Level of Significance Worksheet                                    Key: Solutions in Red/Blue

Introduction to Statistics

Spring, 2008, Dr. Sinn

 

Analyze the level of significance that should be used.  Identify “status quo” or null hypothesis and research hypothesis.  State the effect of both Type I and Type II error and which is most insidious.  State whether alpha should be set high or low, then set it.

 

1.        The state of Georgia is considering spending $350 million on a computerized mathematics curriculum that for grades 3 – 10.  They pilot the program with 250 students in grades 5 and 6 whose end-of-course test scores are compared to the state average of 150.

 

H0:  μ = 150

Ha:  μ > 150

 

Type I:  Falsely claim computerized math works better

Implication: State spends needless $350 million

Type II: Falsely claim no difference between new and old (and spend no money)

Implication: No money spent, computer company (likely) tries to improve product and retest

 

Minimize Type I by setting alpha low: .  (Note, this sample is rather large, which also indicates we should have a low alpha. In practice, we might opt for a “super low” alpha, say: )

 

2.        Pharmaceutical researchers are testing a new drug for acne. Severity of acne is rated on a scale of 1 – 7, 7 being “worst.” Side effects for the new drug may include vomiting, upset stomach, mild fever and/or hyperactivity.  Their sample size is 80, with participants randomly assigned to a control group (n = 40) and treatment group (n = 40).

 

H0:  μT = μC

Ha:  μT < μC

 

Type I:  Falsely claim new drug works well ()

Implication: Patients experience side effects but get no main effect (no reduction in acne)

Type II: Falsely claim new drug doesn’t work

Implication: A working drug goes back to lab for improvements

 

Minimize Type I by setting alpha low:     

 

3.        Researchers are studying the effects of TV watching on cognitive development in toddlers.  Recent surveys have shown that 21% of 3-year-olds watch more than 5 hours of TV per day, and reports from cognitive psychologists indicate TV exposure may “rewire” the developing brain in adverse ways.  They wish to test if toddlers who watch 5 hours or more of TV daily score less than average of 40 on tests of cognitive ability when entering first grade.  Their sample size is 95.

 

        H0:  μTV = 40

        Ha:  μTV < 40

 

        Type I:  Falsely claim TV affects cognition (negatively)

        Implication: Academy of Pediatrics recommends no TV for toddlers when it is NOT in fact harmful.

        Type II: Falsely claim there is no affect by TV on cognition (minimize)

        Implication: Academy of Pediatrics fails to warn parents of a real problem that does exist.

       

        Minimize Type II by setting alpha high: 

 

 

 

4.        Pregnant ladies take a glucose tolerance screening during the 24th week of pregnancy to detect developing gestational diabetes.  This condition can lead to premature labor.

 

H0:  No diabetes (normal or negative test, negative screening)

Ha:  Diabetes (positive test)

 

Type I:  False Positive

Implication: Treatment (diet, exercise, insulin) and more testing initiated when not needed.

Type II: False Negative

Implication: No additional treatment or test recommended when patient truly has condition.

 

Minimize Type II by setting alpha high: 

 

5.        A manufacturer is testing light bulbs in the quality assurance lab.  Each bulb receives a durability score.  100 bulbs from each production run are analyzed.  If 11 or more are found to be defective, the entire run of bulbs is discarded.  Discarding a run costs the company $28,000.

 

        H0:  μ = 10      (Note: a better way to write the null is H0:  μ ≤ 10, but if you’re confused use “ = ”)

        Ha:  μ > 10

 

        Type I:  Falsely claim 11 or more are defective

        Implication: wrongly costs company $28,000

        Type II: Falsely claim bulbs okay

        Implication: a few customers screwed

 

        Minimize Type I by setting alpha low:  (remember, the company is making the decision, and very few customers will buy two types of bulbs and compare lifespans – they’ll never really know)

 

6.        Researchers at NGCSU are testing the political awareness of the undergraduate population.  Due to the Corps of Cadets and an increased level of U. S. military activity, they feel that NGCSU students may be more aware of political policy than average college students.  A political awareness survey that has been given nationwide is administered to 1,200 NGCSU students. The average score nationwide is 12.

 

        H0:  μ NG = 12

        Ha:  μ NG > 12

 

        Type I:  Falsely claim NGCSU more aware than national average

        Type II: Falsely claim NGCSU same as national average

 

        No problem with either implication, so note that n = 1,200 (very large sample). Set alpha low:

 

7.        Crop researchers are testing the yield (bushels/acre) of a hybridized soy bean to see if it produces per acre than former strains (treatment vs. control).  An increase of 2% could mean increased corporate profits of nearly $3 million.  Seventy-two plots are analyzed, half with the new hybrids, half with the former strains.

 

H0:  μT = μC

Ha:  μT < μC

 

Type I:   Falsely claim hybrid yields more bushels/acre

Implication: Company has good marketing campaign for lousy product

Type II:  Falsely claim hybrid does not yield more

Implication: Company bypasses HUGE profits when the seed is actually better.

 

Minimize Type II by setting alpha high: 

 

8.        A hotel chain which, beginning in 1995, has spent several million dollars on a manager development-and-retention plan tests the mean length of employment of its managers.  They are comparing length of service to the nationwide hotel manager average of 3.2.

 

        H0:  μ = 3.2

        Ha:  μ > 3.2

 

Type I:  Falsely claim managers stay longer

Implications: Company spends millions on program that doesn’t work (bad “side effects”)

Type II: Falsely claim managers don’t stay longer

Implications: Company spends no money on a program that does work

 

Minimize Type I by setting alpha low:     

 

9.        The state of Georgia wishes to estimate how much a teacher-preparation program would help prospective teachers meet the guidelines set forth in the No Child Left Behind Act.  They pilot the program at the University of Georgia.  The program would cost several million dollars but would help school districts across the state meet the federal guidelines and qualify for many millions of dollars in federal funds. They use a treatment vs. control design with 50 teachers in each group.

 

H0:  μT = μC

Ha:  μT < μC

 

Type I:  Falsely claim prep program works (is valuable)

Implication: millions $$ spent needlessly

Type II: Falsely claim prep program useless

Implication: Save state money but schools lose out on federal $$

 

Minimize Type II by setting alpha high: