The Excel NORMDIST Function Computes the cumulative probability to the value X Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. 7-1

The Excel NORMDIST Function

Computes the cumulative probability to the value X

Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..

7-1

X

NORMDIST

The Excel NORMDIST Function

Computes the cumulative probability to the value X

No need to convert into a Z value Needs the

mean, andstandarddistribution


7-2

The Excel NORMINV Function

To compute an X value that has a cumulative probability of a certain amount

If mean is 100 and the standard deviation is 20, the X value with a cumulative probability of .95 is 132.9.


7-3

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. 7-4

Chapter 7

Sampling and Sampling Distributions

Business Statistics:A First Course

5th Edition

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..

7-5

Why Sample?

Why not use the entire population? Selecting a sample is less time-consuming than

selecting every item in the population (census).

Selecting a sample is less costly than selecting every item in the population.

An analysis of a sample is less cumbersome and more practical than an analysis of the entire population.


7-6A Sampling Process Begins

With A Sampling Frame

The sampling frame is a listing of items that make up the population

Frames are data sourcessuch as population lists,directories, or maps

Sampling Frame For Population With 850

Items

Item Name Item #Bev R. 001

Ulan X. 002

. .

. .

Joan T. 849

Paul F. 850


7-7A Sampling Process Begins

With A Sampling Frame

The sampling frame is a listing of items that make up the population

Inaccurate or biased results can result if a frame excludes certain portions of the population

Using different frames to generate data can lead to dissimilar conclusions


7-8

Types of SamplesSamplesSamples

Non-Probability Samples


JudgmentJudgment

Probability SamplesProbability Samples

Simple RandomSimple Random

SystematicSystematic

StratifiedStratified

ClusterCluster

ConvenienceConvenience

In nonprobability sampling, items not are selected by probability.

Statistical inference methods cannot be used.

In nonprobability sampling, items not are selected by probability.

Statistical inference methods cannot be used.


7-9




JudgmentJudgment





ClusterCluster


In convenience sampling, items are easy, quickly, convenient, or convenient to sample.

Surveys of users visiting a website.

Representative of what population?

In convenience sampling, items are easy, quickly, convenient, or convenient to sample.

Surveys of users visiting a website.

Representative of what population?

In judgment sampling, experts are selected.

May not be able to generalize to the general public.

In judgment sampling, experts are selected.

May not be able to generalize to the general public.


7-10




JudgmentJudgment





ClusterCluster


In probability sampling, you select frame items based on probabilities.

To be used whenever possible to get unbiased inferences about the population of interest.

In probability sampling, you select frame items based on probabilities.

To be used whenever possible to get unbiased inferences about the population of interest.

There are four types of probability sampling methods.

We will look at two.

There are four types of probability sampling methods.

We will look at two.


7-11Probability Sample:

Simple Random Sample

Every individual or item from the frame has an equal chance of being selected

Selection may be with replacement (selected individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame).

Samples obtained from table of random numbers or computer random number generators.




The book describes tables of random numbers This is prehistoric.

The Excel RAND() function Randomly generates a number between 0 and 1 If you have a sampling frame numbered 1 to 850,

create a table each cell having the equation =850*RAND()

A random number between 0 and 500 will appear in each cell




The Excel RAND() function If you make any changes to the worksheet, the

numbers will be recalculated.

To prevent this, Select the range holding the random numbers Give the command to copy them Select the range again Give the command to paste special


7-14Probability Sample:Stratified Sample

Divide population into two or more subgroups (called strata) according to some common characteristic

A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes

Population

Divided

into 4

strata


7-15Probability Sample:Stratified Sample

Samples from subgroups are combined into one This is a common technique when sampling

population of voters, stratifying across racial or socio-economic lines.

Sampling employees of different levels in an organization.

Population

Divided

into 4

strata


7-16

Evaluating a Survey Design

What is the purpose of the survey?

Is the survey based on a probability sample?

How will various errors be controlled?


7-17

Types of Survey Errors

Coverage error or selection bias Exists if some groups are excluded from the frame

and have no chance of being selected

Example: Calling households during the daytime will exclude many working people

Always consider who is being undercounted

Excluded from frameExcluded from frame


7-18


Non response error or bias People who do not respond may be different from

those who do respond Income level Ethnic background Busier

Try multiple times to reach nonrespondents

Do selective follow-up to determine differences


7-19


Sampling error Variation from sample to sample will always exist Increasing sample size minimizes this

Random differences from sample to sample

Random differences from sample to sample


7-20


Measurement error Problems in the design of questions

Leading questions Hawthorne effect: Answering to please the

interviewer

Ambiguous questions Can be interpreted in different ways

Questions the respondent cannot answer


7-21

Ethics

Purposely create survey errors Coverage errors

Purposely exclude certain groups to give more favorable results

Nonresponse Errors Ignore nonrespondents, whose answers you

probably will not like

Measurement Errors Asking leading questions


7-22

Sampling Distributions A sampling distribution is a distribution of all of the

possible values of a sample statistic for a given size sample selected from a population.

Sample to find the mean age for college students.

If you did this for N times, you would get N sample statistic values for the mean.

The totality of these valueswould constitute the sampling distribution.

xμ


7-23

Sampling Distributions

A sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population.

Most sample valueswill be close to thepopulation mean

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X_


7-24Sample Mean Sampling Distribution:Standard Error of the Mean

A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:

(This assumes that sampling is with replacement or sampling is without replacement from an infinite

population)

Note that the standard error of the mean decreases as the sample size increases

n

σσ

X


7-25Sample Mean Sampling Distribution:

If the Population is Normal

If a population is normally distributed with mean μ

and standard deviation σ, the sampling

distribution of is also normally distributed withX

μμX

n

σσ

X


7-26Z-value for SamplingDistribution of the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

Xμσ

n

σμ)X(

σ

)μX(Z

X

X

X


7-27

Normal Population Distribution

Normal Sampling Distribution (has the same mean)

Sampling Distribution Properties

(i.e. is unbiased )x

x

μμx

xμ

xμ


7-28

Sampling Distribution Properties

As n increases,

decreasesLarger sample size

Smaller sample size

x

(continued)

xσ

μ


7-29Determining An Interval Including A

Fixed Proportion of the Sample Means

Find a symmetrically distributed interval around µ that will include 95% of the sample means when µ = 368, σ = 15, and n = 25.

Since the interval contains 95% of the sample means 5% of the sample means will be outside the interval

Since the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit.

From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96.


7-30Determining An Interval Including A Fixed Proportion of the Sample Means

Calculating the lower limit of the interval

Calculating the upper limit of the interval

95% of all sample means of sample size 25 are between 362.12 and 373.88

12.36225

15)96.1(368

nZX Lσ

μ

(continued)

88.37325

15)96.1(368

nZXUσ

μ


7-31Sample Mean Sampling Distribution:

If the Population is not Normal

We can apply the Central Limit Theorem:

Even if the population is not normal, …sample means from the population

will be approximately normal as longas the sample size is large enough.

Properties of the sampling distribution:

μμx n

σσx


7-32

How Large is Large Enough?

For most distributions, n > 30 will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n > 15 will usually give a sampling distribution is almost normal

For normal population distributions, the sampling distribution of the mean is always normally distributed


7-33

Example

Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.

What is the probability that the sample mean is between 7.8 and 8.2?


7-34

Example

Solution:

Even if the population is not normally distributed, the central limit theorem can be used (n > 30)

… so the sampling distribution of is approximately normal

… with mean = 8

…and standard deviation

(continued)

x

xμ

0.536

3

n

σσx


7-35

Example

Solution (continued):(continued)

0.31080.4)ZP(-0.4

363

8-8.2

nσ

μ- X

363

8-7.8P 8.2) X P(7.8

Z7.8 8.2 -0.4 0.4

Sampling Distribution

Standard Normal Distribution .1554

+.1554

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μX

0μz xX


7-36

Population Proportions

π = the proportion of the population having some characteristic

Sample proportion ( p ) provides an estimate of π:

0 ≤ p ≤ 1

p is approximately distributed as a normal distribution when n is large

(assuming sampling with replacement from a finite population or without replacement from an infinite population)

size sample

interest ofstic characteri the having sample the in itemsofnumber

n

Xp


7-37

Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where π = population proportion)

Sampling DistributionP( ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1 p

πpμn

)(1σp

ππ

5)n(1

5n

and

π

π


7-38

Z-Value for Proportions

n)(1

p

σ

pZ

p

Standardize p to a Z value with the formula:


7-39

Example

If the true proportion of voters who support

Proposition A is π = 0.4, what is the probability

that a sample of size 200 yields a sample

proportion between 0.40 and 0.45?

i.e.: if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?


7-40

Example

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

0.03464200

0.4)0.4(1

n

)(1σp

1.44)ZP(0

0.03464

0.400.45Z

0.03464

0.400.40P0.45)pP(0.40

Find :

Convert to standardized normal:

pσ


7-41

Example

Z0.45 1.44

0.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

Use standardized normal table: P(0 ≤ Z ≤ 1.44) = 0.4251

0.40 0p


7-42

Chapter Summary

Probability and nonprobability samples Two common probability samples Examined survey worthiness and types of errors Introduced sampling distributions Described the sampling distribution of the mean

For normal populations Using the Central Limit Theorem for any

distribution Described the sampling distribution of a proportion Calculated probabilities using sampling distributions

Documents

The Excel NORMDIST Function Computes the cumulative probability to the value X Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. 7-1