IE 3320 Engineering Statisticsie3320.weebly.com/uploads/9/1/3/6/9136035/lecture... · E. Perez IE TXSTATE 32 Sampling Distribution of the Difference between Two Means . E. Perez IE

IE 3320 – Engineering Statistics Lecture 11: Sampling Distributions and Data Descriptions

Eduardo Perez, Ph.D.

Ingram School of Engineering

Texas State University

601 University Dr., San Marcos, TX 78666

[email protected]

http://uweb.txstate.edu/~e_p86

E. Perez IE TXSTATE 1

Ch.8 Fundamental Sampling Distributions and Data Descriptions

mailto:[email protected]

http://uweb.txstate.edu/~e_p86

Today’s Agenda

Chapter 8 Fundamental Sampling Distributions and Data

Descriptions

Statistical Inference

Random Sampling

Some Important Statistics

Sampling Distributions

Sampling Distributions of Means and the Central Limit Theorem

Sampling Distribution of 𝑆2

t-Distribution


Statistics Probability

Measures of Location (mean)

Measures of Variability (variance)

Sample

Event

Counting Sample Points

Probability of an Event

Random Variables

Probability Distributions

Discrete

Continuous

Joint Probability Distributions

Binomial

Hypergeometric

Negative Binomial

Geometric

Poisson

Uniform

Normal

Gamma

Exponential

Lognormal



Learning Objectives

Learning Objectives

Explain the general concepts of estimating the parameters of

a population or a probability distribution

Explain the important role of the normal distribution as a

sampling distribution

Understand the central limit theorem






_________ is concerned with making inferences about

populations and population characteristics.

Population Sample

Deductive Reasoning

(Probability)

Inductive Reasoning

(Inferential Statistics)


Statistical inference is always focused on _______________

about one or more parameters of a __________.

An important part of this process is obtaining __________ of the

_____________.

Suppose that we want to obtain a ____________(a reasonable

value) of a population parameter.

We know that before the data are collected, the observations

are considered to be random variables, say, 𝑋1, 𝑋2, … , 𝑋3

Therefore, any function of the observation, or any _______, is

also a _______________.



For example,

the sample _____ 𝑋 and

the sample ________ 𝑆2

are statistics and they are also _______________.

Since a statistic is a random variable, it has a __________

distribution.

We call the probability distribution of a statistic a _________

distribution.


Random Sampling


Random Sampling

In this lecture, we focus on _________ from distributions or

populations

We provide introduction to the role that:

the sample mean 𝑋 and

the sample variance 𝑆2

will play in statistical inference in later lectures.


Population and Samples

The totality of observations (finite or infinite), constitutes

what we call a ___________

Example: If there are 600 students in the school whom we classified

according to blood type, we say that we have a population of size _____.


Definition 8.1

600

Population and Samples Each __________ in a population is a value of a ________

_________ X having some probability distribution f(x).

Example: If one is inspecting items coming off an assembly line for

defects, then each observation in the population might be a value 0 or 1 of

the Bernoulli random variable X with probability distribution.

where 0 indicates a nondefective item and 1 indicates a defective item.


http://www.williamsonimages.com/#/industry/manufacturing/_Y3C1423

http://www.williamsonimages.com/

Population and Samples In statistical inference, we are interested in

Arriving at __________ concerning a ________ when it is impossible

or impractical to observe the entire _________________.

Example: To determine the average length of life of a certain brand of light

bulb, it would be impossible to test all such bulbs if we are to have any left

to sell.

Therefore, we depend on a _________________from the population

to make inferences concerning that ________________.


http://earthfourenergy.blogspot.com/2012/08/sanjit-bunker-roy-brings-light-to-dark.html















If our ________ from the sample to the population are to be _____,

Obtain samples that are ______________ of the population.

A _________ procedure that produces inferences that consistently

overestimate or

underestimate

some characteristic of the population is said to be _______.

To eliminate any possibility of bias in the sampling procedure,

Choose a random sample were the observations are made ___________ and

at ___________.


Definition 8.2



Definition 8.3



Some Important Statistics Our main purpose in selecting random samples is to draw

information about the __________________ parameters.

It is impossible to question every person in order to compute p= x/N representing the population ______________.

Instead, a large random sample is selected

The proportion 𝒑 = 𝒙/𝒏 of people in this sample favoring the brand of coffee in

question is calculated.

𝑝 is now used to make an ________ concerning the ____ proportion p.


http://www.nih.gov/researchmatters/june2012/06042012coffee.htm

Example: Suppose that we wish to

arrive at a conclusion concerning the

proportion of coffee-drinkers in the

United States who prefer a certain

brand of coffee

http://www.nih.gov/researchmatters/june2012/06042012coffee.htm

Many random samples are possible from the ____ population

we would expect 𝑝 to vary somewhat from ________________

𝑝 is a value of a ________________ that we represent by P.

Such a random variable is called a ________.


Definition 8.4


In Lecture 7, we introduced the two parameters μ and 𝜎2, which

measure the

center of location and

variability

of a probability distribution.

These are ________ population parameters and are not influenced

by the observations of a _______________.

Now, we define some important statistics that describe

corresponding measures of a ________________.

The most commonly used statistics for measuring the center of a set of

data are the ______, _______, and ______


Location Measures of a Sample

Let 𝑋1, 𝑋2, … , 𝑋𝑛 represent n random variables.


Location Measures of a Sample

Sample Mean Sample Median

Sample Mode: is the value of the sample that occurs most often

Example: Suppose a data set consists of the following observations:

0.32, 0.53, 0.28, 0.37, 0.47, 0.43, 0.36, 0.42, 0.38, 0.43

The mode is _______

Let 𝑋1, 𝑋2, … , 𝑋𝑛 represent n random variables.


Variability Measures of a Sample

Sample Variance Sample Standard Deviation Sample Range

Example: A comparison of coffee prices at 4 randomly selected grocery stores in

San Diego showed increases from the previous month of 12, 15, 17, and 20 cents

for a 1-pound bag. Find the variance of this random sample of price increases.



Sampling Distributions The field of statistical inference is concerned with

_____________ and ____________.

The population being sampled here is again finite but very

small.

A ________ is a random variable that depends only on the

_____________, and it must have a probability distribution.


Example: We might state that the average

cost to build a residence in Charleston,

South Carolina, is between $330,000 and

$335,000, based on the estimates of 3

contractors selected at random from the 30

now building in this city http://spot.colorado.edu/~koslicki/photos_house.html

http://spot.colorado.edu/~koslicki/photos_house.html

For example, the probability distribution of 𝑋 is called the

sampling distribution ____________.

The sampling distribution of a statistic depends on:

___________ of the population,

________ of the samples,

_________ of choosing the samples

We will study several of the important _______ distributions

of frequently used statistics.

Applications of the sampling distributions are considered throughout

most of the remaining of this course.


Definition 8.5


The sampling distributions of 𝑋 and 𝑆2 are the mechanisms from

which we will be able to make inferences on the parameters:

The sampling distribution of 𝑋 with sample size n is the

distribution that results when

an experiment is conducted over and over (always with sample size n) and

the ________________ result.

This sampling distribution describes the ___________ of sample averages

around the _______________ μ.

The same principle applies in the case of the _______________.

The sampling distribution produces information about the variability of 𝑠2

values around 𝜎2 in repeated experiments.


What is the sampling distribution of 𝑋 ?

Sampling Distribution of Means and the Central

Limit Theorem


Sampling Distribution of Means and the Central Limit

Theorem

The first important sampling distribution to be considered is that of

the _________

If we are sampling from a population with unknown distribution,

either finite or infinite,

The sampling distribution of 𝑋 will is approximately _______ with mean μ and

variance 𝜎2 /n, provided that the sample size is _______.

This amazing result is an immediate consequence of the following

theorem


Theorem 8.2

Sampling Distribution of Means and the Central Limit

Theorem

The normal approximation for 𝑋 depends on the sample size 𝑛

The figure shows:

the distribution obtained for throws of

a six-sided true die

the distribution of 𝑋 becomes closer

to _______ as n grows _______,

The ______ approximation for 𝑋 will

generally be good if _______

The sample size n = 30 is a

guideline to use for the Central Limit

Theorem.


Montgomery and Runger 5th edition

Example: An electrical firm manufactures light bulbs that

have a length of life that is approximately normally

distributed, with mean equal to 800 hours and a standard

deviation of 40 hours. Find the probability that a random

sample of 16 bulbs will have an average life of less than

775 hours.



Example: Traveling between two campuses of a university in a city via

shuttle bus takes, on average, 28 minutes with a standard deviation of 5

minutes. In a given week, a bus transported passengers 40 times. What

is the probability that the average transport time was more than 30

minutes? Assume the mean time is measured to the nearest minute.


Suppose that we have _____ populations,

the first with mean 𝜇1 and variance 𝜎12, and

the second with mean 𝜇2 and variance 𝜎22.

Let the statistic

𝑋 1 represent the mean of a random sample of size 𝑛1 selected from

the first population

𝑋 2 represent the mean of a random sample of size 𝑛2 selected from

the second population

What can we say about the sampling distribution of the

difference 𝑋 1− 𝑋 2 for repeated samples of size 𝑛1 and 𝑛2?

The ___________________can be easily extended to the two-

sample, two-population case.


Sampling Distribution of the Difference between Two

Means


Sampling Distribution of the Difference between Two

Means

Theorem 8.3

Example: The effective life of a component used in a jet-turbine

aircraft engine is a random variable with mean 5000 hours and

standard deviation 40 hours. The distribution of effective life is

fairly close to a normal distribution. The engine manufacturer

introduces an improvement into the manufacturing process for this

component that increases the mean life to 5050 hours and

decreases the standard deviation to 30 hours. Suppose that a

random sample of 𝑛1 = 16 components is selected from the “old”

process and a random sample of 𝑛2 = 25 components is selected

from the “improved” process. What is the probability that the

difference in the two 𝑋 2− 𝑋 1 sample means is at least 25 hours?

Assume that the old and improved processes can be regarded as

independent populations.


Example: The television picture tubes of manufacturer A have a mean lifetime of 6.5

years and a standard deviation of 0.9 year, while those of manufacturer B have a mean

lifetime of 6.0 years and a standard deviation of 0.8 year. What is the probability that a

random sample of 36 tubes from manufacturer A will have a mean lifetime that is at least

1 year more than the mean lifetime of a sample of 49 tubes from manufacturer B?




If an engineer is interested in the _____________

resistance of a certain type of resistor,

The sampling distribution of 𝑿 will be exploited once the sample

information is gathered.

However, if the __________ in resistance is to be studied,

The sampling distribution of 𝑺𝟐 will be used in learning about the

population variance 𝜎2.

If a random sample of size n is drawn from a normal

population with mean μ and variance 𝜎2,

we obtain a value of the ___________ by computing the sample

variance



The values of the random variable 𝜒2 are calculated from

each ________ by the formula


Theorem 8.4


The _________ that a random sample produces a 𝜒2 value

greater than some specified value is equal to

the area under the curve to the _______________.

It is customary to let 𝜒𝛼2 represent the 𝜒2 value above

which we find an area of α.

Table A.5 gives values of 𝜒𝛼2 for various values of α and v.



Example: A manufacturer of car batteries guarantees that the

batteries will last, on average, 3 years with a standard deviation of

1 year. If five of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5,

and 4.2 years, should the manufacturer still be convinced that the

batteries have a standard deviation of 1 year? Assume that the

battery lifetime follows a normal distribution..


http://www.amazingbattery.com/

http://www.amazingbattery.com/

t-Distribution


Earlier in this lecture, we discussed the utility of the Central

Limit Theorem.

Its applications revolve around inferences on a _________

mean or the difference between ____________ means.

However, it was assumed that the population standard

deviation is __________.

Assumption holds in situations where the engineer is familiar with the

system

In many experimental scenarios, _____________is certainly

no more reasonable than knowledge of

the population mean μ.



Often, an estimate of __ must be _________ by the same sample

information that produced the sample average 𝑥 .

As a result, a natural statistic to consider dealing with inferences

on μ is

since S is the sample analog to 𝜎

If the sample size is large enough, say n≥ 30, the ____________

does not differ considerably from the ________________.




Theorem 8.5



Corollary 8.1


The distribution of T is similar to the distribution of Z in that

they both are

___________________about a mean of zero.

The t-distribution is more variable, owing to the fact that the

T-values depend on the fluctuations of two quantities,

whereas the Z-values depend only on the changes in __ from sample

to sample.

It is customary to let 𝑡𝛼 represent the t-value above which

we find an ______ equal to α.


What does the t-Distribution Look Like?

Since the t-distribution is symmetric about a mean of zero,

we have __________________

The t-value leaving an area of 1− α to the right and

therefore an area of α to the left is equal to

The negative t-value that leaves an area of α in the right tail of the

distribution.

That is, 𝑡0.95 = −𝑡0.05, have 𝑡0.99 = −𝑡0.01 , and so forth.


What does the t-Distribution Look Like?

Example: The t-value with v = 14 degrees of freedom that leaves an area of

0.025 to the left, and therefore an area of 0.975 to the right, is

Example: Find 𝑃(−𝑡0.025 < 𝑇 < 𝑡0.05)


Example: Find k such that 𝑃(𝑘 < 𝑇 < −1.761) = 0.045 for a random sample

of size 15 selected from a normal distribution and 𝑋 −𝜇

𝑆 𝑛


Example: A chemical engineer claims that the population mean yield of a

certain batch process is 500 grams per milliliter of raw material. To check this

claim he samples 25 batches each month. If the computed t-value falls

between−𝑡0.05 and 𝑡0.05, he is satisfied with this claim. What conclusion should

he draw from a sample that has a mean 𝑥 = 518 grams per milliliter and a

sample standard deviation s = 40 grams? Assume the distribution of yields to

be approximately normal.


Today’s Agenda

Chapter 8 Fundamental Sampling Distributions and Data

Descriptions


Random Sampling



Sampling Distributions of Means and the Central Limit Theorem


t-Distribution


Documents

IE 3320 Engineering Statisticsie3320.weebly.com/uploads/9/1/3/6/9136035/lecture... · E. Perez IE TXSTATE 32 Sampling Distribution of the Difference between Two Means . E. Perez IE