Sampling Distributions in Statistics: Understanding the Variability of Sample Statistics, Study notes of Calculus

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Distributions of SamplingStatistics: Part I^ Cyr Emile M’LAN, Ph.D.^ [email protected]^ Distribution of Sampling Statistics: Part I

Introduction

♠^ Text Reference

:^ Introduction to Probability and

Statistics for Engineers and Scientists, Chapter 6. ♠ Reading Assignment

:^ Sections 5.8, 6.1, 6.2, November 5 - November 10 The goal of this chapter is to learn about thedistributions of statistics such as the sample mean sothat our conclusion is accompanied by a statement thatindicates the likelihood that our method is correct.

Distribution of Sampling Statistics: Part I

From Population to Samples to Population^ There are two branches of statistical inference:^ Estimation by means of confidence intervals

estimate a population parameter. Hypothesis testing

: decide whether a claim that is made about a population parameter can withstand ornot. If two different samples are taken from the samepopulation, the values taken by sample statistic may bedifferent for those two samples. Thus,

a statistic is also a random variable

This variability leads to uncertainty as to whether, forexample, our estimate or in general our inference iscorrect.

Distribution of Sampling Statistics: Part I

From Population to Samples to Population^ Therefore, we need a way to assess the reliability of ourinference made about a population based on a singlesample data.^ The measure of reliability is actually a statement ofprobability that describes how likely such an outcome isto occur.^ Because the statistic used to carry out the inference isa random variable, it must have a distribution. Itsdistribution is called

sampling distribution

The statement of probability that accompany anyinferential statistics relies heavily of the samplingdistribution of the statistic.

Distribution of Sampling Statistics: Part I

Hypothesis Testing Example 6.2: A study was designed to test a new blood pressuredrug. In a sample of 50 participants with high bloodpressure, the drug was found to lower blood pressureby 17 points. H: New blood pressure drug does not lower blood 0 pressure H: New blood pressure drug does lower blooda pressureThe sample data shows that for the 50 participants, theaverage drop in blood pressure is 17 points.

Is this one-time performance of the drug significant enough toconclude that the drug effectively lowers bloodpressure?^

Distribution of Sampling Statistics: Part I

The Freshman 15 Problem Do college students really gain weight duringtheir freshman year? This statement expresses a curiosity about something.However,^ how do one express the question posed interms of a population parameter? As we will see, there are many way the question abovecan be answered. The researcher and the statistician will then have to sitand decide what population parameter could be used toanswer this specific question?^ Distribution of Sampling Statistics: Part I

The Freshman 15 Problem We design a question about

p, the population

proportion. What proportion of first year students weighmore at the end of the year than at thebeginning? Take a sample of size

n^ of first year students and measure their weight at the beginning of the study andat the end of the study. Let

X^ represent the number of students that reported some gain in weights. Sample statistic is

x̂ p = n Calculate sample statistic

̂ p, the proportion of students in the sample who gained weight. To estimate^ p^ use the rule:

Distribution of Sampling Statistics: Part I ̂ p ±^ margin of error

The Freshman 15 Problem We design a question about

p−^ p, the difference^1

between two population proportions from independentsamples. What is the difference in populationproportion between the first year collegestudents that gained weight and all thepeople of same age who are not collegestudents who gained weight? Take a sample of first year college students andcalculate the proportion,

̂ p, of these that report a^1 weight gain by comparing their weight at beginning ofthe year to their weight at the end of the year.

Distribution of Sampling Statistics: Part I

The Freshman 15 Problem We design a question about

μ, the mean of pairedd

differences from dependent samples. What is the mean difference of weight gainbetween the first year students at the end ofthe year compared to at the beginning? Take a sample of first year students and record weight Xat the beginning of the year and weight at the end ofb^ the year,^ X.e Calculate the difference

d^ =^ x−^ xfor each first yeare^ b^ student in the sample and then

d^ the mean of the

d values. Use sample statistic

d^ to estimate the population parameter,^ μ, asd

Distribution of Sampling Statistics: Part I d ±^ margin of error

• p. 13/

The Freshman 15 Problem We design a question about

μ−^ μ, the difference^1

between two population means from independentsamples. What is the mean difference of weight gainbetween first year college students and thepopulation of all people of same age who arenot college students? Take a sample of first year students and calculate theirweight gain by comparing their weight at beginning ofthe year to their weight at the end of the year,

x.^1 Take a sample of from the population of all people ofsame age who are not college students and calculateweight gain by comparing their weight at beginning ofthe year to their weight at the end of the year,

x.^2 Distribution of Sampling Statistics: Part I

Joke

I would have completed my work but ... My computer was down. Well, see, I was babysitting and the only way to keepthe kid from crying was to feed them my homework. The phone kept ringing all night, and because no oneelse was home I was forced to answer it. My dog was really sick and fell asleep right on myschoolbooks, and my mom was afraid that the dogmight throw up on the clean rug if I woke him up. I didn’t finish dinner last night, and my mom wouldn’t letme leave the table to do my homework. My uncle who loves me the most says I don’t have to doit, and uncle knows best.

Distribution of Sampling Statistics: Part I

Sampling Distribution The sampling distribution^ is the probability distributionfor a sample statistic. It plays the same role asprobability distributions do for other random variables. The sampling distribution for a statistic describes howvalues of a sample statistic vary across all possiblerandom samples of a specific size that can be takenfrom a population. Three key questions about a sampling distribution for asample statistic come in mind: What is the^ center^ of the sampling distribution? What is the^ spread^ of the sampling distribution? What is the^ shape^ of the sampling distribution?^ Distribution of Sampling Statistics: Part I

Sampling Distribution Unfortunately,^ The probability distribution of anystatistics, in particular, its shape, center and dispersion depends heavily on the population distribution (binomial, poisson, hypergeometric, uniform,exponential, gamma, normal, etc),

the sample size^ n

and^ the sampling method

(whether it is done with or without replacement). The goal of this chapter is to derive the theoreticalsampling distribution of some statistics used to makestatistical inferences about some population parameters(often^ μ,^ μ−^ μ,^ p^1

(^2) , p− p, and σ). (^1 2) Distribution of Sampling Statistics: Part I

• p. 19/

Simple Random Sample The random variables^ X, X,^12

· · ·^ , Xare said to form an^ simple random sample (SRS)

of size^ n^ if

1. The^ X’s are^ independent random variablesi

2. Every^ Xhas the same distribution, i.e they arei^ identically distributed

These two conditions can be combined by saying thatthe^ X’s are^ independent and identically distributedi (i.i.d.). These conditions are automatically satisfied if the sampling is donewith replacement or from an infinite (conceptual) population. Otherwise, under sampling without replacement scheme, it isrequested that the sample size

n^ be at most 5% of the population size^ N^ for conditions 1. and 2. to be satisfied approximately.

Distribution of Sampling Statistics: Part I