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MANE 3332.03

Chapter Seven

Handouts

Chapter 7 Overview

  • Chapter 7 contains a detailed explanation of point estimates for parameters

  • Much of this chapter is of a highly statistical nature and will not be covered in this course

  • Key concepts we will discuss are:

    • Statistical inference

    • Statistic

    • Sampling distribution

    • Point estimator

    • Unbiased estimate

    • MVUE estimator

    • Central limit theorem

    • Sampling distributions

Statistical Inference

  • Montgomery gives the following description of statistical inference.

    The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. There methods utilize the information contained in a sample from the population in drawing conclusions. This chapter begins our study of the statistical methods used for inference and decision making.

  • Statistical inference may be divided into two major areas: parameter estimation and hypothesis testing

Point Estimate

  • Montgomery states that "In practice, the engineer will use sample data to compute a number that is in some sense a reasonable value (or guess) of the true mean. This number is called a point estimate."

  • Discuss examples

  • A formal definition of a point estimate is

    A point estimate of some population parameter \(\theta\) is a single numerical value \(\hat{\theta}\) of a statistic \(\hat{\Theta}\). The statistic \(\hat{\Theta}\) is called the point estimate.

  • Notice the use of the "hat" notation to denote a point estimate

Statistic

  • Point estimate requires a sample of random observations, say \(X_1,X_2,\ldots,X_n\)

  • Any function of the sampled random variables is called a statistic

  • The function of the random variables is itself a random variable

  • Thus, the sample mean \(\bar{x}\) and the sample variance \(s^2\) are both statistics and random variables

Properties of point estimators

  • We would like point estimates to be both accurate and precise

  • An unbiased estimator addresses the accuracy criteria

  • A minimum variance unbiased estimator addresses the precision criteria

Unbiased Estimator

  • The point estimator \(\hat{\Theta}\) is an unbiased estimator for the parameter \(\theta\) if \(\(E\left(\hat{\Theta}\right)=\theta\)\)

  • If the point estimator is not unbiased, then the difference

\[ E\left(\hat{\Theta}\right)-\theta \]

is called the bias of the estimator \(\hat{\Theta}\)

MVUE

  • Montgomery gives the following definition of a minimum variance unbiased estimator (MVUE)

    If we consider all unbiased estimators of \(\theta\), the one with the smallest variance is called the minimum variance unbiased estimator

  • An import fact is that the sample mean \(\bar{x}\) is the MVUE for \(\mu\) when the data comes from a normal distribution

Accuracy vs. Precision

graph of accuracy vs. precision

Sampling Distribution

  • The probability distribution of a statistic is called a sampling distribution

Central Limit Theorem

  • Definition of the Central Limit Theorem is

    If \(X_1,X_2,\ldots,X_n\) is a random sample of size \(n\) taken from a population (either finite or infinite) with mean \(\mu\) and finite variance \(\sigma^2\), and if \(\overline{X}\) is the sample mean, the limiting form of the distribution of \(\(Z=\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\)\) as \(n\rightarrow\infty\), is the standard normal distribution

  • Important result because for sufficiently large \(n\), the sampling distribution of \(\overline{X}\) is normally distribution

  • This is a fundamental result that will be used extensively in the next four chapters of the textbook.