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WEEK 2 Problems
Chapters 6-7
Packet
Pages:
Chapters:
Ch. 6 Introduction to Hypothesis Testing
Ch. 7 The Single-Sample t Test
Refer to the Resource Page on CANVAS for:
Stat Sheets (step by step guides)
Analysis tools/Calculators
Tutorials
Week 2 CH 6-7 Problems
1-6
7-13
Chapter 6
Introduction to Hypothesis Testing
Defining Key Terms:
Provide brief definitions for the following key terms:
ALPHA OR ALPHA LEVEL:
ALTERNATIVE HYPOTHESIS:
BETA:
COMMON ZONE:
CRITICAL VALUE:
HYPOTHESIS:
HYPOTHESIS TESTING:
NONROBUST ASSUMPTION:
NULL HYPOTHESIS:
ONE-TAILED HYPOTHESIS TEST:
P VALUE:
RARE ZONE:
Week 2 CH 6-7 Problems
Page 1 of 13
ROBUST ASSUMPTION:
SIGNIFICANCE LEVEL:
STATISTICALLY SIGNIFICANT:
TWO-TAILED HYPOTHESIS TEST:
TYPE I ERROR:
TYPE II ERROR:
Ch. 6- The 6 Steps in Hypothesis Testing (see textbook p.183 & Stat Sheet Ch.6).
Identify & Describe the 6 Steps in the Hypothesis Testing Process:
STEP 1:
STEP 2:
STEP 3:
STEP 4:
STEP 5:
STEP 6:
Week 2 CH 6-7 Problems
Page 2 of 13
Ch. 6-
Hypothesis Testing with z Tests
(see textbook pp.183-198 & Stat Sheet Ch.6)
For each of the examples below, go through the steps of hypothesis testing using the SingleSample z Test, and answer the questions in each table.
EXAMPLE A:
A health science researcher believes that athletes have a lower resting heartrate than the general
population. According to published health statistics, the general population has an average resting
heart rate of 78, with a Standard Deviation of 11. The researcher obtains a random sample of 20
athletes and collects their resting heart rates, finding a sample average heart rate of 74.
Use the standard decision rule of alpha level, Î± = .05 for a two-tailed hypothesis test.
EXAMPLE A
a.
QUESTIONS:
What groups are being compared in the study?
1. Athletes, 2. General
Population
What is the dependent variable in the study?
b.
c. List the Hypotheses:
Resting heart rate
Null- Athlete resting heart rate is the same as general populations. (H0: mean athletes = 78)
Alt- Athlete resting heart rate differs from the general population. (H1: mean athletes â‰  78)
d. State the decision rule & locate/mark on the distribution below, the critical values of z for
the hypothesis test.
For alpha=.05, 2-tailed test
Critical values = z +/- 1.96
Population:
Mean= 78
Std Dev=11
e.
Sample:
Mean=74
n=20
athletes
Calculate the test statistic for the sample data:
Z test Formula:
Z = m-Âµ
Ïƒm
f.
z = 74-78 = – 1.63
11/âˆš20
Interpret the Results. What decision should be made based on the statistical test?
Determine where the test statistic fallsâ€¦. The z statistic of -1.63 falls in the COMMON ZONE, so it is
not a significant result. To be significant, the test statistic value would need to fall in the RARE
ZONE. -1.63 is less than the required +/- 1.96 critical value. Accept the Null Hypothesis.
g.
Write a SPECIFIC short narrative conclusion about the hypothesis test results.
Not enough evidence to claim that athletes have a different resting heart rate compared to the
general population.
Week 2 CH 6-7 Problems
Page 3 of 13
STUDY 1:
According to 2019 national health statistics, the average Body Mass Index (BMI) for U.S. men is
26.8, with a standard deviation of 4.6. You collect data from a sample of n= 25 men with Type II
Diabetes who are attending a diabetes health promotion program. The average Body Mass
Index (BMI) of the sample is 31.3. Are your data sufficient to conclude that the BMI of the
sample is significantly different from the BMI of the population?
Use the standard decision rule of alpha level, Î± = .05 for a two-tailed hypothesis test.
QUESTIONS:
a.
What groups are being compared in the study?
b. What is the dependent variable in the study?
c.
List the Hypotheses:
NullAlternative-
d.
State the decision rule & locate/mark on the distribution below, the critical values of z
for the hypothesis test.
e.
Calculate the test statistic for the sample data:
f.
Interpret the Results. What decision should be made based on the statistical test?
Week 2 CH 6-7 Problems
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g.
Write a SPECIFIC short narrative conclusion about the hypothesis test results.
STUDY 2:
Recent results suggest that children with Attention Deficit Hyperactivity Disorder (ADHD) tend
to watch more TV than children who are not diagnosed with the disorder. To examine this
relationship, a researcher obtains a random sample of n = 36 children, 8 to 12 years old, who
have been diagnosed with ADHD. The average daily time spent watching tv for the sample is
M = 4.5 hours. It is known that the average time watching TV for the general population of 8
to 12-year-old children is 4.1 hours with a standard deviation of 1.8 hours. Are the data
sufficient to conclude that children with ADHD watch significantly more or less TV than
children without the disorder?
Use the standard decision rule of alpha level, Î± = .05 for a two-tailed hypothesis test.
QUESTIONS:
a.
What groups are being compared in the study?
b. What is the dependent variable in the study?
c.
List the Hypotheses:
Null-
d.
AlternativeState the decision rule & locate/mark on the distribution below, the critical values of z
for the hypothesis test.
Week 2 CH 6-7 Problems
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e.
Calculate the test statistic for the sample data:
f.
Interpret the Results. What decision should be made based on the statistical test?
g.
Write a SPECIFIC short narrative conclusion about the hypothesis test results.
Week 2 CH 6-7 Problems
Page 6 of 13
Chapter 7
The Single-Sample t Test
Defining Key Terms:
Provide brief definitions for the following key terms:
COHENâ€™S D
CRITICAL VALUE OF T:
DEGREES OF FREEDOM:
EFFECT SIZE (D):
REPLICATE:
R2:
SINGLE-SAMPLE T TEST:
UNDERPOWERED:
Week 2 CH 6-7 Problems
Page 7 of 13
Ch. 7-
Deciding When to Use a t Test or a z Test
For each of the following research projects, please indicate the most appropriate
statistical test:
T TEST or Z TEST ?
(see textbook pp. 215-216 & Stat Sheet Ch.7)
Example:
The New Yorker had an article stating that the average person from Manhattan spent 43 hours per year
in airplanes, with a standard deviation of 18 hours. You sample 25 residents of Albany, NY, and find
that they fly an average of 28 hours per year.
ANSWER: Z TEST. The actual population standard deviation was available from the article so the
researcher used that instead of estimating the population variance from the sample data.
RESEARCH PROJECT:
1. An instructor thinks his midterm exams can be completed in an average of 45 minutes. He samples
25 of his Intro Biology students and discover that it takes them an average of 50 minutes with a
standard deviation of 10 minutes.
2. The American Health Society states that you should eat an average of 6 servings of fruits and
veggies per day. You sample 16 friends and discover that they eat an average of 3 servings per day
with a standard deviation of 2 servings.
3. The APA Monitor claims that to get a PhD in psychology you must take an average of 7 statistics
classes with a standard deviation of 3 classes. You sample eight faculty in our Psych Department
and discover that they have had an average of 5 statistics classes.
4. The Mayo Clinic states that healthy cholesterol levels are below 200 mg/dL. You sample 64
middle-aged men who do not exercise or watch their diet. You do blood panels on each person
and find that for the group the average cholesterol level is 260 with a standard deviation of 40.
5. Wired magazine reports findings from a survey indicating that the average college student spends
21 hours a week texting with a standard deviation of 6 hours per week. You sample 20 people in
your class and find that they spend an average of 30 hours a week texting.
Week 2 CH 6-7 Problems
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Ch. 7-
Looking Up Critical Values of t in Table 3 (t Table) of the Appendix
For each study, look up the critical value of t that would cut off the tails of the
distribution. Note: each study specifies the alpha value & whether to use a oneor two-tailed test.
(see textbook pp. 219-228 & Stat Sheet Ch.7)
Example: The CDC reports that the average teenager needs 9 hours of sleep per night. You
sample
16 students from the local high school and find that on average they sleep 6
hours a night with a
standard deviation of 2 hours a night.
Use Î± = .05 for a two-tailed test. What is the cut-off value of t?
STUDY 1:
1. The U.S. Census Bureau reports that the average 3-5 year-old spends 26 hours a week in nonrelative
child care (being cared for by someone who is not related to the family). You think the average might
be lower in upper-income families. You contact 36 upper-income families and find that on average
their children spend 20 hours a week in nonrelative child care with a standard deviation of 12 hours.
Use Î± = .05 for a one-tailed test. What is the cut-off value of t?
Week 2 CH 6-7 Problems
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STUDY 2:
2. USA Today reports that the average preschooler watches an average of 35 hours of TV a week. You
suspect this average is not correct. You sample 81 households with preschoolers and find that on
average they report that their preschool children watch 15 hours a week with a standard deviation of
9 hours a week.
Use Î± = .05 for a two-tailed test. What is the cut-off value of t?
STUDY 3:
3. The CDC is concerned that teenagers are being pushed into too many out-of-school activities. Their
studies conclude that 2 out-of-school activities is the healthiest amount. You sample 49 students
from the local high school and find that on average they participate in 5 activities with a standard
deviation of 2.
Use Î± = .01 for a two-tailed test. What is the cut-off value of t?
STUDY 4:
4. Glamour magazine reports that the average woman in her 30s has tried 5 different diets. You want to
know whether, by age 50, women have tried even more diets. You sample 64 women in their 50s and
find that on average women have tried 7 different diets with a standard deviation of 2.
Use Î± = .01 for a one-tailed test. What is the cut-off value of t?
Week 2 CH 6-7 Problems
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Ch. 7-
Computing Single-Sample t Tests:
For the examples below, go thru steps of hypothesis testing using the Single-Sample t Test, &
answer questions in tables. (see text pp. 215-247 & Stat Sheet Ch.7)
Example Problem:
According to national health guidelines, the target aerobic heart rate for people age 20 years old is 120
bpm. You collect data from 16 male golfers (all age 20 years). Their average heart rate after golfing for 30
minutes is 110 bpm, with a standard deviation of 15.8 bpm. Compute a single-sample t-test to determine
if there is a significant difference between the golfers and the regular 20-year old population on average
bpm. Use the standard decision rule of alpha level, Î± = .05 for a two-tailed hypothesis test.
EXAMPLE A
a.
b.
QUESTIONS:
What groups are being compared in the study? 1. General 20-year old population
2. 20-year old golfers
What is the dependent variable in the study?
Aerobic Heart Rate
c. List the Hypotheses:
Null- Aerobic heart rate is the same for golfers & general population. (H0: m1 = m2)
Alt- Aerobic heart rate differs for golfers & general population. (H1: m1 â‰  m2)
d.
State the decision rule & locate/mark on the distribution below, the critical values of t for
the hypothesis test.
For alpha=.05, 2-tailed test, df=N-1 = 15
Critical values = +/- 2.131
Population:
Mean=120
Sample:
Mean=110
N=16
e.
Calculate the test statistic for the sample data:
f.
t test Formula:
sm = 15.8/âˆš16
t = m-Âµ
= 3.95
sm
Compute the Cohenâ€™s d Effect Size & Interpret
d = 110-120
15.8
t = 110-120
3.95
= – 2.53
= – .63
Medium Effect Size
g.
Interpret the Results. What decision should be made based on the statistical test?
The calculated test statistic was in the critical zone so results are significant. Reject the Null Hyp.
h.
Write a SPECIFIC short narrative conclusion about the hypothesis test results.
Found evidence of significant difference between golfers aerobic heart rate & regular population
heart rate (for 20 yr olds). Specifically, golfers had lower aerobic heart rate (110 vs 120).
Week 2 CH 6-7 Problems
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STUDY 1:
A lightbulb manufacturer believes their new compact fluorescent bulbs last longer than
incandescent bulbs. The manufacturer knows that the average number of hours a 60watt incandescent bulb lasts is 2,350 hours. The manufacturer takes a random sample
of 100 new compact fluorescent bulbs and measures the average number of hours they
last. The compact fluorescent bulbs averaged 2,371 hours with a Standard Deviation of
132 hours. Is there sufficient evidence to conclude the new compact fluorescent bulbs
are significantly different in number of hours they last?
Use the standard decision rule of alpha level, Î± = .05 for a two-tailed hypothesis test.
QUESTIONS:
a.
What groups are being compared in the study?
b. What is the dependent variable in the study?
c.
List the Hypotheses:
Null-
d.
AlternativeState the decision rule & locate/mark on the distribution below, the critical values of t
for the hypothesis test.
e.
Calculate the test statistic for the sample data:
f.
Compute the Cohenâ€™s d Effect Size & interpret.
Week 2 CH 6-7 Problems
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g.
Interpret the Results. What decision should be made based on the statistical test?
h.
Write a SPECIFIC short narrative conclusion about the hypothesis test results.
Week 2 CH 6-7 Problems
Page 13 of 13