Last lecture: Point Estimation

A point estimator is function of the observations in a randomsample which is used to estimate an unknown parameter .A point estimate of some unknown population parameter isa single numerical value of a statistic.

Recall : A Parameter is a number that describes some aspect of a population.

Intuition behind hypothesis testing

We use the statistic from a sample as a point estimate for a population parameter.

• Point estimates will not match population parameters exactly, but they are our best guess, given the data

• Sample statistics vary from sample to sample.

KEY QUESTION: For a given sample statistic, what are plausible values for the population parameter? How much uncertainty surrounds the sample statistic?

KEY ANSWER: It depends on how much the statistic varies from sample to sample!

Lincoln’s Gettysburg Address

“Four score and seven years ago our fathers brought forth, on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battle-field of that war. We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we can not dedicate—we can not consecrate—we can not hallow—this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they here gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.”

Example: Lincoln’s Gettysburg Address• Last class, each of you computed the average length of 30 words

randomly sampled from Lincoln’s Gettysburg Address. The number that you computed is

a) A point estimateb) A point estimator

Ans : A)

Example: Lincoln’s Gettysburg Address• Last class, each of you computed the average length of 30 words

randomly sampled from Lincoln’s Gettysburg Address. The number that you computed is

a) A sample statisticb) A population parameter

Ans : A) A statistic is a function of the random variables in a random sample.

Example: Lincoln’s Gettysburg Address• After you computed the sample average. I made a histogram of the

numbers you reported. This shows us:

a) The population distributionb) The sampling distribution

Ans : B) A sampling distribution is the distribution of sample statistics computed for different samples of the same size from the same population

If we increase the number of words that you select, and repeat above procedures. What do you expect about the distribution of the sample mean?

a) Similar to the population distribution

b) Similar to a normal distribution

Ans: B). This is because Central Limit Theorem. See R code

CLT Example: Lincoln’s Gettysburg Address

Two basic criteria to compare point estimators

Remember the key question is to measure how sample statistic varies from sample to sample

Standard Error

a) higherb) lower

The standard error measures how much the statistic varies from sample to sample.

Are both estimators unbiased estimator?

What is the variance of each estimator?

What is the MSE of each estimator?

Which one is a more efficient estimator?