ES04

Statistical Inference

Parameter Estimation

Hypothesis Testing

Concept

To estimate a parameter, sample statistics will be used (point estimator); once the sample is selected, a numerical value of the statistic is given (point estimate)

Important Questions

Which point estimator to choose?

How to compare unbiased estimators?

How to compare estimators?

Relative Efficiency

e.g. the relative efficiency of a single observation to the sample mean is less than 1, so the sample mean is a better estimator

How do we compare biased estimators?

MSE

e.g. the sample std. dev. is a biased estimator of the population std. dev.

How precise is our estimation?

Standard Error and Estimated Standard Error

MVUE

e.g. the sample median, the sample mean, and a single observation are all unbiased estimator

How do we estimate a parameter form data?

How to decide whether to accept or reject a statement about a parameter?

to make decisions or to draw conclusions about a population using a random sample

Applications

Design and engineering specifications, to verify a theory, to determine if a process has changed

Concept

A statement about the parameters of one or more population is made (a statistical hypothesis). To test it, a random sample from the population is taken. A test statistic is computed and if the information in the sample is consistent with our hypothesis, we conclude it is true

Important Questions

How do we (dis)prove the hypothesis?

We can't, unless we examine the entire population. We can only infer. So there's the possibility that our conclusion is incorrect

What if our conclusion is incorrect?

Type I Error (α)

Type II Error (β)

failing to reject the null hypothesis when it's false

rejecting the null hypothesis when it's true

Which hypothesis can we formulate?

Infer on the population variance

Infer on the population proportion

Confidence Interval

Concept

We are often interested in finding an interval in which we would expect to find the true value of a parameter

Infer on the population mean

If σ is known, then we use the z-test

If σ is unknown, and the sample is small: we assume the distribution is normal and use the t-test

If σ is unknown, and the sample is large (n≳40): s≅σ and we can use the z-test regardless of the population distribution

If σ is unknown, and the sample size is small, and the normal distribution assumption is unreasonable, we use different techniques