenance, and damage fees in addition to thefee that covers the mileage. These leases run for a year. Asample of 150 cars (all were a particular model of four-doorsedan) returned to their dealers early in this program averaged21,000 miles, with standard deviation s = 2,352 miles.Currently this manufacturer has leased approximately 10,000of these vehicles. When the program was launched, theplanning budget projected that the company would earn (indepreciation fees) $6,500 on average per car.

Leasing

Motivation:a. Should the manufacturer assume that if it were to checkevery leased car, the average would be 21,000 miles driven?b. Can the manufacturer use a confidence interval to check onthe claim of $6,500 earnings in depreciation fees?Method:c. Are the conditions using a 95% confidence interval for themean number of miles driven per year satisfied?d. Does the method of sampling raise any concerns?e. Can the manufacturer estimate, with a range, the amount itcan expect to earn in depreciation fees per leased vehicle, onaverage?

Leasing

Mechanics:f. Construct the 95% confidence interval for the number ofmiles driven per year on average for leased cars of this type.g. Construct the 95% confidence interval for earnings over theone-year period of the lease, in a form suitable for presentation.Message:h. Interpret the 95% confidence interval for the number ofmiles driven over the one year period of the lease.i. Interpret the 95% confidence interval for the averageamount earned per vehicle. What is the implication fee for thebudget claim?j. Communicate a range for the total earnings of this program,assuming 10,000 vehicles.

Case: Leasing

Auto manufacturer leases cars to small businesses.

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

Lease runs for one year.

Manufacturer is expecting to earn $6500 on an averageper car.

Currently the manufacturer has leased approximately10000 cars of a particular model.

Business question

Are we going to make profits as expected?

Case: Leasing

A sample of 150 cars returned to the dealers early in thisprogram averaged 21,000 miles with standard deviation 2352miles.

Statistical problem?

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

Let x denote mileage for a leased car and x (, 2).Goal is to estimate .

Estimating population mean using the sample

data

Point Estimation: Point estimator is a sample statistic thatbest describes the population parameter.Ex: = x

Leasing

One estimate of the average mileage for 10000 cars is 21,000miles.

So one estimate of average profit=21000 0.3=$6300.Based on the point estimate of the average profit, what doyou conclude about the average profit from all 10000 cars?

Interval Estimation

Instead of using one number (point estimate) to describe theparameter, we want to use an interval to describe theparameter.

Such intervals are popularly known as confidence intervals.

Sampling distribution of sample mean

Let X150 denote average mileage for 150 cars. ThenX150 N(, 2/150).

X150 follows a normal distribution from the Central LimitTheorem because the sample size 150 is very large.

(1 )% Confidence Interval for population mean

(1 )% confidence interval for population mean is

x z2

n,

where z2

is (1 2

)th percentile of standard normaldistribution.For example: A 95% confidence interval for isx 1.96/n where x is the sample mean and is thepopulation standard deviation.

Confidence Coefficient

Confidence coefficient (such as (1 )%) tells us how muchfaith we should put in the confidence interval.

Popular choices are 90%, 95%, and 99%.

Interpreting a 95% confidence interval

Under repeated sampling, out of 100 confidence intervals 95will contain the true population parameter.

Leasing: x = 21, 000 miles

A 95% confidence interval for average mileage is21000 1.96 2352/150 = [20623.6 miles, 21376.4 miles]

Leasing: Average Profit

[20623.6 0.3, 21376.4 0.3]=[$6187.08,$6412.92]

Understanding terms in the confidence interval

formula

Larger confidence level implies larger z2

. Larger z2

leadsto wider confidence intervals. Narrower confidenceintervals are more informative.

Larger sample size leads to narrower confidence intervalbut a larger sample size comes at a higher cost.

Conclusion

Can we conclude that average earning per car will be $6500?

Caveat

While calculating the 95% confidence interval for the averagemileage of the 10000 cars, we used s instead of . Is thatcorrect?

Ans. No

Caveat

When calculating confidence interval using s instead of , wehave to use percentiles from t-distribution instead ofz-distribution.

t-distribution

Xns/n

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

s/n is also known as standard error.

t-distribution

Find 97.5th percentile for a t-distribution with degrees offreedom

5

11

40

t-distribution

For a larger sample size (say n > 40), t distribution can beapproximated well by a normal distribution. Hence for a largersample size, a z-percentile will do a good job irrespective ofwhether sample standard deviation is used for calculatingconfidence interval or population standard deviation is used forcalculating confidence interval.

What about the Leasing Problem?

We are good because the sample size is very large.

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Point Estimation and Confidence Interval