Statistics and Risk
Management
Confidence Intervals
Performance Objective:
After completing this lesson, the student will understand the concepts of normal
data distributions and their application in quality control environments.
Approximate Time:
When taught as written, this lesson should take 8-10 days to complete.
Specific Objectives:
 The student will discuss the importance of data sampling.
 The student will understand some basic terms and concepts of data
distributions.
 The student will understand the concept of using the Z-Scores scale.
 The student will be able to apply the above concepts to quality control
sampling.
This lesson corresponds with Unit 5 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
Confidence Intervals. Any changes/alterations to the activities may result
in the elimination of any or all of the TEKS listed.
130.169 (g)(6)(H)
…construct and interpret a confidence interval estimate for a single
population mean using standard normal distribution;
130.169 (g)(6)(I)
…establish and interpret a confidence interval estimate for a single
population proportion;
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…
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…
Copyright © Texas Education Agency, 2012. All rights reserved. 2 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/)
Business Continuity Planner 13-1199.04
Similar Job Titles: Business Continuity Management Director, Business
Continuity Planning Director, Business Continuity Strategy Director, IT
Disaster Recovery Manager
Tasks:



Develop disaster recovery plans for physical locations with critical assets
such as data centers.
Test documented disaster recovery strategies and plans.
Analyze impact on, and risk to, essential business functions or
information systems to identify acceptable recovery time periods and
resource requirements
(Soft) Skills:
Oral Expression; Written Comprehension; Deductive Reasoning; Inductive
Reasoning
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:
Normal Distribution
Tutorial of Histograms, Scatter Plots, Central Tendency, Standard Deviation,
and Confidence
Intervals. Previous lessons of Correlation and future lessons on Z Scores
available as well.
http://www.gla.ac.uk/sums/users/lhornibrook/Sensor_Comparisons/normdist1.ht
ml
Basic Statistics Home Page
Mean, median, mode, Standard Deviation, Normal Curve, Comparing Means,
the t-Test, Statistical Significance, the Null Hypothesis, Correlation and
Causation are some of the relevant terms.
http://www.fgse.nova.edu/edl/secure/stats/index.htm
Through the experiment provided on this site teachers can have students
collect data in the field and analyze it to discover the importance and power
statistical sampling has in searching for information. The site provides a through
description of each part of the activity and sample charts that the teacher can
duplicate for the students to use in recording and analyzing their data.
http://slincs.coe.utk.edu/gtelab/learning_activities/33botg.html
Measures of Shape: Skewness and Kurtosis
This site defines the processes of both skewness and kurtosis, breaks down the
concepts behind both of them, gives examples of each, and explains when they
are important to determining the measures of shapes.
http://www.tc3.edu/instruct/sbrown/stat/shape.htm
Copyright © Texas Education Agency, 2012. All rights reserved. 5 Teacher Preparation:
Teacher will:
1.
presentation, and
handouts.
2.
resources and
websites.
3.
websites ready.
Review terms in outline,
Locate and evaluate various
Have assignments and
Learner Preparation:
Break the boring barrier. Probability can be fun and definitely
interesting. 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/Confidence%20Intervals.mov
STUDENTS will take the practice test and review using the Key,
found in Common/Student Documents.
EXHIBIT:
Excitement for Confidence Intervals and Learning
INTRODUCE: The idea that quality control is important, and
using confidence interval techniques is integral to
quality control.
ASK:
Ask students to express how they feel about the
quality of products they buy.
Copyright © Texas Education Agency, 2012. All rights reserved. 6 I. Random Sample
A. Sometimes we need to
understand the
characteristics of the
larger picture.
B. We may take a sample
set of data that may infer
what a larger population
is, will think, do, or say.
II. Assumptions
A. When we collect sample
data it can be assumed
that the data is in the form
of a normal distribution.
B. The larger the sample the
more likely is will be a
normal distribution.
C. It will be Symmetric and
Uni-modal and often
referred to as “Bell
Shaped”.
III. Normal Distribution
IV. Skewed Right
V. Skewed Left
VI. Bimodal
VII. Kurtosis
VIII. Outliers
A. Some data scores do not
fit the Normal Distribution.
B. Trimmed Sample both
ends to eliminate
extremes
C. Winsorized Sample
replace the trimmed
scores with the closest
normal scores.
Use presentation
ConfidenceIntervals_
Distributions.pptx.
Provided .docx files
4.1a
ConfidenceIntervals_
Distributions.docx Provide Assignment sheets and discuss
and answer any questions about
assignment (In class or take homeInstructor’s Option)
Copyright © Texas Education Agency, 2012. All rights reserved. 7 IX. Standardization
A. A “Standard” Normal
Distribution has a Mean of
ZERO and a Standard
Deviation of 1.0.
B. Visualize the
Standard Normal
Distribution
X. The z Score
a. Is based upon Standard
Deviation divisions.
b. Gives you a Standardized
way to measure the distance
left(-) or right(+) of the Mean
value.
c. Works regardless of the
Value of the Mean because
we make the mean ZERO
and the Standard Deviation
1.0
XI. Related PERCENTILES
XII. Related T-SCORES
XIII. Formula for a Z-Score
XIV.
Use presentation
ConfidenceIntervals_
ZScore.pptx
XV. Application
a. We have a 100 test scores
the Mean is 79 and the
standard deviation is 4.
b. If you earned a score of 85
where do you sit within the
distribution?
c. z = (85 - 79) / 4 = 6 / 4 = 1.5
d. You are sitting around the
94th percentile…very good.
XVI. Formula for a Z-Score
Application
We have a 100 test scores the Mean is
79 and the standard deviation is 4.
What score do you need to earn a 90%
to get that “A”?
X = (1.2 x 4) + 79 = 83.8
You need to earn a score of 83.8 or above
for the “A”.
Copyright © Texas Education Agency, 2012. All rights reserved. 8 Provided .docx files
Provide Assignment sheets and discuss and answer any questions about assignment (In class or take home‐
Instructor’s Option) XV.
Critical Values
A. If an obtained value is
more extreme than the
critical value you reject.
B. 5% is a common CI C. 1% CI for drugs D. What would you expect a CI for a heart pacemaker be? XVI. Confidence Intervals XVII. Quality Control A. Suppose we are manufacturing a product and are examining what is our tolerance for defects. B. Example C. Our plant fills up tubs of margarine. We would like the tubs to be filled to 250 grams of product. We will reject tubs filled with less than 247.5 grams or more than 252.5 grams. XVIII. Quality Control XIX. Let’s try Quality Control A. We took a sample of 100 tubs out of a days production of 5000 tubs B. We found a Mean of 251 grams and a Standard Deviation of 2.0 grams. XX. Our 100 Sample of Tubs Provide Assignment sheets and discuss
4.2a
ConfidenceIntervals_
ZScore.docx
Use presentation
ConfidenceIntervals_
Quality.pptx.
Provided .docx files
4.3a
ConfidenceIntervals_
Quality.docx
4.3b
ConfidenceIntervals_
Quality.docx
Copyright © Texas Education Agency, 2012. All rights reserved. 9 Guided Practice:
See teaching outline.
Independent Practice:
See teaching outline.
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. 10 Student Assignment
5.1a Confidence Intervals Distribution
Key
Using a search engine find three examples (images) of various
charts exhibiting various distributions. Explain what the chart is
showing. Identify its characteristics: Normal, Standardized,
Kurtosis, Uni or Bi modal, Skewed, etc.
Find a chart that you believe demonstrates outliers. Be prepared
to explain the chart.
Answers will vary.
Copyright © Texas Education Agency, 2012. All rights reserved. 11 Student Assignment
5.2a Confidence Intervals Z-Score
Key
Assume the above chart is a normal standardized distribution of 100
scores on an exam you took. The exam was worth 200 points. The
mean score for this exam was 125 with a standard deviation of 20.
Your Z-Score was +2.0. What was your point score out of 200
points?
Answer: ________________
ANSWER: 165
Out of the 100 students estimate how many student scored below
you?
Answer: ________________
ANSWER: 70
What is the percentage of the questions you answered correctly?
Answer: ________________
ANSWER: 83%
Approximately, what is the highest exam score earned?
Answer: ________________
or 99.9 percent)
ANSWER: 185 (Z-score of 3.0
Copyright © Texas Education Agency, 2012. All rights reserved. 12 Student Assignment
5.3a Confidence Intervals Quality – Finding
Critical Value
Key
ANSWER = 50 units – (.25+ .25) x 1000
Copyright © Texas Education Agency, 2012. All rights reserved. 13 Student Assignment
5.3b Confidence Intervals Quality – Fast Food
Key
McDonalds fills millions a cup of soda each year. The 22 ounce cup must
be filled to 21 ounce plus or minus 1 ounce with a confidence interval of
95%. The fountain is automated to first fill the glass ¾ full and then 5
seconds later, it will finish topping it off to the specifications. If an employee
has to spent time putting in more soda, customer service will slow and labor
costs will increase. Feeling that the machine was not working correctly, the
store manager ran a test sample. Using a marked test cup, he filled it 10
times and recorded the number of ounces for each fill.
He calculated a Mean of 20.5 ounces and a Standard Deviation of .75
ounces. Using the Z-score of +- 2.0 as the confidence interval points, was
the test acceptable ?
Answer: ____________
NO, there were cups at 19 ounces.
If the SD was .5 ounces would the test be Acceptable?
__________
Answer:
YES if one rounds up the 19.5 To 20 ounces.
Copyright © Texas Education Agency, 2012. All rights reserved. 14 Name:_________________________________ Date:______________________ CLASS:_______________ UNIT 5 CONFIDENCE INTERVALS TEST TRUE and FALSE: 1. The smaller the sample the more likely it will be a normal distribution. A. True B. False 2. A sample is a subset of a population. A. True B. False 3. The z Score is based upon Standard Deviation divisions. A. True B. False 4. When you collect sample data, it should not be assumed that the data is in the form of a normal distribution. A. True B. False 5. If an obtained value is more extreme than the critical value, you reject. A. True B. False 6. All data scores fit the normal distribution. A. True B. False MATCHING: A.
B.
C.
D.
E.
Confidence Interval Z‐Score Normal Distribution Frequency Distribution Critical Value 7. __________ A uni‐modal symmetrical bell style sample distribution. 8. __________ A devised scale with intervals coinciding with the standard deviations points that identifies a point location along a normal distribution. 9. __________ The point in which we reject. 10. __________ The level on which we feel confident that we can accept. 11. __________ The number of times a value of a variable occurs in a data set. MULTIPLE CHOICE: 12. The _________ is used to describe the curve height to width of a bell type distribution. A. Z‐score B. Outliers C. Kurtosis D. Critical Value 13. If the bulk of the data is to the left and the right tail is longer, we say the data is: A. Negatively Skewed B. Positively Skewed C. Bell Shaped D. Normally Distributed 14. To express a confidence interval you need ______ pieces of information. A. 1 B. 2 C. 3 D. 4 15. If the bulk of the data is to the right and the left tail is longer, we say the data is: A. Normally Distributed B. Positively Skewed C. Bell Shaped D. Negatively Skewed Copyright © Texas Education Agency, 2012. All rights reserved. 15 Name:_________________________________ Date:______________________ CLASS:_______________ 16. The larger the sample, the more likely it will be a _____ distribution. A. Skewed left B. Skewed Right C. Normal D. Controlled 17. The _____ gives you a standardized way to measure the distance left or right of the mean value. A. Kurtosis B. Outliers C. Critical Value D. Z‐Score 18. The number of units in a population is a _________. A. Random Sample B. Sample C. Kurtosis D. Sample Size 19. A _____ distribution is a distribution with 2 normal distributions. A. Unimodal B. Bimodal C. Trimodal D. None of the Above 20. This occurs when one tail of the distribution curve is longer than the other. A. Normal Distribution B. Bell Distribution C. Skewed Distribution D. Standard Distribution 21. A “____________” Normal Distribution has a Mean of 0 and a Standard Deviation of 1.0. A. Skewed Left B. Standard C. Skewed Right D. Double Bell Curve SHORT ANSWER: Use the following for questions 22‐24. Assume the above chart is a normal standardized distribution of 100 scores on an exam
you took.
The exam was worth 200 points. The mean score for this exam was 125 with a
standard deviation of 20.
22. Your Z‐Score was +2.0. What was your point score out of 200 points? Copyright © Texas Education Agency, 2012. All rights reserved. 16 Name:_________________________________ Date:______________________ CLASS:_______________ 23. Out of the 100 students estimate how many student scored below you. 24. What is the percentage of the questions you answered correctly? 25. We have 100 test scores. The mean of the test scores is 80 and the standard deviation is 4. If you earned an 87, where do you sit within the distribution? (SHOW WORK FOR CREDIT!) Copyright © Texas Education Agency, 2012. All rights reserved. 17 UNIT 5 CONFIDENCE INTERVALS TEST ANSWER
KEY
1. B
2. A
3. A
4. B
5. A
6. B
7. C
8. B
9. E
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
A
D
C
B
C
D
C
D
D
A
B
C
165
70
83%
1.75 or about 96%
Copyright © Texas Education Agency, 2012. All rights reserved. 18