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How to construct ifhe Sampling Distribution Of The Sample Mean Xbar

n = 36

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I Leading up to the Central Limit Theorem:

• If all samples of a particular size are selected from any population, the sampling distribution of the sample mean Xbar is approximately a normal distribution. This approximation improves with larger samples ~ 5~e tv~

I Central Limit Theorem:

• In selecting random samples of size n from a population, the sampling distribution of the sample mean Xbar can be approximated by a normal distribution as the sample size becomes large

P~~

• If population distribution is symmetrical but not normal, the distribution will converge toward normal when n > 10

• Skewed or thick-tailed distributions converge toward normal when n > 30

• Heavily skew distributions converge n >SO

I Use of Central Limit Theorem:

• We can reason about the Sampling Distribution of Xbar with absolutely no information about the shape of the original distribution from which the sample is taken

• This means that: • We can take one sample and compare it to the Standard Normal

Curve (NORM.S.DIST) or Normal Curve (NORM.DIST) to see if our sample result is reasonable or not.

• If it is reasonable, the process or claim is reasonable • If it is not reasonable, the process or claim is not reasonable

~mpling Methods and the Central Limit Theorem

Populations

I I . I I I

Sarnpling DistribuUons

n= 30 n = 30 n = 30

265

~~ 't.o r" +ex!

n=6

·-_....._,...-~~- x

I I ., I I

n = 30

-__ ........... L..........L..........:...~ x ~--~~----x ~~: ~------x CHART 8-2 ResnHs of tbe Centra) Lirnit Theorem fur Scvcr;rl Populations

Business Decisions Exam pie 1

• History for a food manufacturer shows the weight for a Chocolate Covered Sugar Bombs (popular breakfast cereal) is: • IJ = 14 oz . • 0' = .4 oz.

• If the morning shift sample shows: • Xbar= 14.14 OZ.

• n = 30

• Is this sampling error reasonable, or do we need to shut down the filling operations?

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r Suppose the mean selling price of a gallon of gasoline in the United States is $3.12. (fJ ) Fruther, assume the distribution is positively skewed, 'With a standard deviation of $0.98 (e). VJhat is the probability of selecting a sample of 35 gasoline stations (n = 35) and finding the sample mean within $.33?

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The Crossett Trucking Company The Crossett Trucking Company claims that the mean weight of their delivery trucks when they are fully loaded is 6000 lbs. And the standard deviation is 150 lbs. Assume that the population follows the normal distribution. 40 trucks are randomly selected and weighed.

Within what limits will 95% of the sample means occur?

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Chapter 7:

Population:

All elements of interest

Sample:

A subset of the population

Why do we sample instead of look at whole population?

1) Some populations are impossible to check

Can’t count all the fish in the ocean

2) The cost of studying all the items in a population

General Mills hires firm to test a new cereal:

Sample test: cost ≈ $40,000

Population test: cost ≈ $1,000,000,000

3) Contacting the whole population would often be time-consuming

Political polls can be completed in one or two days

Polling all the USA voters would take nearly 200 years!

4) The destructive nature of certain tests

Test each bottle of wine?!!?

Testing all seeds from Burpee there’d be none left

5] The sample results are usually adequate

Consumer price index constructed from a sample is an excellent estimate for a consumer price index that could be constructed from the population

Why do we need sample data?

We select samples to collect data to answer research question about a population

What do community college presidents think of the new federal proposal to rate colleges?

Is the filling machine filling accurately?

Do Highline College students think that advising should be mandatory?

What is the mean annual accounting salary in Seattle?

Samples are only estimates:

In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter.

We refer to Xbar as the point estimator of the population mean = m = mew.

We refer to s as the point estimator of the population standard deviation = s = sigma.

We refer to Pbar as the point estimator of the population mean p.

What about Sample Error?

Is the Xbar (Sample Mean) and µ (population mean, mew) always equal?

Almost never!

If Xbar (Sample Mean) and µ (population mean, mew) are not equal, is this okay?

It depends.

We will build a model to check to see if our Xbar is a "reasonable" estimate for the population mean!! :)

Random Variable:

Numerical Description of the outcome of an experiment

If we consider the process of selecting a “Random Sample” as an experiment, the Xbar is the numerical description of the outcome of the experiment.

Thus Xbar is the random variable.

If the population is so big, how do we know where to go and get sample data?

We must be careful!!

We have to make sure that the Samples Population is the same as the Target Population.

Sampled Population:

Population from which the sample is drawn

Target Population:

Population we want to make an inference about

Sampled Population and Target Population are not always the same!

Example 1: If your goal was to gather data about students attending South Community College:

If you took a sample from a College Registration List, Sampled Population and Target Population are the same

If you took a sample from people eating at the Culinary Arts restaurant at South Community College you might get some people who are:

students at South (Target Pop)

and

some who are not students at South (NOT Target Pop).

The Sampled Population would be people who eat at Culinary Arts Restaurant.

Sampled Population NOT EQUAL TO Target Population

Example 2: If you take a sample from only matinée movie-goers and you want to make inferences about all movie goers your

Sampled Population (matinée) is different than your Target Population (all movie goers): they are not the same.

Conclusion: When a sample is used to make inferences about the population, make sure that the sampled and target population are in close agreement.

This is not a mathematical calculation, it is a judgment call.

2/27/2015 Page 1 of 4 Excel2013StatisticsChapter07.xlsm

Chapter 7:

Frame:

List of all elements in the population

List of elements that sample will be selected from.

Frames cannot always be created

Finite Population

A population where we can create a Frame

Examples:

List of student names at a college

List of Sales Invoices

“Sampling from a Finite Population”: Use “Simple Random Sampling” method to select a sample

Infinite Population

A population where we can NOT create a Frame

Examples:

Population too big (like all the fish in the sea)

Take a sample of cereal box weights from a cereal box filling machine

Customers entering a store

“Sampling from an Infinite Population or Process”: Use “Random Sampling” method to select a sample

Frame that CAN be constructed:

Take sample at Highline Community College to see how many people have iPods

Sampled Population = List of registered students

Frame = List of registered students

The sampled population has a finite number of elements

This is called “Sampling from a Finite Population”. Use “Simple Random Sampling” method to select a sample

Frame that CANNOT be constructed:

Population is too big (like counting all the fish in the sea)

Take a sample of cereal box weights from a cereal box filling machine

Sampled Population = conceptual population of all boxes that could have been filled at that particular point in time.

In this sense, the sampled population is considered infinite.

Frame = impossible to construct frame from infinite population because all the elements are not present

The sampled population has a conceptually infinite number of elements

This is called “Sampling from an infinite Population or Process”. Use “Random Sampling” method to select a sample

“Random Sampling” is the same as “Simple Random Sampling”, except for two assumptions have to hold true (more later)

Processes (Sampling from an infinite Population):

Examples of processes:

Machine fills boxes of cereal

Machine fills bags of lettuce

Machines make bolts and screws for airplanes

Router makes boomerangs

Transactions occur at bank

Calls arrive at Highline help desk

Customers entering store

All are viewed as coming from a process generating elements from a conceptually infinite population

How a sample can help to decide whether the process is working properly:

Processes not working properly (like machine filling too much) will produce

sample statistics that are not close (statistically significant) to the population parameter

Processes working properly (like machine filling just right) will produce

sample statistics that are close (statistically insignificant) to the population parameter


Chapter 7:

Simple Random Sample (for Finite Populations)

Textbook: A simple random sample of size n from a finite population of size N is a sample selected such that

each possible sample of size n has the same probability of being selected

Each sample element has the same probability of being selected, (if TRUE, then each sample has the same chance)

Think of: Uniform Distribution

You have a frame and you can randomly select from frame.

How to select a sample:

Example 1: Select any n units in a random way

Book method:

Step 1: Assign Random Number to each element in population

Step 2: Select “n” elements that have the “n” smallest (or largest) random numbers

Using Excel’s RAND or RANDBETWEEN functions (each follows a Uniform Distribution between 0 and 1)

Example 2:

RAND, SMALL and VLOOKUP functions can automat the Simple Random Sample

RAND function generates Radom numbers between 0 and 1 with 15 significant digits and generate these numbers based on a Uniform Distribution

Example 3: Names of classmates in a hat, mix up names, select until sample size, “n” is reached

Random Sample (for Infinite Population)

These must hold true:

1) Each element selected comes from same population (Sampled and Targeted Populations are the same).

2) Each element is selected independently: to prevent selection bias (prevent all from similar group, similar attitudes, or choosing to get desired result)

Used for Infinite Populations or Populations where it is not feasible to list all elements

How to select a sample:

Select any n units in a random way

Examples:

Machines filling boxes or bags, choose sample from same point in time

This is done to make sure that each element selected in selected from the same population.

People arriving at a restaurant, choose customer directly after customer who uses coupon (McDonald’s did this to simulate a random selection).

This is done to make sure that each element is selected independently (without bias)

Probability Sample

Each possible sample has a known probability of selection and a random process is used to select the elements for the sample.

Good because we have techniques to help us understand if our sample is “good”, reasonable – more later…

Examples:

1) Simple Random Sample taken (for Finite populations)

2) Random Sample (for Infinite population)

3) Stratified Random Sampling (allows smaller sample size and lower cost)

Population divided into mutually exclusive strata where elements in each strata are similar

Each element in the strata are similar (location, age, department, major)

Works best when the variation among the elements in each strata is relatively small.

4) Cluster Sampling

Elements in clusters are not a like – each cluster is like a min population

Helps to reduce cost.

5) Systematic Sampling

Like with invoices or other ordered populations.

Non-probability Sampling

Not good because we can’t calculate how reasonable the sample results are

Convenience Sampling

Judgment Sampling


Chapter 7:

Leading up to the Central Limit Theorem:

If all samples of a particular size are selected from any population, the sampling distribution of the sample mean Xbar is approximately normal distribution.

This approximation improves with larger samples

Sampling Distribution of Xbar (SDofXbar)

The sampling Distribution of Xbar is the probability distribution of all possible values of sample mean Xbar.

Sample Mean of Sampling Distribution of Xbar = E(Xbar) = m = Pop Mean!!!! Called Unbiased point estimator

Standard Deviation of SDofXbar = sXbar = Standard Error = (s/SQRT(n))*SQRT((N-n)/(n-1)), Don’t need SQRT((N-n)/(n-1)) if population is finite AND n/N <= 0.05

If Pop is Finite AND n/N > 0.05, then use the Finite Population Correction Factor: SQRT((N-n)/(n-1))

As sample size increases, Standard Error will decrease, and thus provide a higher probability that Xbar will fall within a specified distance of the population mean.

Book assumes all problems use s/SQRT(n), unless otherwise stated

The Sampling Distribution can be used to provide probability information about how close the sample statistic is to the population parameter

Sampling Distribution of Pbar (SDofPbar)

Sample Proportion = point estimator of Population Proportion = Pbar = x/n

x = the number of elements in the population that posses the characteristic of interest = binomial variable (bi = 2, nominal = nominal variable)

n = sample size

The sampling Distribution of Pbar is the probability distribution of all possible values of sample proportion Pbar

Sample Proportion of Sampling Distribution of Pbar = E(Pbar) =p = Pop Proportion!! Called Unbiased point estimator

Standard Deviation of SDofPbar = sPbar = Standard Error = SQRT(p*(1-p)/n)*SQRT((N-n)/(n-1)), Don’t need SQRT((N-n)/(n-1)) if pop is finite AND n/N <= 0.05

If n/N > 0.05, then use the Finite Population Correction Factor: SQRT((N-n)/(n-1))

As sample size increases, Standard Error will decrease, and thus provide a higher probability that Pbar will fall within a specified distance of the population mean.

Book assumes all problems use SQRT(p*(1-p)/n), unless otherwise stated

The sampling Distribution of Pbar (and Binomial Distribution) can be approximated by a normal distribution whenever n*p>=5 AND n*(1-p) >=5

Standard Error

Standard Error stands for the Standard Deviation of a point estimator

Examples:

Standard Error of the Mean = sXbar = "Standard Error"

Standard Error of the Proportion = sPbar = "Standard Error"

Central Limit Theorem:

In selecting random samples of size n from a population, the sampling distribution of the sample mean Xbar

can be approximated by a normal distribution as the sample size becomes large

If population distribution is symmetrical but not normal, the distribution will converge toward normal when n > 10

Skewed or thick-tailed distributions converge toward normal when n > 30

Heavily skew distributions converge n > 50

Use of Central Limit Theorem:

We can reason about the Sampling Distribution of Xbar with absolutely no information about the shape of the original distribution from which the sample is taken

This means that:

We can take one sample and compare it to the Standard Normal Curve (NORM.S.DIST) or Normal Curve (NORM.DIST) to see if our sample result is reasonable.

If the sample mean seems reasonable, the original process or claim is reasonable

If the sample mean does not seem reasonable, the original process or claim is not reasonable

The Sampling Distribution can be used to provide probability information about how close the sample statistic is to the population parameter

Statistical Inference:

The process of using data obtained from a sample to make estimates or test hypotheses about characteristics of the population (like mean).

Draw reasonable conclusions about population from statistics

Infer:

Conclude from evidence


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