Chapter 10.2 TESTS OF SIGNIFICANCE. TEST OF SIGNIFICANCE Two basic types of : ”Statistical...
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Transcript of Chapter 10.2 TESTS OF SIGNIFICANCE. TEST OF SIGNIFICANCE Two basic types of : ”Statistical...
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