Estimation(Point Estimation)

Statistical Inference for ManagersLecture- 5

ByImran Khan

Estimation

We are given information of a sample and using this information, we estimate any quantity of population.Point Estimation: The objective of point estimation is to obtain a single number from the sample which will represent the unknown value of the population parameter.Estimator: An estimator is a sample statistic used to estimate a population parameter.

Properties of a good Estimator• Unbiasedness

• Efficiency • Consistency

Unbiasedness:The point estimator θ^ is said to be an unbiased estimator of θ if the expected value or mean of the sampling distribution of θ^ is θ.

Example: The sample mean and sample variance are unbiased estimators of their population parameters.

Bias

The Bias in an estimator θ^ is defined as the difference between its mean and θ Bias (θ^)= E(θ^)- θ

Properties of a good Estimator

Efficiency:Is another property of a good estimate which refers to the size of standard error of statistics. So the most efficient estimator will be the one having smaller standard error.

An estimator with a smaller standard error will produce an estimate closer to the population parameter.

Properties of a good EstimatorConsistency: A statistic is consistent estimator of population parameter if the value of the statistic comes very close to the population parameter as the sample size increases.

Consistency is a large sample property.

Properties of a good Estimator

The sample mean x7 is the best estimator of μ as it is un-biased, consistent and the most efficient estimator. Point Estimate of the population variance:

E(s²)= σ²- Proof required!

Var(x)7= σ²/n

EstimationInterval estimation:An Interval estimate describes a range of values within which a population parameter is likely to lie.

We represent confidence interval by the quantity(1- α).

P= sample proportionπ= population proportion

Estimation

Case-1:Interval for population mean when population standard deviation is known:

Example

Upon collecting a sample of 250 from a population with known standard deviation of 13.7, the mean is found to be 112.4.a) Find a 95% confidence interval for the mean.b) Find a 99% confidence interval for the mean.

Case-2: Interval for Population mean when population standard deviation is unknown and n>30:

Case-3: Interval for population mean when population standard deviation is unknown and n<30

Using t-distribution

Example:

A business school placement officer wants to estimate the mean annual salaries of the school’s former students 5 years after graduation. A random sample of 25 such graduates found a sample mean of $42,740 and a sample standard deviation of $4,780. Assuming that the population distribution is normal, find a 90% confidence interval for the population mean.T-Table= 1.711

Case- 4: Interval for difference of two population means when population standard deviations σ1 & σ2 are known:

Example:The following data is given:X1bar= 4000X2bar= 3500σ1= 500n1= 16σ2= 300n2= 9Find a 95% interval for the difference of two population means?

Case-5: Interval for difference of two population means when σ1 & σ2 are unknown and n1>30, n2>30.

Case-6: Interval for difference of two population means when σ1 & σ2 are unknown and n1<30, n2<30.

Case-7: Interval for population proportion when σ is unknown

Case-8: Interval for difference of two population proportions

Example-1:In a random sample of 120 large retailers, 85 used regression as a method of forecasting. In an independent random sample of 163 small retailers, 78 used regression as a method of forecasting. Find a 99% confidence interval for the difference between the two population proportions?

Example-2:Pair Drug-A Drug-B1 29 262 32 273 31 284 32 275 32 306 29 267 31 338 30 36

Using the above data, estimate with a 99% confidence the mean difference in the effectiveness of the two drugs A & B, to lower cholesterol.

Table values for Interval Estimation:

90% Confidence Interval- z= 1.6495% Confidence Interval- z= 1.9699% Confidence Interval- z= 2.58