Business Statistics: A Decision-Making Approach, 7e © 2008 ...

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 1-1

Descriptive statistics Collecting, presenting, and describing data

Inferential statistics Drawing conclusions and/or making decisions

concerning a population based only on sample data

Tools of Business Statistics


Making statements about a population by examining sample results

Sample statistics Population parameters (known) Inference (unknown, but can

be estimated from

sample evidence)

Sample Population

Inferential Statistics


Inferential Statistics

Estimation e.g., Estimate the population mean

weight using the sample mean weight

Hypothesis Testing e.g., Use sample evidence to test

the claim that the population mean weight is 120 pounds

Drawing conclusions and/or making decisions concerning a population based on sample results.


Tools for Collecting Data

Data Collection Methods

Written questionnaires

Experiments

Telephone surveys

Direct observation and personal interview


A Population is the set of all items or individuals of interest

Examples: All likely voters in the next election All parts produced today

All sales receipts for November

A Sample is a subset of the population Examples: 1000 voters selected at random for interview

A few parts selected for destructive testing

Every 100th receipt selected for audit

Populations and Samples


Key Definitions

A population is the entire collection of things under consideration A parameter is a summary measure computed to

describe a characteristic of the population

A sample is a portion of the population selected for analysis A statistic is a summary measure computed to

describe a characteristic of the sample


Measures of Center and Location

Center and Location

Mean Median Mode Weighted Mean

N

x

n

xx

N

ii

n

ii

1

1

i

iiW

i

iiW

w

xw

w

xwX

Overview


Measures of variation give information on the spread or variability of the data values.

Variation

Same center, different variation


Average of squared deviations of values from the mean Population variance:

Sample variance:

Variance

N

μ)(xσ

N

1i

2i

2

1- n

)x(xs

n

1i

2i

2


Standard Deviation

Most commonly used measure of variation Shows variation about the mean Has the same units as the original data

Population standard deviation:

Sample standard deviation:

N

μ)(xσ

N

1i

2i

1-n

)x(xs

n

1i

2i


The Normal Distribution

‘Bell Shaped’ Symmetrical Mean, Median and Mode

are Equal

Location is determined by the mean, μ

Spread is determined by the standard deviation, σ

The random variable has an infinite theoretical range: + to

Mean = Median = Mode

x

f(x)

μ

σ


By varying the parameters μ and σ, we obtain different normal distributions

Many Normal Distributions


Finding Normal Probabilities

a b x

f(x) P a x b( )

Probability is measured by the area under the curve


Empirical Rules

μ ± 1σencloses about 68% of x’s

f(x)

xμ μσμσ

What can we say about the distribution of values around the mean? There are some general rules:

σσ

68.26%


The Empirical Rule

μ ± 2σ covers about 95% of x’s

μ ± 3σ covers about 99.7% of x’s

xμ

2σ 2σ

xμ

3σ 3σ

95.44% 99.72%

(continued)


The Standard Normal Distribution

Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1

z

f(z)

0

1

Values above the mean have positive z-values, values below the mean have negative z-values


The Standard Normal

Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z)

Need to transform x units into z units


Translation to the Standard Normal Distribution

Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:

σ

μxz


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 long as the sample size is large enough

…and the sampling distribution will have

andμμx n

σσx

Theorem 4


n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x


What is a Hypothesis?

A hypothesis is a claim (assumption) about a population parameter:

population mean

population proportion

Example: The mean monthly cell phone bill of this city is µ = $42

Example: The proportion of adults in this city with cell phones is π = .68


The Null Hypothesis, H0

States the assumption (numerical) to be tested

Example: The average number of TV sets in

U.S. Homes is at least three ( )

Is always about a population parameter, not about a sample statistic

3μ:H0

3μ:H0 3x:H0


The Null Hypothesis, H0

Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until

proven guilty Refers to the status quo Always contains “=” , “≤” or “” sign May or may not be rejected

(continued)


The Alternative Hypothesis, HA

Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S.

homes is less than 3 ( HA: µ < 3 )

Challenges the status quo Never contains the “=” , “≤” or “” sign May or may not be accepted Is generally the hypothesis that is believed

(or needs to be supported) by the researcher – a research hypothesis


Formulating Hypotheses

Example 1: Ford motor company has worked to reduce road noise inside the cab of the redesigned F150 pickup truck. It would like to report in its advertising that the truck is quieter. The average of the prior design was 68 decibels at 60 mph.

What is the appropriate hypothesis test?



Example 1: Ford motor company has worked to reduce road noise inside the cab of the redesigned F150 pickup truck. It would like to report in its advertising that the truck is quieter. The average of the prior design was 68 decibels at 60 mph.

What is the appropriate test?

H0: µ ≥ 68 (the truck is not quieter) status quo

HA: µ < 68 (the truck is quieter) wants to support

If the null hypothesis is rejected, Ford has sufficient evidence to support that the truck is now quieter.



Example 2: The average annual income of buyers of Ford F150 pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim.

What is the appropriate hypothesis test?



Example 1: The average annual income of buyers of Ford F150 pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim.

What is the appropriate test?

H0: µ = 65,000 (income is as claimed) status quo

HA: µ ≠ 65,000 (income is different than claimed)

The analyst will believe the claim unless sufficient evidence is found to discredit it.

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Population

Claim: the population mean age is 50.

Null Hypothesis:

REJECT

Suppose the sample mean age is 20: x = 20

Sample

Null Hypothesis

Is x = 20 likely if µ = 50?

Hypothesis Testing Process

If not likely,

Now select a random sample:

H0: µ = 50


Sampling Distribution of x

μ = 50If H0 is true

If it is unlikely that we would get a sample mean of this value ...

... then we reject the null

hypothesis that μ = 50.

Reason for Rejecting H0

20

... if in fact this were the population mean…

x

Business Statistics: A Decision-Making Approach, 7e © 2008 ...

Documents

Transcript of Business Statistics: A Decision-Making Approach, 7e © 2008 ...