Statistics and Risk
Management
Hypothesis Testing
Performance Objective:
After completing this lesson, the student will understand the concepts of what
hypotheses are, how to state research and null hypotheses, and how to apply
statistics to prove or disprove them.
Approximate Time:
When taught as written, this lesson should take 20-25 days to complete.
Specific Objectives:





The student will discuss what Hypotheses are.
The student will understand how to create Research Hypotheses.
The student will understand how to create Null Hypotheses.
The student will able to apply the Chi Test to a data sample.
The student will able to apply the various t-tests to data samples
This lesson corresponds with Unit 7 of the Statistics and Risk
Management Scope and Sequence. Copyright © Texas Education Agency, 2012. All rights reserved. 1 TEKS Correlations:
This lesson, as published, correlates to the following TEKS for
Hypothesis Testing. Any changes/alterations to the activities may result
in the elimination of any or all of the TEKS listed.
130.169(g)(6)(J)
…carry out an appropriate hypothesis test on a single population mean or
proportion…
130.169(g)(6)(K)
…interpret the p-value of the test statistic…
130.169(g)(6)(L)
…construct a contingency table and perform a chi-squared test of
independence…
InterdisciplinaryTEKS:
English:
110.31 (C) (21) (B)
… organize information gathered from multiple sources to create a
variety of graphics and forms (e.g., notes, learning logs)…
110.31 (C) (22) (B)
…evaluate the relevance of information to the topic and determine the
reliability, validity, and accuracy of sources (including Internet sources)
by examining their authority and objectivity…
110.31 (C) (23) (C)
… use graphics and illustrations to help explain concepts where
appropriate…
Copyright © Texas Education Agency, 2012. All rights reserved. 2 110.31 (C) (23) (D)
… use a variety of evaluative tools (e.g., self-made
rubrics, peer reviews, teacher and expert evaluations)
to examine the quality of the research…
Math:
111.36 (C) (4) (A)
… compare theoretical and empirical probability;
111.37. (C) (3) (B)
… use probabilities to make and justify decisions about risks in everyday
life
Occupational Correlation
(O*Net - http://www.onetonline.org/)
Electrical Engineer 17-2071.00
Similar Job Titles: Electrical Design Engineer, Project Engineer, Electrical
Controls Engineer, Test Engineer, Hardware Design Engineer
Tasks:



Prepare technical drawings, specifications of electrical systems, or
topographical maps to ensure that installation and operations conform to
standards and customer requirements.
Operate computer-assisted engineering or design software or equipment
to perform engineering tasks.
Confer with engineers, customers, or others to discuss existing or
potential engineering projects or products.
(Soft) Skills:
Critical Thinking; Reading Comprehension; Active Listening; Complex Problem
Solving
Copyright © Texas Education Agency, 2012. All rights reserved. 3 Instructional Aids:
1. Display for presentation, websites for assignments
and class discussion
2. Assignment Worksheets
3. Supporting Spreadsheets
Materials Needed:
1. Printer paper
2. Assignments and website information ready to distribute to
students.
Student projects will be displayed to increase interest in Statistics
Equipment Needed:
1. Computer with presentation and Internet Access
2. Computers for Students to Conduct Research and Collect Data
for Projects
Copyright © Texas Education Agency, 2012. All rights reserved. 4 References:
Understanding Hypotheses
This article explores the concepts of creaking and
testing hypothesis. There is no explicit math content,
but links to possible mathematical activities are
provided at the bottom of the article.
http://nrich.maths.org/6178
Chi Squared Statistic
Equations with answers for types of data, contingency table, goodness of fit,
and test of independence.
http://math.hws.edu/javamath/ryan/ChiSquare.html
Type 1 Errors
This three and a half minute video from Khan Academy explains all about Type
1 Errors in hypothesis testing on a level easy for students to follow and
comprehend.
http://www.khanacademy.org/math/statistics/v/type-1-errors
Understanding Hypothesis Testing
Definition, rational decisions, effects, general principles, the model, and
probability.
http://www.psychstat.missouristate.edu/introbook/sbk18.htm
The T-Test
Learn all about the t-test from its definitions to examples in this article from
Research Methods Knowledge Base. With easy to follow sample graphs this
article really helps visual learners who need a picture along with text to
understand a concept.
http://www.socialresearchmethods.net/kb/stat_t.php
Copyright © Texas Education Agency, 2012. All rights reserved. 5 Teacher Preparation:
Teacher will:
1.
Review terms in outline,
presentation, and handouts.
2.
Locate and evaluate various
resources and websites.
3.
Have assignments and
websites ready.
Learner Preparation:
Find examples the student might find interesting; understanding
gaming, designing games, evaluating decision on an ongoing
basis.
Introduction:
STUDENTS will watch the Unit video found here:
jukebox.esc13.net/untdeveloper/Videos/Hypothesis%20Testing.
mov
STUDENTS will take the practice test and review using the Key,
found in Common/Student Documents.
EXHIBIT:
Excitement for using moving opinions to a
testable result.
INTRODUCE:
Testing ideas is what education is all about.
ASK:
Ask students to express how they feel about
some important ideas that have been tested
or need testing.
Copyright © Texas Education Agency, 2012. All rights reserved. 6 I.
What is a Hypothesis?
A. A hypothesis is a tentative
statement about the relationship
between two or more variables.
1. Examples:
As gas prices rise,
consumption of gas will
drop.
Women prefer driving a SUV
over a Pickup Truck.
Bonds provide a better
“Return on Investment”
(ROI) over Stocks.
II. What is a Research Hypothesis?
A. A Research Hypothesis is a testable
statement about there being a
significant difference in a
relationship between two or more
variables.
III. Common Formulas?
A. Research Hypothesis (SUV/PU)
Use ResearchHypothesis.pptx.
B. Research Hypothesis
(Bonds/Stocks)
C. An opinion may not be
supported with facts and may be
difficult to test.
1. Republicans make better
Presidents than
Democrats.
IV. A hypothesis is testable.
A.
In modern times, more
Republicans are elected as
President during recessionary
periods.
B. The reason students drop out
of high school is because they
are failing anyway.
C.
Government Employees are
lazy.
D. Because of the recession,
University enrollment has
plummeted in the last ten years.
E. Student Obesity has been on
the increase for the last 10 years.
Copyright © Texas Education Agency, 2012. All rights reserved. 7 Provide Assignment sheets and discuss
and answer any questions about
assignment (In class or take homeInstructor’s Option)
Provided .docx files
6.1a
HypothesisTesting_
ResearchHypothesis.docx
V. The Null-Hypothesis
A. To prove a Research Hypothesis
as true you must disprove its
corresponding Null-Hypothesis.
B. Why….It is easier to (for sure)
disprove something verses proving
it is true all the time.
C. However….you are not really
proving a hypothesis as true….just
not false…so far.
D. What is a Null-Hypothesis?
E. As gas prices rise consumption of
gas will not significantly drop.
F. Women do not significantly prefer
driving a SUV over a Pickup Truck.
G. Bonds do not significantly provide a
better “Return on Investment” (ROI)
over Stocks.
VI. Common Formulas?
A. Null-Hypothesis (SUV/PU)
Use TheNullHypothesis.pptx.
C.
How would we test this?
Ask randomly 100 adult women
if they prefer driving an SUV or a
Pickup.
E. Analyze the results to see if any
difference is significant for a
sample size of 100 respondents.
F. Can we make Errors?
G. Create a Null-Hypothesis!
H. The reason students drop out of
high school is because they are
failing anyway.
I. Create a Null-Hypothesis!
Null-Hypothesis (Bonds/Stocks)
D.
B.
Copyright © Texas Education Agency, 2012. All rights reserved. 8 VI. Comparison Within
A. We need a way to measure the
difference between obtained
results and what would occur
randomly.
J. Government Employees are lazy.
K. Create a Null-Hypothesis!
L. Because of the recession,
University enrollment has
plummeted in the last ten years.
M. Create a Null-Hypothesis!
N. Student Obesity has been on
the increase for the last 10 years.
O.Provide Assignment sheets and
discuss and answer any questions
about assignment (In class or take
home-Instructor’s Option)
Provided .docx files
6.2a
TheNullHypothesis.docx
B. For Example: At the Mall of America
we asked everyone their age as
they entered the store for one hour
on Saturday. We discarded any
ages under 16 or over 65 (trimmed).
We collected a sample of 100
useable responses. We grouped
them in 10 year blocks.
C. Comparison Within
D. Our null-hypothesis is that we will
get an equal number of people in
each of the five age groups.
H0: P1 = P2 = P3 = P4 = P5
E. Comparison Within
Use HypothesisTesting_
Within_
OneSetOfNumbers.pptx
I. What does that mean?
F. What does this mean?
G. We found that the Chi-Squared
value of 9.80 is larger than the
Critical Value of 9.49
H. So we will reject the Null-Hypothesis
• Because there is a
significant difference
between the obtained results
and what would be expected
as a result of just random
responses.
Notice how close that was!
That there is a significant difference of
Copyright © Texas Education Agency, 2012. All rights reserved. 9 ages in the people we
interviewed beyond what we
would expect (Probability).
J.Chi-Square Formula?
K. Excel Formula
VII. The t Test
A. The t test is used to see if there
is a significant difference
between two groups of data.
This function uses two lists of
Expected (column B) and Observed
(Column C)
Provide Assignment sheets and discuss
and answer any questions about
assignment (In class or take homeInstructor’s Option)
=CHITEST(B1:B5,C1:C5)
B. There are different variations of the
t test for different type of data sets
comparisons.
C. The t Test
D. The t test is a method of comparing
the Means between the groups and
seeing if there is a significant
difference. The Standard
Deviations are an essential part of
it.
E. Sample Vs. Population
F. Example you have the means and
standard deviation of SAT scores
for the whole country (USA).
G. You want to see if you high school’s
SAT scores were significantly
different
H. What is the Null-Hypothesis?
I. Sample Vs. Population
J.Sample Vs. Population
K. Population
• Mean
• Standard Deviation
L. Sample
Provided .docx files
6.3a
HypothesisTesting_
Within_
OneSetOfNumbers.docx
Use HypothesisTesting_
Between_
MultipleSetOfNumbers.pptx.
Copyright © Texas Education Agency, 2012. All rights reserved. 10 M. Standard Deviation
Is there a significant difference?
VIII.
Hypothesis Testing Between
Multiple Sets of Numbers
A. Statistics & Risk Management
B. Samples Same/Related Subjects
C. You are looking to see if there is a
significant difference with the same
study group over a period of time
(like a before and after fix)
D. You are looking to see if there is a
significant difference between a
group of related subject like a
husband and wife or a set of twins.
E. Samples Same/Related Subjects
F. Samples Same Subjects
G. You have a group of 100 Students.
H. You test for their IQ score before
they start college.
I. When they have finished college you
retest their IQ scores.
J.What is the Null-Hypothesis?
K. Samples Related Subjects
L. You have 50 husbands & their
wives.
M. You survey them both at the same
time for their support of public nonsmoking bans.
N. What is the Null-Hypothesis?
O. Hypothesis Testing Between
Multiple Sets of Numbers
P. Statistics & Risk Management
Q. Equal Independent Samples
R. You have two groups (Chicago &
LA) of 100 randomly drawn people.
S. You survey them on who primarily
provides for their healthcare.
• Choices: Government,
Employer, University,
Parents, Self
T. What is the Null-Hypothesis?
U. Unequal Independent Samples
Copyright © Texas Education Agency, 2012. All rights reserved. 11 A. It works like a t test but uses a
GRID or an ARRAY approach
for data analysis.
V. You have two groups (Chicago 100
subjects & LA 50 subjects) of
randomly drawn people.
W. You survey them on who primarily
provides for their healthcare.
• Choices: Government,
Employer, University,
Parents, Self
X. What is the Null-Hypothesis?
Y. Hypothesis Testing Between
Multiple Sets of Numbers
Z. Statistics & Risk Management
AA. More than two sets
IX. The ANOVA (Analysis of Variance)
B. It can identify if there is a significant
difference between the groups of
data, but NOT which one(s) are
different.
C. Just know the name and what it is
used for.
D. Multi-Variant Samples
E. You have a group of 100 randomly
drawn people.
F. You survey them on their Income
Level and who primarily provides
for their Healthcare Coverage.
• Choices: Under $20k, $20$50k, over $50k
• Choices: Government,
Employer, University,
Parents, Self
Provide Assignment sheets and
discuss and answer any questions
about assignment (In class or take
home-Instructor’s Option)
Provided .docx files
6.4a
HypothesisTesting_
Between_
MultipleSetsOfNumbers.docx
Copyright © Texas Education Agency, 2012. All rights reserved. 12 Guided Practice:
See assignments.
Independent Practice:
See assignments.
Review:
Question:
Describe what is meant by a normal distribution?
Question:
What are some main uses of confidence intervals? Informal Assessment:
Instructor should observe student discussion and monitor interaction.
Formal Assessment:
Completion of provided assignments using included rubrics for grading.
Copyright © Texas Education Agency, 2012. All rights reserved. 13 Student Assignment
6.1a Hypothesis Testing – Research
Hypothesis Key
Name: ____________________
Your father has told you that you will get better gas mileage when you
buy premium gasoline for your car. Create a Research Hypothesis
statement that fits with what your father has said.
ANSWER: Burning Premium gasoline in your automobile will increase
its fuel mileage.
TV News has reported that the state crime rates have dropped over last
year. Create a Research Hypothesis statement that fits with what TV
News has reported.
ANSWER: There are less reported crimes this year over last year.
Copyright © Texas Education Agency, 2012. All rights reserved. 14 Student Assignment
6.2a Hypothesis Testing – Null
Hypothesis Key
Name: ____________________
You tend to see more “red” Corvettes driving around. You believe that
Chevrolet sells more red Corvettes than any other color. Create a Nullhypothesis statement that can be used to test your theory.
ANSWER: Total Sales of red Corvettes are NOT significantly different in
numbers from other colors of Corvettes sold.
You think that men do better in math courses than women. Create a
Null-hypothesis statement that can be used to test your theory.
ANSWER: There is NO significant difference between male and female
performances in math courses.
Copyright © Texas Education Agency, 2012. All rights reserved. 15 Student Activity
6.3a Hypothesis Testing – Within
Single Sets Key
Name: _____________________
#1
#2
#3
#4
#5
#6
43
35
29
50
43
40
We asked 240 people which of the 5 Potential Candidates they preferred. Recorded to the left
are their responses.
All things being equal what would be the expected responses per Potential Candidate?
Answer: 40 expected response per candidate or 240/6
What is the degree of Freedom value?
Answer: 5 degrees of freedom (6-1)
What is the Critical Value?
Answer: 11 for 5 df
Using the Chi-Test determine if there is a significant difference in preference per Candidate?
Answer: the Chi-Test value was 6.6 which is not even close to 11 cv. Thus the responses are
not significantly different from what was expected.
Copyright © Texas Education Agency, 2012. All rights reserved. 16 Student Activity
6.4a Hypothesis Testing – Multiple
Sets
Name: _____________________
The state reports that the average score for all high school
students who take this certain exam is 437 points.
Your classroom took the same exam and the 20 students earned the
following scores.
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
375
402
450
425
390
390
425
400
412
380
#11
#12
#13
#14
#15
#16
#17
#18
#19
#20
358
452
385
380
390
402
501
298
405
401
Was there a significant difference with your class scores and those
earned across the state? Show your work.
Answer: Using the spreadsheet 6ss_HypothesisTesting.xlsx and the Students t-test tab, One
Sample Compared to a Population; we found a T-value of 4.09, df=19, CV=1.73. Thus there was
a significant difference in scores
Copyright © Texas Education Agency, 2012. All rights reserved. 17 Name:_________________________ Date:__________________ HYPOTHESIS TESTING Class:___________ MATCHING a. Hypothesis b. Research Hypothesis c. Null Hypothesis d. Type 1 Error e. Type 2 Error f. Chi‐Square g. Degree of Freedom h. T‐Test __________ 1. Rejected the Null Hypothesis when it was TRUE __________ 2. Rejected the Null Hypothesis when it was FALSE __________ 3. Tentative statement about the relationship between variables __________4. The reverse or negative hypothesis to a corresponding research hypothesis __________5. Use to identify a significant difference between two groups of scores __________6. Test for significant difference in scores against the random expected results __________7. A testable hypothesis __________8. Number of scores or groups LESS ONE TRUE/FALSE ___________ 9. A hypothesis is merely an opinion. ___________10. A hypothesis must be testable. ___________11. “Bonds do not significantly provide a better Return on Investment over stocks” could be a hypothesis statement. ___________12. “Republicans make better Presidents than Democrats” could be a hypothesis statement. ___________13. The larger the sample, the more likely it will be a normal distribution ___________14. Sample evidence can prove that a null hypothesis is true. Copyright © Texas Education Agency, 2012. All rights reserved. 18 Name:_________________________ Date:__________________ MULTIPLE CHOICE _____15. What variable does the experimental group receive? a. None b. Controlled c. Dependent d. Independent _____16. What is the name of the “normal” group in an experiment? a. Experimental b. Normal c. Control d. Independent _____17. Which variable is measured by the experimenter? a. Experimental b. Normal c. Dependent d. Independent _____18. How many groups are needed in an experiment? a. One b. It doesn’t matter c. Three d. At least two _____19. What is the order of steps in an experiment? a.
b.
c.
d.
Gather Data, Test Data, Hypothesize, Conclude Research, Gather Data, Test Data, Conclude There is no order Test Data, Gather Data, Research, Conclude Copyright © Texas Education Agency, 2012. All rights reserved. 19 Class:___________ Name:_________________________ Date:__________________ Class:___________ MATCHING a. Normal Distribution b. Skewed c. Kurtosis d. Outliers e. Z‐score f. Critical Value g. Confidence Interval __________20. The point in which we reject __________21. A uni‐modal symmetrical bell style sample distribution __________22. Data collected that would be considered extreme values compared to the rest of the data collected __________23. Those who consider probability to be the relative frequency “in the long run” of outcomes __________24. A devised scale with intervals coinciding with the standard deviations points that identifies a point location along a normal distribution __________25. Used to describe the curve height to width of a bell type distribution curve __________26. The level on which we feel confident that we can accept APPLICATION Write the null and alternative hypotheses you would use to test QUESTIONS #27 AND #28. 27. A politician is concerned about the number of people who express disapproval of him. He hopes to keep the negative opinion of him below 40%. They will use your hypothesis to assess the effectiveness of his ads. 28. According to the U.S. Census Bureau, 11.8% of registered births in the U.S. in 2000 were to teenage mothers. A sociologist believes that this percentage has decreased since then. 29. A survey shows that the proportion of students who own their own cars is higher than it used to be and finds a P value of .019. Is it reasonable to conclude from this that more students own cars? 30. A teacher believes that students whose first language in not English will score worse on the verbal portion of the SAT exam than students whose first language is English. If, according to the College Board, the mean of the students whose first language is English is 515. Suppose a random sample of 20 students whose first language was NOT English results in a sample means SAT score of 458. If SAT verbal scores are normally distributed with a population standard deviation of 112. Why would it be necessary for the SAT verbal scores to be normally distributed in order to test the hypotheses? Use the P‐value approach at the p=.10 level of significance to determine if there is evidence to support the teacher’s belief. Copyright © Texas Education Agency, 2012. All rights reserved. 20 Name:_________________________ Date:__________________ Class:___________ 31. A company’s past antihistamine formula provided relief for 70% of the people who used it. The company tests the new formula to see if it is better than the past formula and gets a P‐value of 0.27. Is it reasonable to conclude that the new formula and the old one are equally effective? Explain your answer. 32. A drug has been tested and found to give relief to 75% of the patients in a large clinical trial. A committee wants to see if the new, improved version works better. What would the null hypothesis be? After testing the new drug, the P‐value is 0.00001. What would you conclude from the results? Copyright © Texas Education Agency, 2012. All rights reserved. 21 Hypothesis Testing Test Key
1. D
2. E
3. A
4. C
5. H
6. F
7. B
8. G
9. F
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
T
T
F
T
F
d
c
c
d
c
f
a
d
b
e
c
g
29.
Yes
27.
28.
Ho: p =.40 HA:p<.40
Ho: p =.118 HA:p<.118
Copyright © Texas Education Agency, 2012. All rights reserved. 22