point estimation and confidence interval
TRANSCRIPT
-
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.
-
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.
-
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.
-
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.
-
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.
-
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?
-
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 .
-
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 .
-
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?
-
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
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
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.