Introduction to Inference

Tests of Significance

Errors in the justice system

Actual truth

Jury decision

Guilty Not guilty

Guilty

Not guilty

Correct decision

Correct decision

Type I error

Type II error

“No innocent man is jailed” justice system

Actual truth

Jury decision

Guilty Not guilty

Guilty

Not guilty

Type I error

Type II error

smaller

larger

“No guilty man goes free” justice system

Actual truth

Jury decision

Guilty Not guilty

Guilty

Not guilty

Type I error

Type II error smaller

larger

Errors in the justice system

Actual truth

Jury decision

Guilty Not guilty

Guilty

Not guilty

Correct decision

Correct decision

Type I error

Type II error

(Ha true) (H0 true)

(reject H0)

(fail to reject H0)

Type I and Type II example

• Water samples are taken from water used for cooling as it

is being discharged from a power plant into a river. It has

been determined that as long as the mean temperature of

the discharged water is at most 150oF, there will be no

negative effects on the river's ecosystem. To investigate

whether the plant is in compliance with regulations that

prohibit a mean discharge water temperature above 150o,

50 water samples will be taken at randomly selected

times, and the temperature of each sample recorded.

• The resulting data will be used to test the hypotheses

Ho: = 150o versus Ha: > 150o.

Type I and Type II example

• Type I error: We think the water

temperature is greater than 150o, but

actually the temperature is equal to 150o.

• Consequence: We falsely accuse the plant

of producing water too hot and harming

the environment when nothing wrongful

was done.

Type I and Type II example

• Type II error: We think the water

temperature is equal to 150o, but actually

the temperature is greater than 150o.

• Consequence: We believe the plant’s

water is a normal temperature, when

actually they are harming the environment.

Type I and Type II errors

• If we believe Ha when in fact H0 is true,

this is a type I error.

• If we believe H0 when in fact Ha is true,

this is a type II error.

• Type I error: if we reject H0 and it’s a

mistake.

• Type II error: if we fail to reject H0 and

it’s a mistake. APPLET

Type I and Type II example

A distributor of handheld calculators receives very large

shipments of calculators from a manufacturer. It is too

costly and time consuming to inspect all incoming

calculators, so when each shipment arrives, a sample is

selected for inspection. Information from the sample is

then used to test Ho: p = .02 versus Ha: p < .02, where p

is the true proportion of defective calculators in the

shipment. If the null hypothesis is rejected, the distributor

accepts the shipment of calculators. If the null hypothesis

cannot be rejected, the entire shipment of calculators is

returned to the manufacturer due to inferior quality. (A

shipment is defined to be of inferior quality if it contains

2% or more defectives.)

Type I and Type II example

• Type I error: We think the proportion of

defective calculators is less than 2%, but

it’s actually 2% (or more).

• Consequence: Accept shipment that has

too many defective calculators so potential

loss in revenue.

Type I and Type II example

• Type II error: We think the proportion of

defective calculators is 2%, but it’s actually

less than 2%.

• Consequence: Return shipment thinking

there are too many defective calculators,

but the shipment is ok.

Type I and Type II example

• Distributor wants to avoid Type I error.

Choose = .01

• Calculator manufacturer wants to avoid

Type II error. Choose = .10

Concept of Power

• Definition?

• Power is the capability of accomplishing

something…

• The power of a test of significance is…

Power Example

In a power generating plant, pressure in a certain line is

supposed to maintain an average of 100 psi over any 4

- hour period. If the average pressure exceeds 103 psi

for a 4 - hour period, serious complications can evolve.

During a given 4 - hour period, thirty random

measurements are to be taken. The standard

deviation for these measurements is 4 psi (graph of

data is reasonably normal), test Ho: = 100 psi versus

the alternative “new” hypothesis = 103 psi. Test at

the alpha level of .01. Calculate a type II error and the

power of this test. In context of the problem, explain

what the power means.

Type I error and

4.73

30x

s

n

100100.73

101.46102.19

for =.01 t*=2.462

is the probability that we think

the mean pressure is above 100 psi,

but actually the mean pressure is

100 psi (or less)

Type I error and

100100.73

101.46102.19

101.80

for =.01 t*=2.462

1002.462

.73

x

4.73

30x

s

n

Type II error and

100100.73

101.46102.19

101.8

103

103

.73z 1.64

.0505

Type II error and

100100.73

101.46102.19

101.8

103

.0505

is the probability that we think the mean pressure is 100 psi,

but actually the pressure is greater than 100 psi.

Power?

100100.73

101.46102.19

103

.0505

Power = 1 .0505 .9495

100100.73

101.46102.19

103

For a sample size of 30, there is a .9495

probability that this test of significance will

correctly detect if the pressure is above

100 psi.

Concept of Power

• The power of a test of significance is

the probability that the null hypothesis

will be correctly rejected.

• Because the true value of is unknown,

we cannot know what the power is for ,

but we are able to examine “what if”

scenarios to provide important

information.

• Power = 1 –