Hypothesis Tests
One Sample Means
How canagency
I tell ifhas
they really
A government
are
underweight?
received numerous complaints
A hypothesis test
that will
a particular
restaurant
has
allow me to
been
selling
underweight
decide
if the
claim
is true or not!
hamburgers.
restaurant
Take The
a sample
& find x.
advertises that it’s patties are
“a quarter
pound”
(4 ounces).
But how
do I know
if this x is one
that I expect to happen or is it one
that is unlikely to happen?
Steps for doing a
hypothesis test
“Since the p-value < (>) a, I reject
1) Assumptions
(fail to reject) the H0. There is (is
not) sufficient evidence to suggest
thathypotheses
Ha (in context).”
2) Write
& define parameter
H0: m = 12 vs Ha: m (<, >, or ≠) 12
3) Calculate the test statistic & p-value
4) Write a statement in the context of the
problem.
Assumptions for t-inference
• Have an SRS from population (or
randomly assigned treatments)
• s unknown
• Normal (or approx. normal)
distribution
– Given
– Large sample size
– Check graph of data
Use only one of
these methods to
check normality
Formulas:
s unknown:
statistic - parameter
test statistic 
standard deviation of statistic
t=
x m
s
n
Calculating p-values
• For z-test statistic –
– Use normalcdf(lb,rb)
– [using standard normal curve]
• For t-test statistic –
– Use tcdf(lb, rb, df)
Draw & shade a curve &
calculate the p-value:
1) right-tail test
t = 1.6; n = 20
P-value = .0630
2) two-tail test
t = 2.3; n = 25
P-value = (.0152)2 = .0304
Example 1: Bottles of a popular cola are
supposed to contain 300 mL of cola.
There is some variation from bottle to
bottle. An inspector, who suspects that
the bottler is under-filling, measures the
contents of six randomly selected bottles.
Is there sufficient evidence that the
bottler is under-filling the bottles?
Use a = .1
299.4 297.7 298.9 300.2 297 301
• I have an SRS of bottles
SRS?
Normal?
•Since the boxplot is approximately symmetrical with
no
outliers, the sampling distribution is approximatelyHow do you
know?
normally distributed
Do you
know s?
What are your
H0: m = 300 where m is the true mean amount
hypothesis
statements? Is
Ha: m < 300 of cola in bottles
there a key word?
299 .03  300
t 
 1.576 p-value =.0880
a = .1
1.503
Plug p-value
values to
Compare your
6
into decision
formula.
a & make
Since p-value < a, I reject the null hypothesis.
Write conclusion in
There is sufficient evidence to suggest
that
the true
context
in terms
of Ha.
mean cola in the bottles is less than 300 mL.
• s is unknown
Example 3: The Wall Street Journal
(January 27, 1994) reported that based
on sales in a chain of Midwestern grocery
stores, President’s Choice Chocolate Chip
Cookies were selling at a mean rate of
$1323 per week. Suppose a random sample
of 30 weeks in 1995 in the same stores
showed that the cookies were selling at
the average rate of $1208 with standard
deviation of $275. Does this indicate that
the sales of the cookies is lower than the
earlier figure?
Assume:
•Have an SRS of weeks
•Distribution of sales is approximately normal due to
large sample size
• s unknown
H0: m = 1323
where m is the true mean cookie sales
error in context?
Ha: m < 1323What is the
per potential
week
What is a consequence of that error?
1208  1323
t 
 2.29 p  value  .0147
275
30
Since p-value < a of 0.05, I reject the null hypothesis.
There is sufficient evidence to suggest that the sales of
cookies are lower than the earlier figure.
Example 9: President’s Choice Chocolate Chip
Cookies were selling at a mean rate of $1323
per week. Suppose a random sample of 30
weeks in 1995 in the same stores showed
that the cookies were selling at the average
rate of $1208 with standard deviation of
$275. Compute a 90% confidence interval for
the mean weekly sales rate.
CI = ($1122.70, $1293.30)
Based on this interval, is the mean weekly
sales rate statistically less than the
reported $1323?
Matched Pairs
Test
A special type of
t-inference
Matched Pairs – two forms
• Pair individuals by
certain
characteristics
• Randomly select
treatment for
individual A
• Individual B is
assigned to other
treatment
• Assignment of B is
dependent on
assignment of A
• Individual persons
or items receive
both treatments
• Order of
treatments are
randomly assigned
before & after
measurements are
taken
• The two measures
are dependent on
the individual
Is this an example of matched pairs?
1)A college wants to see if there’s a
difference in time it took last year’s
class to find a job after graduation and
the time it took the class from five years ago
to find work after graduation. Researchers
take a random sample from both classes and
measure the number of days between
graduation and first day of employment
No, there is no pairing of individuals, you
have two independent samples
Is this an example of matched pairs?
2) In a taste test, a researcher asks people
in a random sample to taste a certain brand
of spring water and rate it. Another
random sample of people is asked to
taste a different brand of water and rate it.
The researcher wants to compare these
samples
No, there is no pairing of individuals, you
have two independent samples – If you would
have the same people taste both brands in
random order, then it would be an example
of matched pairs.
Is this an example of matched pairs?
3) A pharmaceutical company wants to test
its new weight-loss drug. Before giving the
drug to a random sample, company
researchers take a weight measurement
on each person. After a month of using
the drug, each person’s weight is
measured again.
Yes, you have two measurements that are
dependent on each individual.
Stroop Test
Is there an interaction between color &
word?
Or in other words … is there a
significant increase in time?
A whale-watching company noticed that many
customers wanted to know whether it was
better to book an excursion in the morning or
the afternoon.
To test
this question, the
You may subtract
either
company
thewhen
following data on 15
way – collected
just be careful
writing Hadays over the past
randomly selected
month. (Note: days were not
consecutive.)
Day
1
2
Morning
8 9
3
4
5
6
7
8
9
10
11 12 13 14 15
7 9 10 13 10
8
2
5
7 7 6 8 7
After8 10 9 8 9 11 8
noon
Since you have two values for
10
4 7 8 9 6 6 9
First, you must find
the differences for
each day.
each day, they are dependent
on the day – making this data
matched pairs
Day
1
2
3
Morning
8
9
7 9 10 13 10
Afternoon
8 10
4
5
9 8 9
6
7
8
9
10
11 12 13 14 15
8
2
5
7 7 6 8 7
11
8 10 4 7 8 9 6 6 9
I subtracted:
Differenc
0 -1 -2 1 1 Morning
2 2 – -2
-2 -2 -1 -2 0 2 -2
afternoon
es
You could subtract the other
way!
• Have an SRS of days for whale-watching
You need to state assumptions using the
• s unknown
differences!
Assumptions:
•Since the normal probability plot is approximately
linear, the distribution of difference is approximately
Notice the granularity in this
normal.
plot, it is still displays a nice
linear relationship!
Differences
0
-1
-2
1
1
2
2
-2
-2
-2
-1 -2
0
2
Is there sufficient evidence that more whales are
sighted in the afternoon?
H0: mD = 0
Ha: mD < 0
Be careful writing your Ha!
Think about
how you–
If you subtract
afternoon
subtracted: M-A
Hdifferences
mD>0should
Notice morning;
we
mthen
a:more
D foris
Ifused
afternoon
& it equals
since the nullbeshould
the0 differences
+ or -?
be that there
NOat
difference.
Don’t islook
numbers!!!!
Where mD is the true mean
difference in whale sightings
from morning minus afternoon
-2
Differences
0
-1
-2
1
1
2
2
-2
finishing the hypothesis test:
x m
.4  0
t 

 .945
s
1.639
n
15
p  .1803
df  14
a  .05
-2
-2
-1 -2
0
2
In your calculator,
perform
t-test
Notice athat
if
the
youusing
subtracted
differences
(L3)
A-M, then your
test statistic
t = + .945, but pvalue would be
the same
Since p-value > a, I fail to reject H0. There
is
How could
I
insufficient evidence to suggest that more
whales
increase
theare
sighted in the afternoon than in the morning.
power of this
test?
-2
Example 2: The Degree of Reading Power
(DRP) is a test of the reading ability of
children. Here are DRP scores for a random
sample of 44 third-grade students in a
suburban district:
(data on note page)
At the a = .1, is there sufficient evidence to
suggest that this district’s third graders
reading ability is different than the national
mean of 34?
• I have an SRS of third-graders
SRS?
Normal?
•Since the sample size is large, the sampling distribution
is
How do you
approximately normally distributed
know?
OR
Do you
•Since the histogram is unimodal
withs?no outliers, the
know
What are your
sampling distribution is approximately normally
hypothesis
distributed
• s is unknown
statements? Is
H0: m = 34
a key word?
where m is the true mean there
reading
Ha: m ≠ 34
ability of the district’s third-graders
35.091  34
Plug values
t
 .6467
into formula.
11.189
44
p-value = tcdf(.6467,1E99,43)=.2606(2)=.5212
Use tcdf to
calculate p-value.
a = .1
Compare your p-value to
a & make decision
Since p-value > a, I fail to reject the null
hypothesis.
Conclusion:
There is not sufficient evidence to suggest that the
true mean reading ability of the district’s third-graders
is different than the national mean of 34.
Write conclusion in
context in terms of Ha.
A type II error – We decide that the true mean
reading ability is not different from the national
What type of error could you
average when it really is different.
potentially have made with this
decision? State it in context.
What confidence level should you
use so that the results match this
hypothesis test?
90%
Compute the interval.
(32.255, 37.927)
What do you notice about the
hypothesized mean?