Intro to Stats, Test 2 Review

Sinn, Fall 2007 Solutions in Red Below

Section A: Which Test to Run?

For each of the following research scenarios, select the appropriate statistical test.

1. Researchers are wondering if hypertension and smoking are related. They divide their 500 participants into 3 groups: non-smokers, moderate smokers and heavy smokers. They measure the blood pressure of the participants in each category.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

3 Groups/Samples => ANOVA compares 3 or more groups on the same variable (blood pressure).

2. The following data represents the running time of recent movies from two top motion-picture companies. Test the hypothesis that company 2’s movies have on average a longer running time.

Company	Running Time (Minutes)
1	102	86	98	109	92
2	81	165	97	134	92	87	114

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. Can’t be matched pairs (n1 not = n2). Ind. Samples

3. A college infirmary conducted an experiment to determine the degree of relief provided for each of three cold remedies: NyQuil, Robitussin and Triaminic. They had 30 students with colds. Ten students tried each remedy and reported the level of relief they experienced on a scale of 1 – 10 with 10 being perfect relief and 1 being no relief.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

3 Groups/Samples => ANOVA compares 3 or more groups on the same variable (relief experienced).

4. Joe wants to see if high school GPA is related to college GPA. He asks 50 NGCSU upperclassmen to report both on a survey.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Variables => Regression or Chi-Square. The variables are NUMERIC, so Linear Regression

5. A large automobile manufacturer is deciding between two brands of tires to purchase for its newly designed SUV. They test 50 sets of each type by putting sets of tires on the SUV’s and driving them until the tires are worn out.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. Even though n₁=n₂, this is like a demographics comparison, with tires from completely different manufacturers. Ind. samples

6. To determine if a new serum will arrest leukemia, 9 mice with advanced leukemia are selected, 5 of which receive the serum, 4 of which do not. Survival times (in months) are recorded.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. This is treatment vs. control – prototype for ind. samples.

7. A manufacturer of rice cereal (baby food) claims the average fat content does not exceed 1.5 milligrams per serving. A consumer group purchases 40 jars of the food and tests the average fat content. They are concerned that the food has a higher fat content than advertised.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

1 Sample => T-Test, comparing a subgroup (sample) to overall population.

8. Joe and Moe are two Marine Corps drill instructors at Quantico, VA. Joe doesn’t think Moe works his Officer Candidate School (OCS) classes hard enough. Moe gives all 25 of his OCS candidates a physical fitness when they arrive. The same test is repeated after 4 weeks of Moe’s gentle persuasions.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. Pretest/posttest design is dep. samples prototype.

9. Jack Erwin, professional fisherman extraordinaire, catches a large trout and nearly has the doggone thing reeled in. He caught a 26” trout earlier that day that weighed in at a hefty 7.5 lbs, and this one looks even bigger. But then, doggone it, his 15 lb. test fishing line breaks, and he suffers the agony of watching one get away. He purchases 10 more packages of fishing line to test his hypothesis that the mean breaking strength of the fishing line is less than the advertised 15 lbs. of force.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

1 Sample => T-Test, comparing a subgroup (sample) to overall population

10. Researchers were trying to determine if the material in a college physics course was better understood by students when the course had an accompanying lab section. Of the 28 participants, 11 were randomly selected for the lab course and 17 for the course without a lab. The exact same end-of-course test was administered to all 28.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. This is treatment vs. control – prototype for ind. samples.

11. Joe and Moe, over beers in the NCO, decide to estimate the fitness levels of incoming OCS candidates. They give 100 future Marine Corps officers a physical fitness test and estimate with a 95% level of confidence.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

If these aren’t free points…. J ESTMIATE is another key word, here.

12. Researchers in Boulder, CO, believe that running decreases the resting heart rate (RMR) in older women. The average RMR of 30 elderly women runners was lower than the average RMR of 30 sedentary elderly women.

 Linear Regression  Confidence Interval  t-Test  t-Test (Dep. Samples)  t-Test (Ind. Samples)  ANOVA

2 Samples => Ind. vs. Dep. samples T-Test. This is treatment vs. control – prototype for ind. samples.

Section B: Symbolic Hypotheses, Error Rates and Choosing α

For each of the following research scenarios, (a) set up the null and alternate hypotheses in correct mathematical symbols, (b) describe both Type I and II errors in words and (c) choose an appropriate level for α.

13. Environmental engineering students at Georgia Tech have found a new way to cure concrete. The old method (developed in Athens, Georgia) generated concrete with an average strength of 5000 kg/cm². They cure and test 42 samples. Statistically test the hypothesis that the new curing method generates stronger cement.

H₀ : μ = 5000

H_a : μ > 5000

Type I: Falsely claim Tech concrete is better than UGA.

Type II: Falsely claim there is no difference between UGA and Tech concrete.

If concrete is NOT better (or, perhaps, worse), we could be putting people at risk by building with it. Minimize Type I. Set α low, say α = .01.

14. Pharmaceutical researchers are testing allergy medications. Their new drug SuperCureAll has some rather nasty side effects. They compare 31 adults with allergies who take SuperCureAll to 22 adults with allergies who take a placebo. The allergy sufferers rate their levels of relief from 1 – 10, 10 being complete relief. The placebo group average is 3.9.

H₀ : μ_t = μ_c (T = Treatment, C = Control)

H_a : μ_t > μ_c Higher score (closer to 10) would mean MORE allergy relief

Type I: Falsely claim treatment works

Type II: Falsely claim there is no difference between treatment and control

If treatment is NOT significantly better, allergy sufferers take medication (and get nasty side effects) for no reason. Minimize Type I. Set α low, say α = .01.

15. Psychological researchers in Northern Georgia are studying the paranormal, wondering if the high proportion of religious individuals will mean the residents are more likely to have psychic abilities. They administer a test of psychic ability to 12,304 adult residents of outlying rural communities and compare the results to the national population whose average score on the test is 21 (out of 112).

H₀ : μ = 21 Comparing a sub-group to population – 1 Sample t-Test

H_a : μ > 21

Type I: Falsely claim NG residents more likely to be psychic

Type II: Falsely claim there is no difference between NG residents and rest of nation

There is no real danger attached to either option, and n = 12,304 > 250. This sample size is HUGE, and we need to set α low, say α = .01 or even α = .001.

16. A consumer advocacy group is testing the shock absorbency of an infant car seat they believe may be defective. They purchase 12 of the seats whose mean absorbency is rated at 1000 lbs.

H₀ : μ = 1000 Comparing a sub-group to population – 1 Sample t-Test

H_a : μ < 1000

Type I: Falsely claim car seat is defective (absorbs less than 1000 lbs.)

Type II: Falsely claim car seat is OK (absorbs 1000 lbs.)

It should be clear that making Type II error could cost lives. Minimize Type II. Set α high, say α = .1.

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By the way, this hypothesis can be tested in the opposite direction:

H₀ : μ = 1000

H_a : μ > 1000

If you set it up this way, your Type I and I error statement are the reverse of what typed above, and you would set low, say α = .01 or even α = .001.

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Section C: Hypothesis Testing

Conduct all relevant hypothesis testing steps for each of the following scenarios.

17. Dr. Olsen is concerned about her pH meter. She finds a neutral substance which should give meter readings of 7.0 on the pH scale. She conducts 10 sample measurements (data given below) of the substance on her balky meter. At the α = .1 level, test her hypothesis that the meter is faulty. Assume normality.

7.07

7.0

7.1

6.97

7.0

7.03

7.01

7.01

6.98

7.08

H₀ : μ = 7 Comparing a sub-group to population

H_a : μ ≠ 7 1 Sample t-Test

Type I: Falsely claim car meter is faulty.

Type II: Falsely claim meter is OK.

Since α = .1 is given, no Type I/Type II error analysis is

necessary. (Note: I performed error analysis to provide you with additional study material – you would NOT do this step on the test!!)

Normality is assumed, so no graphics checks (box plot or

histogram) are needed.

Run Test (see graphics for steps)

Since p = .1062 > .1 = α, we FAIL TO REJECT NULL.

In real world terms, this means we believe the meter is OK.

18. You are a consultant for a manufacturer who asks you to compare the abrasive wear for two types of lamination. For Laminate X, 12 pieces of material are tested and earn an average rating of 85 on a scale of 0 - 100 where 100 indicates “no visible wear” and 0 indicates “completely destroyed” (s.d. = 4). For Laminate Z, 10 pieces are tested and average an 81 (s.d. = 5). Given that the two laminates cost roughly the same to produce, test the hypothesis that X is significantly better than Z at an appropriate level. Assume normality.

H₀ : μ_X = μ_Z Comparing two samples

H_a : μ_X > μ_Z 2 Sample t-Test Independent

Type I: Falsely claim X “wears” better.

Type II: Falsely claim X and Z are not different.

Because productions costs are similar, the company can sell a better product (X) and have better customer satisfaction. Hence, they want to minimize Type II. Set α high, for example, α = .1.

Normality is assumed, so no graphics checks (box plot or

histogram) are needed.

Run Test (see graphics for steps)

Since p = .0284 < .1 = α, we REJECT NULL.

In real world terms, this means we believe X is a better product (both for customers and the company) than Z.

19. Researchers are wondering if hypnosis will influence mathematics test anxiety. They give 9 students a mathematics test un-hypnotized. Three weeks later, the participants take the same test after being hypnotized by a trained professional (do NOT try this at home!). Their scores are given below. Test their hypothesis about hypnosis at the α = .05 level assuming normality.

Test 1	58	67	79	59	66	71	70	52	65
Test 2	59	70	78	66	68	66	81	71	71

H₀ : μ_d = 0 Pretest/posttest (remember we make 3^rd list: L3 = L2 – L1)

H_a : μ_d > 0 2 Sample t-Test, Dependent

Type I: Falsely claim hypnosis helps.

Type II: Falsely claim hypnosis does not help.

Since α = .05 is given, no Type I/Type II error analysis is necessary. (Note: I performed error analysis to provide you with additional study material – you would NOT do this step on the test!!)

Recall that, for dependent samples, we need to construct the “Difference List.” The steps are shown below on your TI graphing calculator.

Normality is assumed, so no graphics checks (box plot or histogram) are needed.

Run Test (see graphics for steps)

Since p = .0388 < .05 = α, we REJECT NULL.

In real world terms, this means we believe that hypnosis helps alleviate test anxiety.

20. The following data represents the running time of recent movies from two top motion-picture companies. Test the hypothesis that Company 2’s movies have on average a longer running time at the α = .025 level. Assume normality.

Company	Running Time (Minutes)
1	102	86	98	109	92
2	81	165	97	134	92	87	114

H₀ : μ₁ = μ₂ Comparing two samples

H_a : μ₁ < μ₂ 2 Sample t-Test Independent

Again, α is given, so there is NO NEED to perform error analysis. This is just me working extra hard so you can study more thoroughly!!

Type I: Falsely claim there is a difference in running times.

Type II: Falsely claim there is no difference in running times.

Given: α = .025

Normality is assumed, so no graphics checks (box plot or

histogram) are needed.

Run Test (see graphics for steps, except I used incorrect alternative hypothesis – it should be “less than” rather than “not equal to.”)

Since p = .1652 > .025 = α, we FAIL TO REJECT NULL. (again, graphic is wrong, but the p-value should be .1652)

In real world terms, this means we have no evidence that either company’s movies run longer than the other’s.