Statistical techniques in business ecohomics chap010

Step Three: Select the test statistic.distributed when the sample is reasonably large recall Central Limit Theorem... Using the p-Value in Hypothesis Testing If the p-Value is larger t

Ten One-Sample Tests of

Compute the probability of a Type II error.

One-Sample Tests of H ypothesis

What is a Hypothesis?

Twenty percent of all customers at Bovine’s Chop House return for another

meal within a month

What is a Hypothesis?

The mean monthly

income for systems

analysts is $6,325

What is Hypothesis Testing?

a reasonable statement and should not be rejected, or is unreasonable and should

be rejected

Hypothesis Testing

D o n ot reject n u ll R eject n u ll an d accep t altern ate

S tep 5 : Take a sam p le, arrive at a d ecision

S tep 4 : F orm u late a d ecision ru le

S tep 3 : Id en tify th e test statistic

S tep 2 : S elect a level of sig n ifican ce

Alternative Hypothesis H1 :

A statement that is accepted if the sample data provide evidence that the null hypothesis is false

3 hypotheses about means

Step Two: Select a Level of

Step Two: Select a Level of Significance.

Type I error



Type II Error



Correct decision

Risk table

Step Three: Select the test statistic.

distributed when the sample

is reasonably large (recall Central Limit Theorem).

Step Four: Formulate the decision rule.

where the null hypothesis is rejected and the region

where it is not rejected.

0 1.65

D o not reject [Probability =.95]

R egion of rejection [Probability=.05]

Reject the null

Using the p-Value in Hypothesis

Testing

If the p-Value is larger

than or equal to the

significance level, , H0

is not rejected.

the test statistic at least as extreme as the computed value for the test

Calculated from the

Movie

One-Tailed Tests of Significance

H1 : The mean speed of trucks traveling on I-

95 in Georgia

is less than 60 miles per hour

(µ<60)

H1 : Less than 20 percent of the

customers pay cash for their gasoline purchase

20)

One-Tailed Test of Significance

.

0 1.65

D o not reject [Probability =.95]

R egion of rejection [Probability=.05]

Two-Tailed Tests of Significance

H1 : The mean price for a gallon of gasoline is not equal to $1.54 (µ ne $1.54).

No direction is specified in the alternate hypothesis H1

H1 : The mean

amount spent by

customers at the

Wal-mart in Georgetown is

not equal to $25

(µ ne $25).

Two-Tailed Tests of Significance

R egion of rejection [Probability=.025]

C ritical value

-1.96

R egion of rejection [Probability=.025]

C ritical value

Testing for the Population Mean: Large Sample, Population Standard Deviation

Known

n /

Test for the population

mean from a large sample

with population standard

deviation known

Example 1

indicate on the label that the

bottle contains 16 ounces of

catsup The standard deviation

of the process is 0.5 ounces A

sample of 36 bottles from last

hour’s production revealed a

mean weight of 16.12 ounces per

bottle At the 05 significance

level is the process out of

control? That is, can we

conclude that the mean amount

per bottle is different from 16

ounces?

Identify the test statistic Because

we know the population standard

deviation, the test statistic is z.

Make a decision and

interpret the results.

Example 1

44

1 36

5 0

00 16 12

decision and

interpret the results.

We cannot conclude the mean is different from 16 ounces.

Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown

Example 2

chain issues its own

credit card Lisa, the

credit manager, wants to

find out if the mean

monthly unpaid balance

is more than $400 The

level of significance is set

at 05 A random check

of 172 unpaid balances

revealed the sample

mean to be $407 and the

sample standard

deviation to be $38

Should Lisa conclude that the population mean is greater than

$400, or is it reasonable

to assume that the difference of $7 ($407-

$400) is due to chance?

Step 3

Because the sample is large

we can use the z

distribution as the test

2172

38

$

400

$407

X

The p(z > 2.42) is

.0078 for a tailed test.

Lisa can conclude that

the mean unpaid balance

is greater than $400.

Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown

n s

X t

Example 3

Co is 250 per hour A new

machine has been purchased

and installed that, according

to the supplier, will increase

the production rate The

production hours are normally

distributed A sample of 10

randomly selected hours from

last month revealed that the

mean hourly production on

the new machine was 256

units, with a sample standard

deviation of 6 per hour

At the 05 significance level can Neary conclude that the new machine is faster?

Step 4 State the decision rule.

There are 10 – 1 = 9

degrees of freedom

Step 1 State the null and

the t distribution since 

is not known and n < 30 The null hypothesis is rejected if t > 1.833 or, using the

p-value, the null hypothesis is rejected if p < 05.

Example 3

162

3 10

hour.

p z

) 1

sample

in the successes

Example 4

In the past, 15% of the

mail order solicitations for

a certain charity resulted in

a financial contribution A

new solicitation letter that

has been drafted is sent to

a sample of 200 people

and 45 responded with a

contribution At the 05

significance level can it be

concluded that the new

letter is more effective?

Step 5

Make a decision and interpret the results.

Example 4

97.2200

)15.1(15

15

.200

45)







Because the computed z of 2.97 > critical z of

1.65, the p of 0015 <  of 05, the null

hypothesis is rejected More than 15 percent

responding with a pledge The new letter is