Chapter 8: Hypothesis Testing
for Population Proportions
The basics of Significance Testing
Statistical Inference
• Already discussed confidence intervals for unknown
population parameter, p
• Confidence Intervals used when the goal is to estimate an
unknown population parameter like ρ (like when we
estimated the true proportion of all 5,000 COC students
who have at least one tattoo)
• This chapter... statistical inference through significance
tests
• Evaluate evidence (a statistic) provided by sample data
about some claim concerning an unknown population
parameter like ρ
The Main Ingredients of Hypothesis
Testing
• There once were four students who missed
the midterm for their statistics class. They
went to the professor together and said,
“Please let us make up the exam. We carpool
together, and on our way to the exam, we got
a flat tire. That’s why we missed the exam.”
The professor didn’t believe them, but instead
of arguing he said, “Sure, you can make up the
exam. Be in my office tomorrow at 8.”
The Main Ingredients of Hypothesis
Testing
• The next day, they met in his office. He sent each
student to a separate room and gave them an
exam. The exam consisted of only one question:
“Which tire?”
• Let’s imaging all four students answered, “left
rear tire.”
• So... what do you think? Were students most
likely telling the truth? Lying?
The Main Ingredients of Hypothesis
Testing
• Surprised or not?
• The professor suspected they had been lying. That’s why he did
what he did.
• Maybe they just got lucky ... just by chance they all guess the same
tire.
• The probability that all four students would guess the same tire is
only about 1.6%.
• Do you consider that likely/typical or unlikely/rare that they could
have just simply, by chance guessed the same tire?
I’m a great free-throw shooter...
• I claim that in the last 5 years of playing
basketball, I, on average, make 95% of my
basketball free throws.
• To test my claim, I am asked to shoot 50 free
throws. I make only 10 of the 50 (only 20%).
• Do you still believe my claim that I make 95%
of my free throw? Why or why not?
I claim 95% ... I actually made only 20%...
• Do you still believe my claim that I make 95% of my free
throw? Why or why not?
• Do you agree that statistics vary from sample to sample?
So if I attempted another 50 free throws, chances are I
would make something other than 10 of them?
• So the question is... would me actually making only 10
out of 50 (or 20%) happen so, so very rarely (assuming
my 95% claim were true) that now you are starting to
question/doubt if my claim really is true.
Hypothesis Testing or Significance
Testing ...
•
A formal procedure that enables us to choose between two hypotheses when we
are uncertain about our measurements.
•
Basic idea... An outcome that would rarely happen if a claim were really true is
good evidence that the claim is not true.
•
Example... I claim that 99% of adult humans are 6 feet tall or taller.
•
So, I’m going to take a SRS of 1000 humans and measure their heights. Then I
calculate the sample mean height and get 5’ 8”. Do you agree with my claim or
disagree with my claim, based on my sample statistic?
•
If my claim was true, it would be very rare to get most of the adult humans in a
SRS of 1000 that are shorter than 6 feet. OR on the flip side, it would be very
common for me to get humans in a SRS of 1000 that are 6 feet tall or taller.
Start with: A Pair of Hypotheses
• If we flip a penny, we can agree that the
probability of heads or tails is 0.50; fair.
• However, some claim if we spin a penny on a
table, because the heads side bulges outwards,
the lack of symmetry will cause the spinning coin
to land on one side more often than the other;
probability is not 0.50 for each side; unfair
• Some people might find this claim outrageous;
completely false
Start with a research hypothesis...
Null hypothesis, Ho, p = 0.50
-null hypothesis is always neutral, no change, always =
-null hypothesis is always in terms of population
parameter (like p or μ)
Alternative hypothesis, Ha, p ≠ 0.50
- alternative hypothesis is always <, >, or ≠
-alternative hypothesis is always in terms of population
parameter (like p or μ)
Like in a criminal trial...
Null hypothesis, Ho, p = 0.50
- In the beginning, we assume null is true (like
defendant is assumed not guilty in the beginning
of a trial) until there is overwhelming evidence
that suggests this is not so; then we may reject
this believe if/when the evidence is clearly
against it
Alternative hypothesis, Ha, p ≠ 0.50
Null Hypothesis... Ho ...
• The null hypothesis always gets the benefit of
the doubt and is assumed to be true
throughout the hypothesis-testing procedure.
If we decide at the last step that the observed
outcome is extremely unusual under this
assumption, then and only then do we reject
the null hypothesis.
Ho: p = 0.50 Ha: p ≠ 0.50
• If null hypothesis is correct, then when we spin a
coin a number of times, about ½ of the outcomes
should be heads. If null hypothesis is wrong, we
will see either a much larger or much smaller
proportion.
• Let’s spin some pennies. Spin (on desk) 20 times.
Count the # of heads
• Calculate the sample proportion,
write on board
pˆ of heads and
Ho: p = 0.50 Ha: p ≠ 0.50
• Let’s look at our sampling distribution; describe
using SOCS (review)
• If we did this again, would be get different
results?
• How ‘extreme’ of a result would we need for you
to not believe our null hypothesis/to reject null?
• We will come back to this later in the chapter...
Practice with null and alternative
hypotheses...
• What’s wrong with ...
•
•
•
•
•
Ho:
Ho:
Ho:
Ho:
Ho:
p = 0.17
p = - 0.20
p > 0.45
p = 1.50
pˆ = 0.92
Ha:
Ha:
Ha:
Ha:
Ha:
p ≠ 0.19
p < 0.15
p = 0.45
p > 1.50
pˆ < 0.92
Practice: State the appropriate null hypothesis and
alternative hypothesis in each case. Be sure to define
your parameter each time.
A recent Gallup Poll report on a national survey of 1028
teenagers revealed that 72% of teens said they rarely
or never argue with their friends. You wonder
whether this national result would be different in
your school. So you conduct your own survey of a
random sample of students at your school.
Practice: State the appropriate null hypothesis and
alternative hypothesis in each case. Be sure to define
your parameter each time.
The proportion of people who live after
suffering a stroke is 0.85. A drug
manufacturer has just developed a new
treatment that they claim will increase the
survival rate.
Explain what is wrong in each situation and why
it is wrong
A change is made that should improve student
satisfaction with the parking situation at COC.
The null hypothesis, that there is an
improvement, is tested versus the alternative,
that there is no change.
Explain what is wrong in each situation and why
it is wrong
• A researcher tests the following null
hypothesis
• Ho: pˆ = 0.80

Explain what is wrong in each situation and why it is
wrong
A statistics instructor at COC read that 90% of all
college students use social media on a regular basis.
She wonders if the percent of COC students who use
social media on a regular basis is different.
Ho:
p = 0.90
Ha:
p > 0.91
Explain what is wrong in each situation and why
it is wrong
The Census Bureau reports that households spend
an average of 31% of their total spending on
housing. A homebuilders association in Cleveland
believes that this average is lower in their area.
They interview a sample of 40 households in the
Cleveland metropolitan area to learn what
percent of their spending goes toward housing.
Take p to be the mean percent of spending
devoted to housing among all Cleveland
households.
H0: p = 31%
Ha: p < 31%
The Main Ingredient: Surprise
• Surprise itself; when something unexpected occurs (like
only making 20% of free throws when we claimed to make
95%)
• Null hypothesis tells us what to expect; it’s what we believe
throughout the process until we see evidence otherwise
• If we see something unexpected, then we should doubt the
null hypothesis
• If we are really surprised, then we should rejected it
altogether
Let’s go back to our penny spinning...
We have a way to measure our
“surprise”...
• Instead of just not surprising, kind of surprising, very
surprising, etc., we have...
• p-value
• A p-value is a probability. Assuming the null hypothesis
is true, the p-value is the probability that if the
experiment were repeated many times, we would get
as extreme or more extreme outcome than the one we
actually got (our statistic). A small p-value suggests
that a surprising outcome has occurred and discredits
the null hypothesis.
P-Values
A p-value is a quantitative measure of rarity
of/how unlikely a finding
Small p-values are evidence against Ho
Large p-values fail to give evidence against Ho
P-value... all about extremes...
• Understanding how to interpret a p-value is
crucial to understanding hypothesis testing.
• Minitab will calculate the p-value, but we need to
understand how the computer did the calculation
• The meaning of the phrase, “as extreme as or
more extreme than’ depends on the alternative
hypothesis
P-value... all about extremes... Three basic pairs
of hypotheses...
Let’s go back to spinning coins... Ho: p = 0.50 Ha: p ≠ 0.50
• Note: the closer the number of heads is to 10, the larger the p-value
• Also note the p-value for an outcome of 11 heads is the same as for 9
heads, etc.
Statistical Significance...
• Most of the time, we take one more step to
assess evidence against Ho
• We compare the p-value to some predetermined value (versus ‘unlikely’) called a
significance level, symbol α (alpha)
• Can think of this as a rejection zone (sketch)
Statistical Significance
• Significance level makes ‘not likely’ more
exact, more informative
• Most common α levels are α = 0.05 or α = 0.01
• Interpretation:
– At α = 0.05, data give evidence against Ho so
strong it would happen no more than 5% of the
time
Statistical Significance
• If p-value is as small or smaller than α, we say
data are statistically significant at level α
• Note: ‘significant’ in statistics doesn’t mean
important (like in English); it means not likely
to happen by chance
Let’s sketch some pictures of rejection
zones and p-values...
• Ho: p = ...
Ha: p ...
• I gathered sample data, and calculated a pvalue based on sample data (probability of
getting that value or more extreme assuming
that null hypothesis is true)
• 1-sided
• 2-sided
Statistically Significant Sketches
• If p-value is p = 0.03... this is significant at α =
0.05 level (in rejection zone)
• If p-value is p = 0.03... this is not significant at
α = 0.01 level (not in rejection zone)
Interpretation/Wording
Reject Ho (Null Hypothesis):
This happens when sample statistic is statistically
significant, p-value is too unlikely to have
occurred by chance (we don’t believe null
hypothesis), in the rejection zone
Wording must reference all of the following for a
complete interpretation... p-value, α level, reject
Ho, and conclusion in context (caution about
using the word ‘cause’ or ‘prove’).
Interpretation/Wording
Fail to Reject Ho (Null Hypothesis):
This happens when sample statistic could have
occurred by chance (we do believe null
hypothesis; we don’t believe the alternative), not
in rejection zone
Wording must reference all of the following for a
complete interpretation... p-value, α level, fail to
reject Ho, and conclusion in context (caution
about using the word ‘cause’ or ‘prove’)
Conditions for Tests about a population proportion...
• Random Sample ... randomly selected or randomly
assigned
• Large Sample Size; Normality (see next slide) ...
npo ≥ 10 and n(1 – po) ≥ 10; the sample has at least 10
expected successes and at least 10 expected failures
• Big Population (Independence) ... Population at least 10
times sample size; and each observation has no
influence on any other
• ...if these conditions are satisfied, then we can
use the Central Limit Theorem for sample
proportions; distribution is ≈ Normal! That’s a
great thing!
• When doing a hypothesis test, you MUST
check conditions... this is an essential part of
the hypothesis testing process
So many p’s... Caution!
Hypothesis testing in four steps...
Work stress...
According to the National Institute for Occupational Safety and Health, job
stress poses a major threat to the health of workers. A national survey
of restaurant employees found that 75% said that work stress had a
negative impact on their personal lives.
A simple random sample of 100 employees from a large restaurant chain
finds that 68 answer “Yes” when asked, “Does work stress have a
negative impact on your personal life?” Is this good reason to think that
the proportion of all employees in this chain who would say “Yes”
differs from the national proportion p0 = 0.75?
H0: p = 0.75
Ha: p ≠ 0.75
We want to test a claim about p, the true proportion of this chain's
employees who would say that work stress has a negative impact on
their personal lives.
Work stress...
Conditions: 1-sample proportion; α = 5% (rejection zone)
Random Sample – stated in problem
Large Sample Size/Normality - The expected number of
“Yes” and “No” responses are (100)(0.75) = 75 and
(100)(0.25) = 25, respectively. Both are at least 10.
Big Population (Independence) - Since we are sampling
without replacement, this “large chain” must have at
least (10)(100) = 1000 employees.
H0: p = 0.75
Ha: p ≠ 0.75
sample statistic = 68/100
Calculations for 1-sample proportion 2-sided hypothesis
test; use Minitab
1 sample, proportion; change options and data as needed
z = -1.6165
P-value = 0.1059
Work stress...
Interpretation:
Fail to reject Ho. With a P-value of 0.1059 and
an α = 5%, we fail to reject the null hypothesis
and conclude that there is not enough
evidence to suggest that the proportion of this
chain restaurant's employees who suffer from
work stress is different from the national
survey result, 0.75.
We want to be rich...
• In a recent study, 73% of first-year college students
responding to a national survey identified “being very
well-off financially” as an important personal goal. A
state university finds that 132 of a random sample of
200 of its first-year students say that this goal is
important.
• Is there evidence that the proportion of all first-year
students at this university who think being very well-off
is important differs from the national value, 73%? Carry
out a significance test to help answer this question.
n = 200; x = 132; SRS; p = .73; 𝑝 =
0.66
We want to test Ho: p = 0.73 versus Ha: p ≠ 0.73
regarding the proportion of all first-year
students at this university who think being
very well-off is important differs from the
national value of 73%.
n = 200; x = 132; SRS; p = .73; 𝑝 =
0.66
Conditions: 1-sample proportion; α = 5% (rejection zone)
Random Sample/SRS – stated in problem
Large Sample Size/Normality – np ≥ 10 & n (1 – p) ≥ 10
(200)(0.73) ≥ 10 & (200) (1 -0.73) ≥ 10
Big Population (Independence) – We must assume at least
(10)(200) first-year students in the population.
n = 200; x = 132; SRS; p = .73; 𝑝 =
0.66
Calculations... Minitab; 1-sample proportion
z = -2.22
P-value = 0.0258
Interpretation...
Reject Ho. With a p-value of 0.0258, and
assuming an α = 0.05, we conclude that we do
have statistically significant evidence that the
proportion of all first-year students at this
university who think being very well-off is
important differs from the national value.
(determination, p-value, α, and context...
always)
Interpretation...
Reject Ho. With a p-value of 0.0258, and
assuming an α = 0.05, we conclude that we do
have statistically significant evidence that the
proportion of all first-year students at this
university who think being very well-off is
important differs from the national value.
What if.... Our alpha had been 1%? Would our
decision have changed?
Dreaming in color...
Researchers wondered whether a greater proportion of
people now dream in color than did so before color
television and movies became as prominent as they are
today. In the past, before color TV and movies, this
proportion was 0.29. Researchers took a random
sample of 113 people. Of these 113 people, 92
reported dreaming in color.
Is there evidence (at a significance level of 1%) that more
people today dream in color than in the past (before
color TV and movies became as prominent as they are
today)? Carry out an appropriate hypothesis test to
help answer this question.
Dreaming in color...
What are our null and alternative hypotheses?
Conditions: 1-sample proportion; α = 1% (rejection zone)
Random Sample –
Large Sample Size/Normality Big Population/Independence –
Calculations –
Determination and interpretation -
Dreaming in color...
What are our null and alternative hypotheses
Ho: p = 0.29 Ha: p > 0.29
Conditions 1-sample proportion; α = 1% (rejection zone)
Random Sample
Large Sample Size/Normality
Big Population/Independence
Calculations
z = 12.28, P-value ≈ 0
Determination and interpretation
Reject null hypothesis. At an alpha level of 1%, and a P-value of
about zero, there is sufficient evidence to suggest that more
people today dream in color than in the past (before color TVs,
etc.)
Two Proportion Hypothesis Testing
• Ho: p1 = p2
• Ha: p1 ≠ or > or < p2
Minitab will calculate this for us; no need to
memorize
Two Proportion Hypothesis Testing
Conditions...
• SRS – Each of the two samples must be SRSs
from their respective populations or they
must each be randomized experiments
• Normality – Each of the following are all ≥ 10
(n1)(𝑝c)
(n1)(1 – 𝑝c)
(n2)(𝑝c)
(n2)(1 – 𝑝c)
Two Proportion Hypothesis Testing
Conditions...
• Big Population/Independence
Each of the populations must be at least (10)
times each of the corresponding sample sizes;
and one sample does not influence the other
Two Proportions... Conditions...
On our tests, I will not ask you to check conditions
for 2-sample proportion hypothesis testing
We will assume all conditions have been checked
and met for 2-sample proportion hypothesis
testing
For your project, however, you will have to check
conditions
Does Pre-School Help?
To study the long-term effects of preschool programs for poor
children, a research foundation has followed two groups of
Michigan children since early childhood. A control group of 61
children represents population 1, poor children with no pre-school.
Another group of 62 from the same area and similar backgrounds
attended pre-school as 3- and 4-year-olds represents population 2,
poor children who attend pre-school. Sizes are n1 = 61 and n2 = 62.
One response variable of interest is the need for social services as
adults. In the past ten years, 38 of the preschool sample and 49 of
the control sample have needed social services (mainly welfare).
Carry out an hypothesis test to determine if there is significant
evidence that pre-school reduces or increases the later need for
social services?
You may assume all conditions have been checked and met. Use an
alpha level = 5%.
n pre-school = 62
nno pre-school = 61
38 of pre-school needed social services;
49 of no pre-school needed social services
State null and alternative hypothesis
Ho: pno pre-school = ppre-school
Ha: pno pre-school ≠ ppre-school
Conditions: Stated in the problem that they
have been checked and met. We are doing a
2-proportion hypothesis test.
Ho: pno pre-school = ppre-school
Ha: pno pre-school ≠ ppre-school
Minitab to calculate test statistic, p-value, etc.
Two Sample, Proportion, Options & Data
z = -2.3201
P-value = 0.0203
Ho: pno pre-school = ppre-school
Ha: pno pre-school ≠ ppre-school
Interpretation:
Reject null hypothesis. At a significance level of
5% (α = 0.05), and a p-value of approximately
0.02 there is sufficient evidence to show that
p no pre-school ≠ p pre-school (or evidence that
pre-school reduces or increases (changes) the
later need for social services
Fear of Crime...
The elderly fear crime more than younger people, even
though they are less likely to be victims of crime. One
of the few studies that looked at older blacks recruited
random samples of 56 black women and 63 black men
over the age of 65 from Atlantic City, New Jersey. Of
the women, 27 said they “felt vulnerable” to crime; 46
of the men said this.
What proportion of women in the sample feel
vulnerable? Of men? (Note: Men are victims of crime
more often than women, so we expect a higher
proportion of men to feel vulnerable.)
Fear of Crime...
Test the hypothesis that the true, unknown
population proportion of all elderly black
males who feel vulnerable is higher than that
of all elderly black women who feel
vulnerable. You may assume that all
conditions have been checked and met.
Hypothesis, Conditions/Name of
Procedure/Alpha Level, Computations,
Interpretation
Ho: p men = p women or p men – p women = 0
Ha: p men > p women or p men – p women > 0
• sample statistics: 46/63 men & 27/56 women
• z = 2.7731
P-value = 0.0028
• Reject null hypothesis. At any reasonable alpha
level, with a P-value less than 1%, we have
evidence to suggest that the proportion of all
black men who feel vulnerable is higher than the
proportion of all black women who feel
vulnerable.
Cholesterol & Heart Attacks...
• High levels of cholesterol in the blood are associated
with higher risk of heart attacks. Will using a drug to
lower blood cholesterol reduce heart attacks? The
Helsinki Heart Study looked at this question. Middleaged men were assigned at random to one of two
treatments: 2,051 men took the drug gemfibrozil to
reduce their cholesterol levels, and a control group of
2,030 men took a placebo. During the next five years,
56 men in the gemfibrozil group and 84 men in the
placebo group had heart attacks.
• Is the apparent benefit of gemfibrozil statistically
significant? Assume all conditions have been checked
and met. Use a 1% alpha level.
Ho: pgemfibrozil = pplacebo Ha: pgemfibrozil < pplacebo
OR
Ho: pgemfibrozil - pplacebo = 0 Ha: pgemfibrozil – pplacebo < 0
We want to use this comparative randomized experiment
to draw conclusions about p1, the proportion of
middle-aged men who would suffer heart attacks after
taking gemfibrozil, and p2, the proportion of middleaged men who would suffer heart attacks if they only
took a placebo. We hope to show that gemfibrozil
reduces heart attacks, so we have a one-sided
alternative.
Ho: pgemfibrozil - pplacebo = 0
Ha: pgemfibrozil – pplacebo < 0
n gemfibrozil = 2,051
n placebo = 2,030
x gemfibrozil = 56
x placebo = 84
Sample statistic for gemfibrozil = 56/2051 ≈2.7% had
heart attacks
Sample statistic for placebo = 84/2030 ≈ 4.1% had heart
attacks
Is this difference just due to chance? Or is there really a
difference between the medication and the placebo?
Ho: pgemfibrozil - pplacebo = 0
Ha: pgemfibrozil – pplacebo < 0
n gemfibrozil = 2,051
n placebo = 2,030
x gemfibrozil = 56
x placebo = 84
Sample statistic for gemfibrozil = 56/2051 ≈2.7% had heart attacks
Sample statistic for placebo = 84/2030 ≈ 4.1% had heart attacks
z = - 2.47
P-value = 0.0068
Reject null hypothesis. With an alpha level of 1%
and a P-value of 0.0068, we have evidence that
shows there is an apparent benefit to gemfibrozil
(i.e., the drug appears to reduce heart attacks).
Three-Strikes Law...
California’s controversial ‘three strikes law’ requires
judges to sentence anyone convicted of three
felony offenses to life in prison. Supporters say
that this decreases crime; opponents argue that
people serving life sentences have nothing to
lose, so violence within the prison system
increases.
Assume all conditions have been checked and met.
Three-Strikes Law...
• Researchers looked at data from the California
Department of Corrections.
• Of 734 prisoners who had three strikes, 163 of them
had committed ‘serious’ offenses
• Of 3,188 prisoners who did not have three strikes, 974
had committed ‘serious’ offenses
• Determine whether those with three strikes tend to
have more offenses than those who do not. Use a 5%
significance level.
Ho: p 3 strikes offenses = p no 3 strikes offenses
Ha: p 3 strikes offenses > p no 3 strikes offenses
Sample statistic for prisoners who had three strikes was
163/734 ≈ 22.2%
Sample statistic for prisoners who did not have three
strikes was 974/3188 ≈ 30.6%
z = - 4.49
P-value = 0.9999
Fail to reject null hypothesis. At a 5% alpha level and a Pvalue ≈ 1, there is not sufficient evidence to conclude
that prisoners who have three strikes commit more
serious offences than those prisoners who do not have
three strikes.
We check conditions for a reason...
• Again you do not have toworry about checking
conditions for 2-sample proportion... It’s
basically a mess to check conditions for 2sample tests...
• But know that... in the real world, if conditions
are not satisfied, our results may not be
accurate, reliable, trustworthy, etc.
Use & Abuse of Tests...
• Significance tests are used in a variety of settings... Marketing,
FDA drug testing, discrimination court cases, etc.
• Significance tests quantify event that is unlikely to occur
simply by chance
• Different levels of significance (α) are chosen depending on
the given situation; typically α = 0.10, 0.05, or 0.01
• Continue to use caution when using “prove” or “cause”... even
when doing hypothesis testing
Use & Abuse of Tests...
• P-values allow us to decide individually if
evidence is sufficiently strong
• But, there is still no practical distinction between
p-values of, say, 0.049 and 0.051 if our alpha level
was, say, 5%
• Statistical inference does not correct basic flaws
in survey or experimental design, such as ...
Using Inference to Make Decisions...
Sometimes we do everything correctly... data collection,
conditions, calculations, interpretation... but we still make
an incorrect decision/determination... perhaps we just
happen to get a sample statistic that is very extreme... that
really doesn’t represent our population accurately
... we reject the null hypothesis when we really should have
failed to reject (Ho was really true)
OR we fail to reject the null hypothesis when we really should
have rejected the null hypothesis (Ho was really false)
... we make an error
Making errors when using inference...
• Type I Error
We reject Ho (null hypothesis) when Ho is really true
In other words, we determine Ha (alternative hypothesis)
is true when, in actuality, Ho (null hypothesis) is true
• Type II Error
We fail to reject Ho (null hypothesis) when Ho is really
false
In other words, we determine Ho (null hypothesis) is true,
when, in reality, Ha (alternative hypothesis) is true
Type I and Type II Errors...
Probabilities of Type I and Type II
Errors...
• Probability of Type I Error (rejecting Ho when
null is really true): α, your significance level
for the hypothesis test.
• Probability of Type II Error (failing to reject Ho
when alternative is really true): β. Very
complicated to calculate. Beyond scope of
this course.
Power of a Test...
• Power: Probability that a test will reject Ho
when Ha is true
• Think of power as making the correct decision,
not making an error, not making a mistake
• High level of power is a good thing
• Power = 1 – β (remember β is probability of
making a type II error); so ‘power’ and β are
complimentary
Power of a Test...
• How can we increase power (making the correct
decision)?
• Increase α
• Increase n
• Decrease standard deviation (same effect as
increasing the sample size, n)
Chapter 8 HW Quiz ...