Estimation (Point Estimation) Statistical Inference for Managers Lecture- 5 By Imran 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 Estimator Consistency: 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 x̅ 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)̅= σ²/n

Estimation Interval 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 proportion π= 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. Find a 95% confidence interval for the mean. 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= 4000 X2bar= 3500 σ1= 500 n1= 16 σ2= 300 n2= 9 Find 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?

Using the above data, estimate with a 99% confidence Example-2: Pair Drug-A Drug-B 29 26 32 27 31 28 32 30 31 33 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. T-distribution

Table values for Interval Estimation: 90% Confidence Interval- z= 1 Table values for Interval Estimation: 90% Confidence Interval- z= 1.64 95% Confidence Interval- z= 1.96 99% Confidence Interval- z= 2.58