Download Chapter 10 Notes: Hypothesis Tests for two Population Parameters

Chapter 10 Notes: Hypothesis Tests for two Population Parameters
(Tests involving data from Two Samples)
Chapter 8: Confidence Interval Estimates for an unknown population parameter
 μ mean when population standard deviation is known
 μ mean when population standard deviation is not known
 p proportion
Chapter 9: Testing a hypothesis about an unknown population parameter
 μ mean when population standard deviation is known
 μ mean when population standard deviation is not known
 p proportion
Chapters 8 and 9 investigate one parameter for one population
Chapter 10 investigates two populations to compare how the values of a parameter in these
populations are related to each other.
Chapter 10: Testing a hypothesis about an unknown population parameter in two populations
Characteristics of this type of test:
Test of two means compares the population means for 2 populations for the same variable:
Example: Comparing the average age of male and female community college students
Test of two proportions compares population proportions for 2 populations for the same variable:
Example: Comparing proportions of high school students and college students who take math classes.
Inappropriate uses of this type of test:
 Comparing the percent of males who like action movies to the percent of females who
like romance movies (the variable is different for the two populations)
 Comparing the percent of community college students who intend to transfer to the
percent of community college students who already have degrees and are only
attending college to get additional job training (one population, two variables)
Look for Data from 2 Samples to recognize this type of hypothesis test
Examples:
a. We want to perform a hypothesis test to determine if the average length of time that boys and
girls play sports daily is different. A table is given showing data for a sample of boys and a
sample of girls.”
Is this a test of two population averages? Why or why not? Write the hypotheses.
b. We want to perform a hypothesis test to determine if the average number of children in a
family for Surf City is different from Ocean County.
For Ocean County, the average number of children in a family is 2.5.
In a sample of Surf City families, the average was 2.1 and the standard deviation was 0.3.
Is this a test of two population averages? Why or why not? Write the hypotheses.
Chapter 10: Testing a hypothesis about an unknown population parameter in two populations
Type of tests:
 Test of two proportions
 Test of two independent means – when both population standard deviations are known
 Test of two independent means – when at least one population standard deviation is not known
 Test of means for dependent (matched, paired) samples
a. A hypothesis test is performed to determine if the proportion of patients undergoing a particular
surgery who are cured is greater than 60%
b. A hypothesis test is performed to determine if the proportion of patients undergoing a particular
surgery who are cured is greater than the proportion who are cured by the current non-surgical
standard treatment.
c. A hypothesis test is performed to determine if the average time that a medication lasts is more
than 3 hours.
d. A hypothesis test is performed to determine if the average times that medication A and
medication B last are the same.
e. A hypothesis test is performed to determine if the average time to graduation is at Hudson
University is 4 years.
f.
A hypothesis test is performed to determine if the average time to graduation is longer at a public
university than at a private college.
g. A hypothesis test is performed to determine if taking an SAT training class increases SAT scores
of those who take the training, on average.
h. A hypothesis test is performed to determine if recent female college graduates are subject to pay
discrimination, earning less on average for similar work than recent male college graduates with
similar qualifications.
Examples for Chapter 10: Hypothesis Tests comparing 2 unknown population parameters
Hypothesis Tests involving data from TWO SAMPLES
Some but not all these examples will be used as class lecture examples.
Those examples with references noted for Introductory Statistics at OpenStax
or for Collaborative Statistics by Illowsky and Dean at Connexions.org can be downloaded for free at
https://openstaxcollege.org/textbooks/introductory-statistics
or http://cnx.org/content/col10522/1.21/
Types of Tests: Two population proportions
Two population means, independent samples, population standard deviation unknown
Two population means, independent samples, population standard deviation known
Two population means, dependent (matched, paired) samples
Example 10.8 in OpenStax Introductory Statistics:
Two medications for hives are being tested to determine if there is a difference in the percentage of
adult patient reactions. 20 people in a random sample of 200 adults given medication A still had hives
thirty minutes after taking the medication. 12 people in another random sample of 200 adults given
medication B still had hives thirty minutes after taking the medication. At a 1% level of significance, is
there a difference in the "non-response" rate for medication A and medication B?
Example 10.1 in OpenStax Introductory Statistics:
The average amount of time boys and girls ages 7 through 11 spend playing sports each day is believed
to be the same. Data is collected for a random sample of boys and a random sample of girls, resulting
in the data given below
Sample
Sample Average Hours Playing
Sample
Size
Sports Per Day
Standard Deviation
Girls
9
2
0.866
Boys
16
3.2
1.00
Do the data show that there is a difference in the average amount of time that boys and girls ages 7
through 11 play sports each day? Perform a hypothesis test using a 5% level of significance.
(This is Example 10.1 in the textbook; as of spring 2015 the Introductory Statistics textbook by
OpenStax shows the sample standard deviation for girls as 0.866 which is an error. The square root
symbol should not be there; the sample standard deviation for girls should be 0.866 as shown in the
table above in these notes).
Assume that the populations of times that individual boys play sports and that individual girls play
sports are approximately normally distributed.
Example 10.2 in OpenStax Introductory Statistics:
A study is done of two colleges to see if students from College A graduate with more math classes on
average than students who graduate from College B.
Sample
Sample Mean
Sample
Size
number of math classes
Standard Deviation
College A
11
4
1.5
College B
9
3.5
1.0
Do the data show that graduates from College A have more math classes on average than graduates
from College B? Perform a hypothesis test using a 1% level of significance.
Assume that the populations of numbers of math classes for individual students at each college are
approximately normally distributed.
Example 10.6 in OpenStax Introductory Statistics:
The mean lasting times of two floor waxes is to be compared.
20 floors are randomly assigned to test each wax. The data are given in the following table
Sample Mean time in months
Population
that the wax lasts
Standard Deviation
Wax 1
3
0.33
Wax 2
2.9
0.36
Do the data indicate that wax 1 is more effective than wax 2?
Perform a hypothesis test using a 5% level of significance.
Example 10.11 in OpenStax Introductory Statistics
A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for
randomly selected subjects are shown in the table. The "before" value is matched to an "after" value.
Are the sensory pain measurements, on average, lower after hypnotism?
Perform a hypothesis test using a 5% significance level.
Subject
A
B
C
D
E
F
G
H
Before
6.6
6.5
9
10.3
11.3
8.1
6.3
11.6
After
6.8
2.5
7.4
8.5
8.1
6.1
3.4
2
Example 10.12 in OpenStax Introductory Statistics
A college football coach was interested in whether the college’s strength development class increased
his players’ maximum lift (in pounds) on the bench press exercise. He asked 4 players to participate
in a study. The amount of weight they could each lift was recorded before and after they took the
strength development class. The coach wants to know if the strength development class makes his
players stronger, on average. The data are:
Weight (in pounds)
Player 1
Player 2
Player 3
Player 4
Weight lifted before the class
205
241
338
368
Weight lifted after the class
295
252
330
360
Perform a hypothesis test to determine if the strength development class makes players stronger, on
average.
Example E1: ( not in textbook) :
A local in-home health care service has ten nurse's aides in the company that visit patients' homes to
provide assistance. Under the old assignment system, appointments were scheduled on a first come
first serve basis to fill available time. The company director decides to try a new scheduling system
based on location. This table presents the number of visits before and after the new system is
implemented on randomly selected days for each aide.
Is there sufficient evidence to conclude that there is an average increase in the population number of visits
per day made by the nurse's aides using the new schedule, as compared to the old schedule?
Perform a hypothesis test using a 5% level of significance.
Number of
Visits/Day
Old Schedule
Number of
Visits/Day New
Schedule
Ana
Binh
Cyd
Dina
Ed
Fran
Greg
Hal
Ido
Juna
6
7
8
6
8
7
11
9
10
12
10
10
9
11
5
10
10
13
8
15
Example E2 (not in textbook):
In a study of 15600 patients, patients were randomly assigned to a treatment group receiving the
medication plavix or to a control group receiving aspirin. Assume that the 15, 600 patients were equally
divided between the two groups. (data source: Mercury News 3/13/2006)
In the treatment group, 6.8% suffered heart attack or stroke.
In the control group, 7.3% suffered a heart attack or stroke.
Perform a hypothesis test at the 2% level of significance to determine whether the treatment is
effective at reducing the occurrence of heart attacks and strokes.
Example E3 (not in textbook):
In 1998,the FDA approved the drug tamoxifen to prevent breast cancer in high risk women, stopping
the study earlier than planned based on the strength of the data obtained thus far.
According to data contained in an article in the San Jose Mercury News opinion column on 11/16/98,
13,175 women were randomly assigned to the treatment (tamoxifen) or control (placebo) groups. Of
the 6,576 women in the tamoxifen group, 89 developed invasive breast cancer. Of the 6,599 women
in the placebo group, 175 developed invasive breast cancer.
Perform the appropriate hypothesis test to determine whether the sample data provides sufficient
evidence that the incidence of invasive breast cancer is lower in the tamoxifen group than in the
placebo group. Use a 1% level of significance.
Example E4: ( not in textbook) “A/B” Testing:
When internet users visit websites, the companies and organizations running the websites collect data
about how customers use the website. One way in which they use this data is to test different
appearances or formats of the website to determine which gets better responses; responses can be
measured in the form of more time spent at the website, or more purchases made at the website, or
more donations made at the website, or other metrics meaningful to the company or organization. This
is called A/B testing and is commonly used in many varied contexts, such as trying to increase sales on
shopping sites, increasing advertising viewership on sites that show advertising, or trying to increase
donations to political campaigns made through candidates websites. The example below gives one
view of how A/B testing may sometimes be conducted; however the sites that conduct A/B testing
usually have very large data sets.
To determine if there is a difference in the amount of time that people spend on a social networking
website depending on the appearance of the site, the social networking company conducts a hypothesis test
to compare the average time spent at the site for samples of users seeing one of two interfaces being tested.
For 124 randomly selected users seeing interface A, the average time spent was 3 minutes with a standard
deviation of 0.85 minutes.
For 82 randomly selected users seeing interface B, the average time spent was 2.25 minutes with a standard
deviation of 1.05 minutes.
Conduct a hypothesis test of determine if there is a difference between the average times that users spend
on the site when using interface A vs when using interface B.
For this problem, assume that the populations of times spent at the site by individual users for interface
A and interface B are approximately normally distributed. (Note that if this assumption were not true,
then other statistical methods beyond the scope of Math 10 would be used to perform the testing.)
Example E5 (not in textbook):
A frozen pizza manufacturer wants to determine whether the average time needed to cook its low
fat pizza is less than for its regular pizza.
For a sample of 15 low fat pizzas, the average cooking time was 14.8 minutes with a standard
deviation of 2.3 minutes.
For a sample of 15 regular pizzas, the average cooking time was 16.1 minutes with a standard
deviation of 2.8 minutes.
All pizzas were cooked in identical ovens at the same temperature.
Can we conclude that the true average cooking time is less for low fat pizzas?
Example E6 (not in textbook):
Strain A:
Strain B:
24
9
13
13
26
27
15
12
22
15
14
21
10
9
8
28
10
18
18
15
33
23
21
25
30
12
20
18
26
29
23
7
16
8
17
10
14
14
9
15
30
18
24
16
25
28
7
6
19
15
A biologist is studying the average germination times of
two strains of seeds to determine whether the two strains
of seeds have the same mean germination time.
The number of days until germination are given for a
random sample of 25 seeds of strain A and a random
sample of 25 seeds of strain B.
All seeds are grown in identical greenhouse conditions.
Assume that the underlying populations of germination
times of individual plants is approximately normally
distributed.
At a 2% level of significance, is there sufficient
evidence of a difference in mean germination times for
the two strains of seeds?
What type of hypothesis test is appropriate for this
problem? Why?
Sample
A
B
Sample
Mean
18.96
16.44
Sample Standard
Deviation
7.35
6.98
Chapters 8, 9, 10: Summary of Intervals and Tests
Chapter 8: Confidence Intervals
Unknown Other
Random
Parameter Conditions Variable
µ
µ
 known
 NOT
known
p
X
X
Distribution
used to
calculate
Interval
Point
Estimate
N  x ,  


n

x
t n1
x
Error
Bound
Confidence Interval
Z/2   


x  Z/2   


t/2  s 


x  t/2  s 


 n
 n
P'
 n
 n
p'
N  p' , p' q' 
p'  Z/2
Z/2 p ' q '


n
n


Note: Both notations p' and p̂ are used to represent the value of the sample proportion.
Chapter 9:
Unknown
Parameter
µ
µ
Hypothesis Tests
Other
Random
Conditions Variable
 known
X
 NOT
known
p
Distribution used to
calculate pvalue
Point
Estimate
Test Statistic
N   0 ,  


x
Z = x  0
 n
X
t n1
x
P'
N  p ,
 0

t = x  0
s n
Z = p ' p 0

n
p0 q0
n
p'




p' q'
n
Calculator
Test
Z-Test
T-Test
1PropZTest
p0 q0
n
Note: Both notations p' and p̂ are used to represent the value of the sample proportion.
Note: Symbols µ0, and p0 represent the numerical values in the null hypothesis
Chapter 10: Hypothesis Tests Comparing Two Population Parameters (using 2 sets of sample data)
Unknown Other Conditions
Random
**Distribution used to
Point
Calculator
Parameter
Variable
calculate pvalue
Estimate
Test
Independent Samples
2SampZTest
2
2 

µ1  µ2 1 AND 2 both known X 1 X 2 N  0,  1   2 
x 1 x 2


Independent Samples
µ1  µ2
µd
p 1 p 2
1 OR 2 NOT known
Dependent, Matched,
Paired Samples
Samples must be
independent.
(Matched samples for
proportions require
other tests not covered
in Math 10)
X 1 X
X
n1
n2 
t with df as given by
your calculator
2
d
P1  P2
t n1
xd



N  0, pˆ C qˆ C  1  1  




2SampTTest
x 1 x 2
 n1
n2  

p1  p2
TTest
2PropZTest
where
x1  x 2 n1 pˆ 1  n2 pˆ 2

n1  n2
n1  n2
Note: Both notations p' and p̂ are used to represent sample proportions.
** These distributions are based on the assumption that in Math 10 we are testing whether the difference
in parameters for the two populations is 0 (i.e. whether the parameters are equal).
pˆ C 