Chapter 7 Sampling and Point Estimation Sample This Chapter 7A

This Week in Prob & Stat today fine print warning: while today’s presentation is mostly conceptual, Thursday’s presentation will be much more mathematical.

7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing Fundamental Problem: Given that X 1, X 2, …, X n is a random sample from some unknown population, what can be said about the population? For example, what is the distribution, mean, variance, median, range, etc.

The Big Picture Again population descriptive statistics parameter (e.g. mean) sample descriptive statistics statistic (e.g. sample mean) probability theory deduction induction (inferential statistics)

Statistics and Sampling Statistical Inference: Draw conclusions about a population based on sample. Hypothesis tests and parameter estimation. Population: Generally impossible or impractical to observe an entire population. Be aware that population may change over time. Sample: A subset of observations from a population. Must be representative of the population. Must be chosen randomly to avoid bias.

Parameter Estimation a population

Estimators

Sampling – A Pictorial Presentation X f(x) Population Random Sample X 1, X 2, …, X n  Sample  22 X i ~ Population( ,  2 )

Sampling Distributions The probability distribution of a statistic is called a sampling distribution. - Definition makes sense. Statistic is a property of a sample from a population. - Depends on the population distribution, sample size, and method of sample selection. - Key statistics are things like the sample mean, variance, proportion, and difference of two means.

Definition of a Statistic Statistic – any function of the observations in a random sample. Examples of point estimates: A Sampling Distribution is the probability distribution of a statistic.

Sampling Distributions cont’d If the X i have a normal distribution, then so does the sample mean. The X i are I.I.D.R.V.

7.2 Sampling Distributions and the Central Limit Theorem Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definitions:

The Central Limit Theorem

Figure 7-1 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).]

What does it take to become normal? Who are you calling normal?

More Normalcy

The CLT in Action Ten people from a population having a mean weight of 190 lb. with a variance of 400 lb 2 get on an elevator having a weight capacity of 2000 lb. What is the probability that their average weight exceeds 200 lb. and they all fall to their death?

Some Normal Thoughts on the Central Limit Theorem (CLT) CLT tells us that a distribution of means will always be nearly normal if the sample is large enough. Heights of adults are a result of both genetic and environmental factors. they are polygenic – influenced by many different genes also many environmental factors; e.g. nutrition and childhood diseases eventual height is then an average sample from the large population of “height factors” Height therefore has a normal distribution The tall and the short of it.

However All is not Normal Weights of individuals are not normally distribution It is not just the cumulative result of many small factors In addition, there may be one or two dominant causes of obesity; e.g. a glandular disturbance In a similar manner, income is not normally distributed as a result of a dominant factor – e.g. inherited wealth On the other hand, mental test scores tend to be normally distributed due to many determinants: e.g. genetics and long-term environmental conditions

CLT Revisited The Main Result (of all time!):

The sum of all Random Variables A doctor spends an average (mean) of 20 minutes with each patient with a standard deviation of 8 minutes. Today’s appointment book shows 10 patients scheduled this morning (8 – 12). The good doctor has a luncheon appointment at noon before her afternoon golf outing. What is the probability she will make the luncheon on time?

Difference in Sample Means Approximate Sampling Distribution

More Mean Differences

A Normal Difference Example The section 1 class in ENM 661 consisting of 24 students had an average score of 82.7 on their midterm while section 2 consisting of 16 students scored an average of What is the probability that their average scores would differ by at least 1.3 if  1 -  2 = 0. Assume the population standard deviations are known where  1 = 10 and  2 = 12.

Now begins the discussion on point estimation The discussion on the central limit theorem has now ended.

Definition of Point Estimate s 2 is a population parameter, S 2 is a point estimator of s 2. The estimate of S 2 is s 2. S 2 has a sampling distribution. But s 2 does not – it is just a number. Point Estimate  ˆ  is a point estimate of some population parameter  of a statistic ˆ .  ˆ  is apoint estimator of . After a sample has been selected ˆ  takes on a particular value ˆ .  ˆ  is a random variable, ˆ  is not, e.g.

Properties of Estimators What makes a good estimator? What is the best estimator for a population parameter? Bias - does it hit the target? Variance – estimate is based on a sample Standard Error and Estimated Standard Error Mean Squared Error and Efficiency Consistency – how does the estimator behave as the sample size increases? Sufficiency – does the estimator use all the information that is available?

Bias of the Estimator Is the sample mean unbiased?

7-3 General Concepts of Point Estimation Unbiased Estimators Definition

Example 7-1

Example 7-1 (continued)

A Biased Estimator Define an estimator for the population variance to be:

7-3.2 Variance of a Point Estimator Definition Figure 7-5 The sampling distributions of two unbiased estimators

Variance of Estimator Sample mean is the MVUE for the population mean for a population with normal distribution. Generally, the stat package you use is making the reasonable choices for you. Example of bad choice: sample size n=2 Method 1: estimate mean as (X 1 + X 2 )/2 Method 2: estimate mean as (X 1 + 2X 2 )/3 Variance of method 1 is  2 /2 Variance of method 2 is 5 2 /9

7-3.2 Variance of a Point Estimator I just knew it was going to be the sample mean.

BLUE Estimator Best Linear Unbiased Estimator (BLUE) Best is defined as the minimum variance estimator from among all unbiased linear estimators Is the sample mean a BLUE estimator for the population mean?

An Engineering Management Bonus Round!!!! A real world example of sampling, parameter estimation, and fishing for the correct answer.

How many Fish are in the Lake? Let N = the number of fish in the lake k = the number of fish caught and tagged, and released back into the lake allow for the tagged fish to be uniformly dispersed within the lake X = a RV, the number of tagged fish caught in the follow- on sample of size n Then an estimate for the number of fish in the lake is found by assuming

Is N-hat unbiased, BLUE, or MVUE? counting the fish in the lake

Point Estimation - to be continued next time - making standard errors - those magic moments - maximizing likelihoods ENM 500 students engaged in random sampling. ENM 500 students caught discussing the central limit theorem.