Introduction to Inference

Tests of Significance

Proof

925 950 975 1000

1000x 125

2525

x

979x

xz

sn

979 1000

25

.84

( 979) .2005P x

Proof

925 950 975 1000

1000x 125

2525

x

920x

xz

sn

920 1000

25

3.2

( 920) .0007P x

Definitions

• A test of significance is a method for using sample data to make a decision about a population characteristic.

• The null hypothesis, written H0, is the starting value for the decision (i.e. H0 : =m 1000).

• The alternative hypothesis, written Ha, states what belief/claim we are trying to determine if statistically significant (Ha : <m 1000).

Note: population characteristic could

be , or hypothesized value

Examples

• Chrysler Concord– H0: = 8m

– Ha: > 8m

• K-mart– H0: = 1000m

– Ha: < 1000m

Chrysler

8

8.7x

xz

sn

8.7 81

10

2.21

( 8.7) .0134P x

K-mart

1000

982x

xz

sn

982 100065

20

1.24

( 982) .1078P x

Phrasing our decision

• In justice system, what is our null and alternative hypothesis?

• H0: defendant is innocent

• Ha: defendant is guilty

• What does the jury state if the defendant wins?

• Not guilty• Why?

Phrasing our decision• H0: defendant is innocent

• Ha: defendant is guilty

• If we have the evidence:– We reject the belief the defendant is innocent

because we have the evidence to believe the defendant is guilty.

• If we don’t have the evidence: – We fail to reject the belief the defendant is

innocent because we do not have the evidence to believe the defendant is guilty.

Chrysler Concord

• H0: 8• Ha: 8• p-value = .0134• We reject H0 since the probability is so

small there is enough evidence to believe the mean Concord time is greater than 8 seconds.

K-mart light bulb

• H0: 1000• Ha: 1000• p-value = .1078• We fail to reject H0 since the probability is

not very small there is not enough evidence to believe the mean lifetime is less than 1000 hours.

Remember:Inference procedure overview

• State the procedure• Define any variables• Establish the conditions (assumptions)• Use the appropriate formula• Draw conclusions

Test of Significance Example

• A package delivery service claims it takes an average of 24 hours to send a package from New York to San Francisco. An independent consumer agency is doing a study to test the truth of the claim. Several complaints have led the agency to suspect that the delivery time is longer than 24 hours. Assume that the delivery times are normally distributed with standard deviation (assume s for now) of 2 hours. A random sample of 25 packages has been taken.

Example 1

test of significance

= true mean delivery time

Ho: = 24

Ha: > 24

Given a random sample

Given a normal distribution

Safe to infer a population of at least 250 packages

Example 1 (look, don’t copy)

22.8 23.2 23.6 24 24.4 24.8 25.2

24x 2

0.425

x

24.85x

24.85

xz

sn

24.85 24

.4

2.125

Example 1

let a = .05

test of significance = true mean delivery timeHo: = 24 Ha: > 24Given a random sampleGiven a normal distributionSafe to infer a population of at least 250 packages.

24.85 242.125

225

z

p-value 1 .9834 .0166

Example 1test of significance = true mean delivery timeHo: = 24 Ha: > 24Given a random sampleGiven a normal distributionSafe to infer a population of at least 250 packages.

let = .05a

We reject Ho. Since p-value<a there is enough evidence to believe the delivery time is longer than 24 hours.

p-value .016624.85 242.125

225

z

Wording of conclusion revisit

• If I believe the statistic is just too extreme and unusual (P-value < a), I will reject the null hypothesis.

• If I believe the statistic is just normal chance variation (P-value > a), I will fail to reject the null hypothesis.

We rejectfail to reject

Ho, since the p-value<a, there is p-value>a, there is not

enough evidence to believe…(Ha in context…)

Example 3test of significance = true mean distance Ho: = 340 Ha: > 340Given random sampleGiven normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=.1038

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

z

Familiar transition

• What happened on day 2 of confidence intervals involving mean and standard deviation?

• Switch from using z-scores to using the t-distribution.

• What changes occur in the write up?

Example 3test of significance = true mean distance Ho: = 340 Ha: > 340Given random sample.Given normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=.1038

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

z

Example 3t-test = true mean distance Ho: = 340 Ha: > 340Given random sample.Given normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=.1038

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

z

Example 3t-test = true mean distance Ho: = 340 Ha: > 340Given random sampleGiven normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=.1038

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

t

9df

Example 3t-test = true mean distance Ho: = 340 Ha: > 340Given random sample.Given normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

t

9df

t-chart

1.26t

Example 3t-test = true mean distance Ho: = 340 Ha: > 340Given random sample.Given normally distributed.Safe to infer a population of at least 100 missiles.

let = .05a.10<p-value<.15

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

t

9df

Example 3t-test = true mean distance Ho: = 340 Ha: > 340Given random sample.Given normally distributed.Safe to infer a population of at least 100 missiles.

let = .05ap-value=.1188

We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

348 3401.26

2010

t

9df