Chapter 10.2Chapter 10.2

TESTS OF TESTS OF SIGNIFICANCESIGNIFICANCE

TEST OF SIGNIFICANCETEST OF SIGNIFICANCE• Two basic types of : ”Statistical Inference”• The One we have just studied: –CONFIDENCE INTERVALS–Goal: To estimate a population parameter

• The 2nd type: – TEST OF SIGNIFICANCETEST OF SIGNIFICANCE– Goal: To assess evidence provided by data

about a claim regarding a population parameter

Example 10.8: Example 10.8: I’M A GREAT FREE-THROW SHOOTERI’M A GREAT FREE-THROW SHOOTER• Lynch claims: “I am an 80% free-thrower”• To test this … you ask me to shoot 20 free-throws• I ONLY make 8 out of 20• So you REJECT (disbelieve) my initial claim• Your logic is based on how rare it would be for me

to only go 8/20 (40%) IF I were in reality p = 0.8• In statistical reality: • This small probability causes you to reject my claim

( 8) 0.0001P X

Example 10.9 SWEETENING COLASExample 10.9 SWEETENING COLAS

• A sample of n = 10. Sweetness tasters taste a batch of cola before … and then after high temperature storage for a month (simulating 4 months of storage)

• Matched pairs design … each tester gives sweetness score on 1- 10 scale … before, then after. A “difference” of Before minus After is shown below.

2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3

Example 10.9 SWEETENING COLASExample 10.9 SWEETENING COLAS

• Are these data strong enough evidence to conclude that the cola lost sweetness during storage?

• Find: • One of two things must be true:– A) The average of 1.02 reflects a real loss in sweetness

(or)

– B) We could achieve a loss of 1.02 by chance

2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3

x 1.02x

NULL HYPOTHESIS - HNULL HYPOTHESIS - H00

• The statement being tested in a test of significance is called the

null hypothesis. The test of significance is designed to assess

the strength of the evidence is against H0(h naught). Usually the null hypothesis is a statement of

“no effect” or “no difference”.

• FIRST STEP: Identification of the population being concluded about in this case the population parameter μ – sweetness loss that all consumers will experience.

• SECOND STEP: A statement of the HYPOTHESES:– H0 (null hypothesis) and Ha (alternative hypothesis)

• If H0 is true … the “difference” is just due to chance … and there is NO REAL CHANGE in the population

• If Ha is true … the suspected drop in sweetness … the “difference” is NOT due to chance … and so there IS A REAL CHANGE in the population!

0 : 0 : 0aH H

Example 10.9 SWEETENING COLASExample 10.9 SWEETENING COLAS

• Assume that the standard deviation of sweetness rankings is

• So the standard deviation of the sampling distribution would be…?

• Now, how does an look like now?

• Z-score?

• P(Z > 3.228) = ?• 0.00062 (a very low P-Value … it is statistically

significant) • So … we would reject H0 in favor of Ha

1

10.316

10x

1.02x 1.02 0

3.2280.316x

xz

Example 10.9 SWEETENING COLASExample 10.9 SWEETENING COLAS

Exercise 10.28: Exercise 10.28: SPENDING ON HOUSINGSPENDING ON HOUSING

ONE-SIDED AND ONE-SIDED AND TWO-SIDED ALTERNATIVESTWO-SIDED ALTERNATIVES• One sided:– Is there a loss? …. A gain? … more than? … less

than?

• Two-sided:– Is there a difference? … a change? Was there an

effect?

0 : 0 : 0aH H

0 : 0 : 0aH H

0 : 0 : 0aH H

Example 10.10: Example 10.10: STUDYING JOB SATISFACTIONSTUDYING JOB SATISFACTION

• Does job satisfaction DIFFER for assembly workers if their work is machine-paced vs. self-paced?

• 28 subjects … 14 to group I … 14 to group II• Job Diagnosis Survey (JDS) after two weeks• Switched groups … two more weeks of work• JDS again after two more weeks• Matched Pairs: “Difference X” = Self-paced minus Machine-

paced satisfaction score• The authors of the study want to know, do the working

conditions have different levels of satisfaction?

0 : 0 : 0aH H

Exercise 10.30: Exercise 10.30: HOUSEHOLD INCOMEHOUSEHOLD INCOME

Exercise 10.32: Exercise 10.32: SERVICE TECHNICIANSSERVICE TECHNICIANS

P-VALUEP-VALUE• The probability, computed assuming that

H0 is true, that the observed outcome would take on a value as extreme or

more extreme than that actually observed is called the

P-Value of the test. The smaller the P-Value is, the stronger the evidence is

against H0 provided by the data.

Example 10.11: CALCULATING Example 10.11: CALCULATING ANOTHER ONE-SIDED TESTANOTHER ONE-SIDED TEST

• This time the taste-testers examined a “new” cola.• The “new cola” sample mean: • Hypotheses?• So, what is:• Draw it!• z? Recall• P-Value?• Normalcdf(0.95, 10) = 0.1711

0 : 0; : 0aH H 0.3x

( 0.3)P x

10.316

10x 0.95z

TEST FOR A POPULATION MEANTEST FOR A POPULATION MEAN

• .

• One-sample z-statistic

0 0:H

0xz

n

STATISTICAL SIGNIFICANCESTATISTICAL SIGNIFICANCE

• If the P-Value is as small or smaller than alpha (), we say that the data are statistically

significant at the level

STATISTICAL SIGNIFICANCESTATISTICAL SIGNIFICANCE

Example 10.13: Example 10.13: EXECUTIVES’ BLOOD PRESSUREEXECUTIVES’ BLOOD PRESSURE• NCHS reports that the mean systolic blood pressure for all

males 35-44 is 128 with standard deviation 15• 72 subjects … executives in this age group• • Is this evidence to conclude that the company’s execs have

a different mean than that of the general population?• Population/parameter of interest? Set up hypotheses.• Choose inference procedure.• z? … P?• Interpret.

126.07x

Example 10.13: Example 10.13: EXEUTIVES’ BLOOD PRESSUREEXEUTIVES’ BLOOD PRESSURE

Example 10.14: CAN YOU BALANCE Example 10.14: CAN YOU BALANCE YOUR CHECKBOOK?YOUR CHECKBOOK?

• NAEP (National Assessment of Education Progress) survey reports that a score of 275 on its quantitative test is sufficient to indicate skill needed

• 840 subjects … young Americans• .• Is this evidence to conclude that the mean of ALL young

men is below 275?• Population/parameter of interest? Set up hypotheses.• Choose inference procedure.• z? … P?• Interpret.

272;x 60

Example 10.14: CAN YOU Example 10.14: CAN YOU BALANCE YOUR CHECKBOOK?BALANCE YOUR CHECKBOOK?

Example 10.15: Example 10.15: DETERMINING SIGNIFICANCEDETERMINING SIGNIFICANCE

• Back to this past example again … where we examined whether the mean of ALL young men is below 275?

• We can look at this problem from a slightly different perspective

• Assuming alpha = 0.05 – and that we have a one tail test• With 0.05 in ONE TAIL … z* = 1.645 … think … Why?• All we need to do then is to examine if the z-score is

“further out” than z*• Since z = 1.45 in this case, then NO, we fail to reject H0

Example 10.15: Example 10.15: DETERMINING SIGNIFICANCEDETERMINING SIGNIFICANCE

Example 10.16: Example 10.16: IS THE SCREEN TENSION OK?IS THE SCREEN TENSION OK?

• Recall the problem from a while ago with 20 TVs.• • Is there evidence at the = 0.01 level to conclude a

difference from the proper prescribed tension of 275mV? • Population/parameter of interest? Set up hypotheses.• One-tail test or a two-tail?• Choose inference procedure.• z? … P? What is the area in each tail? What is z*?• Interpret.

306.3; 43x

Example 10.16: Example 10.16: IS THE SCREEN TENSION OK?IS THE SCREEN TENSION OK?

CONFIDENCE INTERVALS CONFIDENCE INTERVALS AND TWO-SIDED TESTSAND TWO-SIDED TESTS

• A level significance test rejects the hypothesis: exactly when the value of falls outside the 1 – confidence interval.

• Compare the very last example to the 99% CI we created last week: (281.5, 331.1)

• Consider instead.

0 0:H 0

0 : 290H