This document provides an outline and overview of Chapter 9 from a statistics textbook. The chapter covers hypothesis testing for single populations, including:
- Establishing null and alternative hypotheses
- Understanding Type I and Type II errors
- Testing hypotheses about single population means when the standard deviation is known or unknown
- Testing hypotheses about single population proportions and variances
- Solving for Type II errors
The chapter teaches students how to implement the HTAB (Hypothesis, Test Statistic, Accept/Reject regions, Boundaries, Conclusion) system to scientifically test hypotheses using statistical techniques like z-tests and t-tests. Key concepts covered include one-tailed and two-tailed tests, critical values, p
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Chapter 10: Correlation and Regression
10.2: Regression
Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests
Chapter Topic:
Hypothesis Testing Methodology
Z Test for the Mean ( Known)
p-Value Approach to Hypothesis Testing
Connection to Confidence Interval Estimation
One-Tail Tests
t Test for the Mean ( Unknown)
Z Test for the Proportion
Potential Hypothesis-Testing Pitfalls and Ethical Issues
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
I’m a young Pakistani Blogger, Academic Writer, Freelancer, Quaidian & MPhil Scholar, Quote Lover, Co-Founder at Essar Student Fund & Blueprism Academia, belonging from Mehdiabad, Skardu, Gilgit Baltistan, Pakistan.
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Test of significance (t-test, proportion test, chi-square test)Ramnath Takiar
The presentation discusses the concept of test of significance including the test of significance examples of t-test, proportion test and chi-square test.
Answer questions Minimum 100 words each and reference (questions.docxamrit47
Answer questions Minimum 100 words each and reference (questions #1-2) KEEP questions WITH ANSWER
1. A key point to get out of this topic is the idea that these errors are theoretical. You won't be sure as to whether one occurred or not. Why are they theoretical in nature? Hint: think about a study and knowing the "truth"
2. Pick a study of interest and identify the null and alternative hypothesis. How does this fit in with regards to the topic of a type I and type II error? Always keep this in mind when you are trying to identify what a type I and type II error are.
A minimum of 75 words each question and References (IF NEEDED)(Response #1 – 7) KEEP RESPONSE WITH ANSWER
Make sure the Responses includes the Following: (a) an understanding of the weekly content as supported by a scholarly resource, (b) the provision of a probing question. (c) stay on topic
1) According to the reading, we set the alpha which is the largest probability for type I error. To increase the power of a hypothesis researchers can use larger samples which provides more information and raise the significance level which increases the probability that the hypothesis will be rejected.
2) A Type 1 error occurs when individuals involved in research make the decision to reject the belief of truth when in actuality the hypothesis is true. Type 1 errors are errors in research when the researcher makes the wrong decision to reject a true null hypothesis. Type II errors are considered less of a problem than Type 1 errors, but can prove to be detrimental in the field of medicine. This type of error occurs when researchers decide to keep a false null hypothesis, when in fact the hypothesis is true. The method to avoid making Type 1 decisions is to test the null hypothesis at the highest level (Alpha Level). This will lessen the possibilities of making this type of error (Privitera, 2018).
3) According to Privitera (2018) a type 1 error is the probability of rejecting a null hypotheses that is actually true, researchers purposely make this error. A null hypotheses is a statement about a population parameter that is assumed to be true, this hypotheses is a starting point (Privitera, 2018). The type 2 error or beta error is the probability of retaining a null hypotheses that is actually false (Privitera, 2018). The type 1 error is committed when a researcher decides to reject previous notions of truth that are in fact true (Privitera, 2018). The best way to avoid these types of errors is to be open minded and not reject notions if there is fact to back the notions up. In my opinion a type 1 error is something committed with bias by the researcher. I say this because as a researcher it is their job to find all facts or at least most all facts and apply them to their study or research, especially if they commit a type one error knowingly. If a researcher does this error then they are not following through with basic research guidelines.
4) A one-tailed test is used in a hypothesi.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests
Chapter Topic:
Hypothesis Testing Methodology
Z Test for the Mean ( Known)
p-Value Approach to Hypothesis Testing
Connection to Confidence Interval Estimation
One-Tail Tests
t Test for the Mean ( Unknown)
Z Test for the Proportion
Potential Hypothesis-Testing Pitfalls and Ethical Issues
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
I’m a young Pakistani Blogger, Academic Writer, Freelancer, Quaidian & MPhil Scholar, Quote Lover, Co-Founder at Essar Student Fund & Blueprism Academia, belonging from Mehdiabad, Skardu, Gilgit Baltistan, Pakistan.
I am an academic writer & freelancer! I can work on Research Paper, Thesis Writing, Academic Research, Research Project, Proposals, Assignments, Business Plans, and Case study research.
Expertise:
Management Sciences, Business Management, Marketing, HRM, Banking, Business Marketing, Corporate Finance, International Business Management
For Order Online:
Whatsapp: +923452502478
Portfolio Link: https://blueprismacademia.wordpress.com/
Email: arguni.hasnain@gmail.com
Follow Me:
Linkedin: arguni_hasnain
Instagram : arguni.hasnain
Facebook: arguni.hasnain
Test of significance (t-test, proportion test, chi-square test)Ramnath Takiar
The presentation discusses the concept of test of significance including the test of significance examples of t-test, proportion test and chi-square test.
Answer questions Minimum 100 words each and reference (questions.docxamrit47
Answer questions Minimum 100 words each and reference (questions #1-2) KEEP questions WITH ANSWER
1. A key point to get out of this topic is the idea that these errors are theoretical. You won't be sure as to whether one occurred or not. Why are they theoretical in nature? Hint: think about a study and knowing the "truth"
2. Pick a study of interest and identify the null and alternative hypothesis. How does this fit in with regards to the topic of a type I and type II error? Always keep this in mind when you are trying to identify what a type I and type II error are.
A minimum of 75 words each question and References (IF NEEDED)(Response #1 – 7) KEEP RESPONSE WITH ANSWER
Make sure the Responses includes the Following: (a) an understanding of the weekly content as supported by a scholarly resource, (b) the provision of a probing question. (c) stay on topic
1) According to the reading, we set the alpha which is the largest probability for type I error. To increase the power of a hypothesis researchers can use larger samples which provides more information and raise the significance level which increases the probability that the hypothesis will be rejected.
2) A Type 1 error occurs when individuals involved in research make the decision to reject the belief of truth when in actuality the hypothesis is true. Type 1 errors are errors in research when the researcher makes the wrong decision to reject a true null hypothesis. Type II errors are considered less of a problem than Type 1 errors, but can prove to be detrimental in the field of medicine. This type of error occurs when researchers decide to keep a false null hypothesis, when in fact the hypothesis is true. The method to avoid making Type 1 decisions is to test the null hypothesis at the highest level (Alpha Level). This will lessen the possibilities of making this type of error (Privitera, 2018).
3) According to Privitera (2018) a type 1 error is the probability of rejecting a null hypotheses that is actually true, researchers purposely make this error. A null hypotheses is a statement about a population parameter that is assumed to be true, this hypotheses is a starting point (Privitera, 2018). The type 2 error or beta error is the probability of retaining a null hypotheses that is actually false (Privitera, 2018). The type 1 error is committed when a researcher decides to reject previous notions of truth that are in fact true (Privitera, 2018). The best way to avoid these types of errors is to be open minded and not reject notions if there is fact to back the notions up. In my opinion a type 1 error is something committed with bias by the researcher. I say this because as a researcher it is their job to find all facts or at least most all facts and apply them to their study or research, especially if they commit a type one error knowingly. If a researcher does this error then they are not following through with basic research guidelines.
4) A one-tailed test is used in a hypothesi.
Hypothesis Testing Definitions A statistical hypothesi.docxwilcockiris
Hypothesis Testing
Definitions:
A statistical hypothesis is a guess about a population parameter. The guess may or not be
true.
The null hypothesis, written H0, is a statistical hypothesis that states that there is no
difference between a parameter and a specific value, or that there is no difference between
two parameters.
The alternative hypothesis, written H1 or HA, is a statistical hypothesis that specifies a
specific difference between a parameter and a specific value, or that there is a difference
between two parameters.
Example 1:
A medical researcher is interested in finding out whether a new medication will have
undesirable side effects. She is particularly concerned with the pulse rate of patients who
take the medication. The research question is, will the pulse rate increase, decrease, or
remain the same after a patient takes the medication?
Since the researcher knows that the mean pulse rate for the population under study is 82
beats per minute, the hypotheses for this study are:
H0: µ = 82
HA: µ ≠ 82
The null hypothesis specifies that the mean will remain unchanged and the alternative
hypothesis states that it will be different. This test is called a two-tailed test since the
possible side effects could be to raise or lower the pulse rate. Notice that this is a non
directional hypothesis. The rejection region lies in both tails. We divide the alpha in two
and place half in each tail.
Example 2:
An entrepreneur invents an additive to increase the life of an automobile battery. If the
mean lifetime of the automobile battery is 36 months, then his hypotheses are:
H0: µ ≤ 36
HA: µ > 36
Here, the entrepreneur is only interested in increasing the lifetime of the batteries, so his
alternative hypothesis is that the mean is greater than 36 months. The null hypothesis is
that the mean is less than or equal to 36 months. This test is one-tailed since the interest
is only in an increased lifetime. Notice that the direction of the inequality in the alternate
hypothesis points to the right, same as the area of the curve that forms the rejection
region.
Example 3:
A landlord who wants to lower heating bills in a large apartment complex is considering
using a new type of insulation. If the current average of the monthly heating bills is $78,
his hypotheses about heating costs with the new insulation are:
H0: µ ≥ 78
HA: µ < 78
This test is also a one-tailed test since the landlord is interested only in lowering heating
costs. Notice that the direction of the inequality in the alternate hypothesis points to the
left, same as the area of the curve that forms the rejection region.
Study Design:
After stating the hypotheses, the researcher’s next step is to design the study. In designing
the study, the researcher selects an appropriate statistical test, chooses a level of
significance, and formulates a plan for conducting the study..
Hypothesis TestingThe Right HypothesisIn business, or an.docxadampcarr67227
Hypothesis Testing
The Right Hypothesis
In business, or any other discipline, once the question has been asked there must be a statement as to what will or will not occur through testing, measurement, and investigation. This process is known as formulating the right hypothesis. Broadly defined a hypothesis is a statement that the conditions under which something is being measured or evaluated holds true or does not hold true. Further, a business hypothesis is an assumption that is to be tested through market research, data mining, experimental designs, quantitative, and qualitative research endeavors. A hypothesis gives the businessperson a path to follow and specific things to look for along the road.
If the research and statistical data analysis supports and proves the hypothesis that becomes a project well done. If, however, the research data proved a modified version of the hypothesis then re-evaluation for continuation must take place. However, if the research data disproves the hypothesis then the project is usually abandoned.
Hypotheses come in two forms: the null hypothesis and the alternate hypothesis. As a student of applied business statistics you can pick up any number of business statistics textbooks and find a number of opinions on which type of hypothesis should be used in the business world. For the most part, however, and the safest, the better hypothesis to formulate on the basis of the research question asked is what is called the null hypothesis. A null hypothesis states that the research measurement data gathered will not support a difference, relationship, or effect between or amongst those variables being investigated. To the seasoned research investigator having to accept a statement that no differences, relationships, and/or effects will occur based on a statistical data analysis is because when nothing takes place or no differences, effects, or relationship are found there is no possible reason that can be given as to why. This is where most business managers get into trouble when attempting to offer an explanation as to why something has not happened. Attempting to provide an answer to why something has not taken place is akin to discussing how many angels can be placed on the head of a pin—everyone’s answer is plausible and possible. As such business managers need to accept that which has happened and not that which has not happened.
Many business people will skirt the null hypothesis issue by attempting to set analternative hypothesis that states differences, effects and relationships will occur between and amongst that which is being investigated if certain conditions apply.Unfortunately, however, this reverse position is as bad. The research investigator might well be safe if the data analysis detects differences, effect or relationships, but what if it does not? In that case the business manager is back to square one in attempting to explain what has not happened. Although the hypothesis situation may seem c.
UNIT 3
SUCCESS GUIDE
1 | GB 513 Unit 3 Success Guide v.6.13.17
UNIT 3 SUCCESS GUIDE
This unit is the other “most difficult” one. Hypothesis testing has two parts: setting-up
the hypotheses and calculating the critical values to determine results. They both
pose difficulty for a lot of students. The seminar will be on the first and the recorded
lecture will be on the second. You need to make sure you understand both,
otherwise you will not be able to get to the right conclusions.
1. As always, start by reading the chapters and studying the solved examples.
2. Watch the lecture video in document sharing. It focuses on why we do
hypothesis testing, how to do it with Excel and solves two sample problems.
3. Watch this from Khan Academy:
https://www.khanacademy.org/math/statistics-probability/significance-
tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-
values
This one talks more about how to write the null and alternative hypotheses
(which a lot of students get wrong) and also solves the problem using
formulas.
4. Watch the sample problem solutions in Course Resources.
5. If you still want more videos, search YouTube for “hypothesis testing.” Several
introductory level videos are available, such as
https://www.youtube.com/watch?v=HmMjS88eSVE and
https://www.youtube.com/watch?v=0zZYBALbZgg
Email your instructor if you find any of these links to be broken.
Avoid these mistakes!
GENERAL NOTES
RESOURCES
COMMON MISTAKES IN THE ASSIGNMENT
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=0zZYBALbZgg
http://www.youtube.com/watch?v=0zZYBALbZgg
http://www.youtube.com/watch?v=0zZYBALbZgg
2 | GB 513 Unit 3 Success Guide v.6.13.17
Students commonly get the null and alternative hypotheses reversed, or
get them completely wrong.
Students also commonly do not state the hypothesis fully. This is correct:
“null hypothesis: there is no difference between the average salary for
group 1 and the average salary of group 2.” This is not sufficient: “ho:
x1=x2”
Students sometimes compare the averages of the two groups and base
their determination on which one is greater, rather than properly doing a
hypothesis test.
Students sometimes do the calculations correctly, but do not write out
what the conclusion is. This is correct: “We therefore reject the null
hypothesis, which means we conclude that there i ...
TEST #1Perform the following two-tailed hypothesis test, using a.docxmattinsonjanel
TEST #1
Perform the following two-tailed hypothesis test, using a .05 significance level:
· Intrinsic by Gender
· State the null and an alternate statement for the test
· Use Microsoft Excel (Data Analysis Tools) to process your data and run the appropriate test. Copy and paste the results of the output to your report in Microsoft Word.
· Identify the significance level, the test statistic, and the critical value.
· State whether you are rejecting or failing to reject the null hypothesis statement.
· Explain how the results could be used by the manager of the company.
TEST #2
Perform the following two-tailed hypothesis test, using a .05 significance level:
· Extrinsic variable by Position Type
· State the null and an alternate statement for the test
· Use Microsoft Excel (Data Analysis Tools) to process your data and run the appropriate test.
· Copy and paste the results of the output to your report in Microsoft Word.
· Identify the significance level, the test statistic, and the critical value.
· State whether you are rejecting or failing to reject the null hypothesis statement.
· Explain how the results could be used by the manager of the company.
GENERAL ANALYSIS (Research Required)
Using your textbook or other appropriate college-level resources:
· Explain when to use a t-test and when to use a z-test. Explore the differences.
· Discuss why samples are used instead of populations.
The report should be well written and should flow well with no grammatical errors. It should include proper citation in APA formatting in both the in-text and reference pages and include a title page, be double-spaced, and in Times New Roman, 12-point font. APA formatting is necessary to ensure academic honesty.
Be sure to provide references in APA format for any resource you may use to support your answers.
Making Inferences
When data are collected, various summary statistics and graphs can be used for describing data; however, learning about what the data mean is where the power of statistics starts. For example, is there really a difference between two leading cola products? Hypothesis testing is an example of making these types of inferences on data sets.
Hypothesis Tests
Claims are made all the time, such as a particular light bulb will last a certain number of hours.
Claims like this are tested with hypothesis testing. It is a straight forward procedure that consists of the following steps:
1. A claim is made.
2. A value for probability of significance is chosen.
3. Data are collected.
4. The test is performed.
5. The results are analyzed.
Hypothesis tests are performed on the mean of the population. µ
It is not possible to test the full population. For example, it would be impossible to test every light bulb. Instead, the hypothesis test is performed on a sample of the population.
Setting up a Hypothesis Test
When performing hypothesis testing, the test is setup with a null hypothesis (or claim) and the alternative hypothesis. ...
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1. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 1
Chapter 9
Statistical Inference:
Hypothesis Testing for Single Populations
LEARNING OBJECTIVES
The main objective of Chapter 9 is to help you to learn how to test hypotheses on single
populations, thereby enabling you to:
1. Understand the logic of hypothesis testing and know how to establish null and
alternate hypotheses.
2. Understand Type I and Type II errors and know how to solve for Type II errors.
3. Know how to implement the HTAB system to test hypotheses.
4. Test hypotheses about a single population mean when σ is known.
5. Test hypotheses about a single population mean when σ is unknown.
6. Test hypotheses about a single population proportion.
7. Test hypotheses about a single population variance.
CHAPTER TEACHING STRATEGY
For some instructors, this chapter is the cornerstone of the first statistics course.
Hypothesis testing presents the logic in which ideas, theories, etc., are scientifically
2. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 2
examined. The student can be made aware that much of the development of concepts to
this point including sampling, level of data measurement, descriptive tools such as mean
and standard deviation, probability, and distributions pave the way for testing hypotheses.
Often students (and instructors) will say "Why do we need to test this hypothesis when
we can make a decision by examining the data?" Sometimes it is true that examining the
data could allow hypothesis decisions to be made. However, by using the methodology
and structure of hypothesis testing even in "obvious" situations, the researcher has added
credibility and rigor to his/her findings. Some statisticians actually report findings in a
court of law as an expert witness. Others report their findings in a journal, to the public,
to the corporate board, to a client, or to their manager. In each case, by using the
hypothesis testing method rather than a "seat of the pants" judgment, the researcher
stands on a much firmer foundation by using the principles of hypothesis testing and
random sampling. Chapter 9 brings together many of the tools developed to this point
and formalizes a procedure for testing hypotheses.
The statistical hypotheses are set up as to contain all possible decisions. The
two-tailed test always has = and ≠ in the null and alternative hypothesis. One-tailed tests
are presented with = in the null hypothesis and either > or < in the alternative hypothesis.
If in doubt, the researcher should use a two-tailed test. Chapter 9 begins with a two-tailed
test example. Usually, that which the researcher wants to demonstrate true or prove true
is usually set up as an alternative hypothesis. The null hypothesis is that the new theory
or idea is not true, the status quo is still true, or that there is no difference. The null
hypothesis is assumed to be true before the process begins. Some researchers liken this
procedure to a court of law where the defendant is presumed innocent (assume null is true
- nothing has happened). Evidence is brought before the judge or jury. If enough
evidence is presented, the null hypothesis (defendant innocent) can no longer be accepted
or assume true. The null hypothesis is rejected as not true and the alternate hypothesis is
accepted as true by default. Emphasize that the researcher needs to make a decision after
examining the observed statistic.
Some of the key concepts in this chapter are one-tailed and two-tailed test and
Type I and Type II error. In order for a one-tailed test to be conducted, the problem must
include some suggestion of a direction to be tested. If the student sees such words as
greater, less than, more than, higher, younger, etc., then he/she knows to use a one-tail
test. If no direction is given (test to determine if there is a "difference"), then a two-tailed
test is called for. Ultimately, students will see that the only effect of using a one-tailed
test versus a two-tailed test is on the critical table value. A one-tailed test uses all of the
value of alpha in one tail. A two-tailed test splits alpha and uses alpha/2 in each tail thus
creating a critical value that is further out in the distribution. The result is that (all things
being the same) it is more difficult to reject the null hypothesis with a two-tailed test.
Many computer packages such as MINITAB include in the results a p-value. If you
designate that the hypothesis test is a two-tailed test, the computer will double the p-value
so that it can be compared directly to alpha.
In discussing Type I and Type II errors, there are a few things to consider. Once
a decision is made regarding the null hypothesis, there is a possibility that the decision is
3. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 3
correct or that an error has been made. Since the researcher virtually never knows for
certain whether the null hypothesis was actually true or not, a probability of committing
one of these errors can be computed. Emphasize with the students that a researcher can
never commit a Type I error and a Type II error at the same time. This is so because a
Type I error can only be committed when the null hypothesis is rejected and a Type II
error can only be committed when the decision is to not reject the null hypothesis. Type I
and Type II errors are important concepts for managerial students to understand even
beyond the realm of statistical hypothesis testing. For example, if a manager decides to
fire or not fire an employee based on some evidence collected, he/she could be
committing a Type I or a Type II error depending on the decision. If the production
manager decides to stop the production line because of evidence of faulty raw materials,
he/she might be committing a Type I error. On the other hand, if the manager fails to
shut the production line down even when faced with evidence of faulty raw materials,
he/she might be committing a Type II error.
The student can be told that there are some widely accepted values for alpha
(probability of committing a Type I error) in the research world and that a value is
usually selected before the research begins. On the other hand, since the value of Beta
(probability of committing a Type II error) varies with every possible alternate value of
the parameter being tested, Beta is usually examined and computed over a range of
possible values of that parameter. As you can see, the concepts of hypothesis testing are
difficult and represent higher levels of learning (logic, transfer, etc.). Student
understanding of these concepts will improve as you work your way through the
techniques in this chapter and in chapter 10.
CHAPTER OUTLINE
9.1 Introduction to Hypothesis Testing
Types of Hypotheses
Research Hypotheses
Statistical Hypotheses
Substantive Hypotheses
Using the HTAB System to Test Hypotheses
Rejection and Non-rejection Regions
Type I and Type II errors
9.2 Testing Hypotheses About a Population Mean Using the z Statistic (σ known)
Using a Sample Standard Deviation
Testing the Mean with a Finite Population
Using the p-Value to Test Hypotheses
Using the Critical Value Method to Test Hypotheses
Using the Computer to Test Hypotheses about a Population Mean Using
the z Test
9.3 Testing Hypotheses About a Population Mean Using the t Statistic (σ unknown)
Using the Computer to Test Hypotheses about a Population Mean Using
the t Test
4. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 4
9.4 Testing Hypotheses About a Proportion
Using the Computer to Test Hypotheses about a Population Proportion
9.5 Testing Hypotheses About a Variance
9.6 Solving for Type II Errors
Some Observations About Type II Errors
Operating Characteristic and Power Curves
Effect of Increasing Sample Size on the Rejection Limits
KEY TERMS
Alpha(α ) One-tailed Test
Alternative Hypothesis Operating-Characteristic Curve (OC)
Beta(β ) p-Value Method
Critical Value Power
Critical Value Method Power Curve
Hypothesis Rejection Region
Hypothesis Testing Research Hypothesis
Level of Significance Statistical Hypothesis
Nonrejection Region Substantive Result
Null Hypothesis Two-Tailed Test
Observed Significance Level Type I Error
Observed Value Type II Error
SOLUTIONS TO PROBLEMS IN CHAPTER 9
9.1 a) Ho: µ = 25
Ha: µ ≠ 25
x = 28.1 n = 57 σ = 8.46 α = .01
For two-tail, α/2 = .005 zc = 2.575
z =
57
46.8
251.28 −
=
−
n
x
σ
µ
= 2.77
observed z = 2.77 > zc = 2.575
5. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 5
Reject the null hypothesis
b) from Table A.5, inside area between z = 0 and z = 2.77 is .4972
p-value = .5000 - .4972 = .0028
Since the p-value of .0028 is less than
2
α
= .005, the decision is to:
Reject the null hypothesis
c) critical mean values:
zc =
n
s
xc µ−
± 2.575 =
57
46.8
25−cx
x c = 25 ± 2.885
x c = 27.885 (upper value)
x c = 22.115 (lower value)
9.2 Ho: µ = 7.48
Ha: µ < 7.48
x = 6.91 n = 24 σ = 1.21 α =.01
For one-tail, α = .01 zc = -2.33
z =
24
21.1
48.791.6 −
=
−
n
x
σ
µ
= -2.31
observed z = -2.31 > zc = -2.33
Fail to reject the null hypothesis
9.3 a) Ho: µ = 1,200
Ha: µ > 1,200
6. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 6
x = 1,215 n = 113 σ = 100 α = .10
For one-tail, α = .10 zc = 1.28
z =
113
100
200,1215,1 −
=
−
n
x
σ
µ
= 1.59
observed z = 1.59 > zc = 1.28
Reject the null hypothesis
b) Probability > observed z = 1.59 is .0559 which is less than α = .10.
Reject the null hypothesis.
c) Critical mean value:
zc =
n
s
xc µ−
1.28 =
113
100
200,1−cx
x c = 1,200 + 12.04
Since calculated x = 1,215 which is greater than the critical x = 1212.04, reject
the null hypothesis.
9.4 Ho: µ = 82
Ha: µ < 82
x = 78.125 n = 32 σ = 9.184 α = .01
z.01 = -2.33
z =
32
184.9
82125.78 −
=
−
n
x
σ
µ
= -2.39
7. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 7
Since observed z = -2.39 < z.01 = -2.33
Reject the null hypothesis
Statistically, we can conclude that urban air soot is significantly lower. From a
business and community point-of-view, assuming that the sample result is
representative of how the air actually is now, is a reduction of suspended particles
from 82 to 78.125 really an important reduction in air pollution? Certainly it
marks an important first step and perhaps a significant start. Whether or not it
would really make a difference in the quality of life for people in the city of St.
Louis remains to be seen. Most likely, politicians and city chamber of commerce
folks would jump on such results as indications of improvement in city
conditions.
9.5 H0: µ = $424.20
Ha: µ ≠ $424.20
x = $432.69 n = 54 σ = $33.90 α = .05
2-tailed test, α/2 = .025 z.025 = + 1.96
z =
54
90.33
20.42469.432 −
=
−
n
x
σ
µ
= 1.84
Since the observed z = 1.85 < z.025 = 1.96, the decision is to fail to reject the
null hypothesis.
9.6 H0: µ = $62,600
Ha: µ < $62,600
x = $58,974 n = 18 σ = $7,810 α = .01
1-tailed test, α = .01 z.01 = -2.33
z =
18
810,7
600,62974,58 −
=
−
n
x
σ
µ
= -1.97
Since the observed z = -1.97 > z.01 = -2.33, the decision is to fail to reject the
null hypothesis.
8. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 8
9.7 H0: µ = 5
Ha: µ ≠ 5
x = 5.0611 n = 42 σ = 0.2803 α = .10
2-tailed test, α/2 = .05 z.05 = + 1.645
z =
42
2803.0
50611.5 −
=
−
n
x
σ
µ
= 1.41
Since the observed z = 1.41 < z.05 = 1.645, the decision is to fail to reject the
null hypothesis.
9.8 Ho: µ = 18.2
Ha: µ < 18.2
x = 15.6 n = 32 σ = 2.3 α = .10
For one-tail, α = .10, z.10 = -1.28
z =
32
3.2
2.186.15 −
=
−
n
x
σ
µ
= -6.39
Since the observed z = -6.39 < z.10 = -1.28, the decision is to
Reject the null hypothesis
9.9 Ho: µ = $4,292
Ha: µ < $4,292
x = $4,008 n = 55 σ = $386 α = .01
For one-tailed test, α = .01, z.01 = -2.33
z =
55
386$
292,4$008,4$ −
=
−
n
x
σ
µ
= -5.46
Since the observed z = -5.46 < z.01 = -2.33, the decision is to
Reject the null hypothesis
9. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 9
The CEO could use this information as a way of discrediting the Runzheimer
study and using her own figures in recruiting people and in discussing relocation
options. In such a case, this could be a substantive finding. However, one must
ask if the difference between $4,292 and $4,008 is really an important difference
in monthly rental expense. Certainly, Paris is expensive either way. However, an
almost $300 difference in monthly rental cost is a non trivial amount for most
people and therefore might be considered substantive.
9.10 Ho: µ = 123
Ha: µ > 123
α = .05 n = 40 40 people were sampled
x = 132.36
This is a one-tailed test. Since the p-value = .016, we
reject the null hypothesis at α = .05.
The average water usage per person is greater than 123 gallons.
9.11 n = 20 x = 16.45 s = 3.59 df = 20 - 1 = 19 α = .05
Ho: µ = 16
Ha: µ ≠ 16
For two-tail test, α/2 = .025, critical t.025,19 = ±2.093
t =
20
59.3
1645.16 −
=
−
n
s
x µ
= 0.56
Observed t = 0.56 < t.025,19 = 2.093
The decision is to Fail to reject the null hypothesis
9.12 n = 51 x = 58.42 s2
= 25.68 df = 51 - 1 = 50 α = .01
Ho: µ = 60
Ha: µ < 60
For one-tail test, α = .01 critical t.01,50 = -2.403
10. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 10
t =
51
68.25
6042.58 −
=
−
n
s
x µ
= -2.23
Observed t = -2.33 > t.01,50 = -2.403
The decision is to Fail to reject the null hypothesis
9.13 n = 11 x = 1,235.36 s = 103.81 df = 11 - 1 = 10 α = .05
Ho: µ = 1,160
Ha: µ > 1,160
For one-tail test, α = .05 critical t.05,10 = 1.812
t =
11
81.103
160,136.236,1 −
=
−
n
s
x µ
= 2.44
Observed t = 2.44 > t.05,10 = 1.812
The decision is to Reject the null hypothesis
9.14 n = 20 x = 8.37 s = .189 df = 20-1 = 19 α = .01
Ho: µ = 8.3
Ha: µ ≠ 8.3
For two-tail test, α/2 = .005 critical t.005,19 = ±2.861
t =
20
189.
3.837.8 −
=
−
n
s
x µ
= 1.66
Observed t = 1.66 < t.005,19 = 2.861
The decision is to Fail to reject the null hypothesis
11. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 11
9.15 n = 12 x = 1.85083 s = .02353 df = 12 - 1 = 11 α = .10
H0: µ = 1.84
Ha: µ ≠ 1.84
For a two-tailed test, α/2 = .05 critical t.05,11 = 1.796
t =
12
02353.
84.185083.1 −
=
−
n
s
x µ
= 1.59
Since t = 1.59 < t11,.05 = 1.796,
The decision is to fail to reject the null hypothesis.
9.16 n = 25 x = 1.1948 s = .0889 df = 25 - 1 = 24 α = .01
Ho: µ = $1.16
Ha: µ > $1.16
For one-tail test, = .01 Critical t.01,24 = 2.492
t =
25
0889.
16.11948.1 −
=
−
n
s
x µ
= 1.96
Observed t = 1.96 < t.01,24 = 2.492
The decision is to Fail to reject the null hypothesis
9.17 n = 19 x = $31.67 s = $1.29 df = 19 – 1 = 18 α = .05
H0: µ = $32.28
Ha: µ ≠ $32.28
Two-tailed test, α/2 = .025 t.025,18 = + 2.101
t =
19
29.1
28.3267.31 −
=
−
n
s
x µ
= -2.06
The observed t = -2.06 > t.025,18 = -2.101,
The decision is to fail to reject the null hypothesis
12. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 12
9.18 n = 26 x = 19.534 minutes s = 4.100 minutes α = .05
H0: µ = 19
Ha: µ ≠ 19
Two-tailed test, α/2 = .025, critical t value = + 2.06
Observed t value = 0.66
Since the observed t = 0.66 < critical t value = 2.06,
The decision is to fail to reject the null hypothesis.
Since the Excel p-value = .256 > α/2 = .025 and MINITAB p-value =.513 > .05,
the decision is to fail to reject the null hypothesis.
She would not conclude that her city is any different from the ones in the
national survey.
9.19 Ho: p = .45
Ha: p > .45
n = 310 pˆ = .465 α = .05
For one-tail, α = .05 z.05 = 1.645
z =
310
)55)(.45(.
45.465.ˆ −
=
⋅
−
n
qp
pp
= 0.53
observed z = 0.53 < z.05 = 1.645
The decision is to Fail to reject the null hypothesis
9.20 Ho: p = 0.63
Ha: p < 0.63
n = 100 x = 55
100
55
ˆ ==
n
x
p = .55
For one-tail, α = .01 z.01 = -2.33
13. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 13
z =
310
)55)(.45(.
45.465.ˆ −
=
⋅
−
n
qp
pp
= -1.66
observed z = -1.66 > zc = -2.33
The decision is to Fail to reject the null hypothesis
9.21 Ho: p = .29
Ha: p ≠ .29
n = 740 x = 207
740
207
ˆ ==
n
x
p = .28 α = .05
For two-tail, α/2 = .025 z.025 = ±1.96
z =
740
)71)(.29(.
29.28.ˆ −
=
⋅
−
n
qp
pp
= -0.60
observed z = -0.60 > zc = -1.96
The decision is to Fail to reject the null hypothesis
p-Value Method:
z = -0.60
from Table A.5, area = .2257
Area in tail = .5000 - .2257 = .2743
.2743 > .025
Again, the decision is to Fail to reject the null hypothesis
Solving for critical values:
z =
n
qp
ppc
⋅
−ˆ
14. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 14
±1.96 =
740
)71)(.29(.
29.ˆ −cp
cpˆ = .29 ± .033
.257 and .323
Sample p = pˆ = .28 not outside critical values in tails
Again, the decision is to Fail to reject the null hypothesis
9.22 Ho: p = .48
Ha: p ≠ .48
n = 380 x = 164 α = .01 α/2 = .005 z.005 = +2.575
380
164
ˆ ==
n
x
p = .4316
z =
380
)52)(.48(.
48.4316.ˆ −
=
⋅
−
n
qp
pp
= -1.89
Since the observed z = -1.89 is greater than z.005= -2.575, The decision is to fail to
reject the null hypothesis. There is not enough evidence to declare that the
proportion is any different than .48.
9.23 Ho: p = .79
Ha: p < .79
n = 415 x = 303 α = .01 z.01 = -2.33
415
303
ˆ ==
n
x
p = .7301
z =
415
)21)(.79(.
79.7301ˆ −
=
⋅
−
n
qp
pp
= -3.00
15. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 15
Since the observed z = -3.00 is less than z.01= -2.33, The decision is to reject the
null hypothesis.
9.24 Ho: p = .31
Ha: p ≠ .31
n = 600 x = 200 α = .10 α/2 = .05 z.005 = +1.645
600
200
ˆ ==
n
x
p = .3333
z =
600
)69)(.31(.
31.3333.ˆ −
=
⋅
−
n
qp
pp
= 1.23
Since the observed z = 1.23 is less than z.005= 1.645, The decision is to fail to
reject the null hypothesis. There is not enough evidence to declare that the
proportion is any different than .48.
Ho: p = .24
Ha: p < .24
n = 600 x = 130 α = .05 z.05 = -1.645
600
130
ˆ ==
n
x
p = .2167
z =
600
)76)(.24(.
24.2167.ˆ −
=
⋅
−
n
qp
pp
= -1.34
Since the observed z = -1.34 is greater than z.05= -1.645, The decision is to fail to
reject the null hypothesis. There is not enough evidence to declare that the
proportion is less than .24.
9.25 Ho: p = .18
Ha: p > .18
n = 376 pˆ = .22 α = .01
one-tailed test, z.01 = 2.33
16. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 16
z =
376
)82)(.18(.
18.22.ˆ −
=
⋅
−
n
qp
pp
= 2.02
Since the observed z = 2.02 is less than z.01= 2.33, The decision is to fail to reject
the null hypothesis. There is not enough evidence to declare that the proportion
is greater than .18.
9.26 Ho: p = .32
Ha: p < .32
n = 118 x = 22
118
22
ˆ ==
n
x
p = .186 α = .01
For one-tailed test, z.05 = -1.645
z =
118
)68)(.32(.
32.186.ˆ −
=
⋅
−
n
qp
pp
= -3.12
Observed z = -3.12 < z.05 –1.645
Since the observed z = -3.12 is less than z.05= -1.645, The decision is to reject the
null hypothesis.
9.27 Ho: p = .47
Ha: p ≠ .47
n = 67 x = 40 α = .05 α/2 = .025
For a two-tailed test, z.025 = +1.96
67
40
ˆ ==
n
x
p = .597
z =
67
)53)(.47(.
47.597.ˆ −
=
⋅
−
n
qp
pp
= 2.08
Since the observed z = 2.08 is greater than z.025= 1.96, The decision is to reject the
null hypothesis.
17. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 17
9.28 a) H0: σ2
= 20 α = .05 n = 15 df = 15 – 1 = 14 s2
= 32
Ha: σ2
> 20
χ2
.05,14 = 23.6848
χ2
=
20
)32)(115( −
= 22.4
Since χ2
= 22.4 < χ2
.05,14 = 23.6848, the decision is to fail to reject the null
hypothesis.
b) H0: σ2
= 8.5 α = .10 α/2 = .05 n = 22 df = n-1 = 21 s2
= 17
Ha: σ2
≠ 8.5
χ2
.05,21 = 32.6705
χ2
=
5.8
)17)(122( −
= 42
Since χ2
= 42 > χ2
.05,21 = 32.6705, the decision is to reject the null hypothesis.
c) H0: σ2
= 45 α = .01 n = 8 df = n – 1 = 7 s = 4.12
Ha: σ2
< 45
χ2
.01,7 = 18.4753
χ2
=
45
)12.4)(18( 2
−
= 2.64
Since χ2
= 2.64 < χ2
.01,7 = 18.4753, the decision is to fail to reject the null
hypothesis.
d) H0: σ2
= 5 α = .05 α/2 = .025 n = 11 df = 11 – 1 = 10 s2
= 1.2
Ha: σ2
≠ 5
χ2
.025,10 = 20.4831 χ2
.975,10 = 3.24697
χ2
=
5
)2.1)(111( −
= 2.4
Since χ2
= 2.4 < χ2
.975,10 = 3.24697, the decision is to reject the null hypothesis.
18. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 18
9.29 H0: σ2
= 14 α = .05 α/2 = .025 n = 12 df = 12 – 1 = 11 s2
= 30.0833
Ha: σ2
≠ 14
χ2
.025,11 = 21.92 χ2
.975,11 = 3.81575
χ2
=
14
)0833.30)(112( −
= 23.64
Since χ2
= 23.64 < χ2
.025,11 = 21.92, the decision is to reject the null hypothesis.
9.30 H0: σ2
= .001 α = .01 n = 16 df = 16 – 1 = 15 s2
= .00144667
Ha: σ2
> .001
χ2
.01,15 = 30.5779
χ2
=
001.
)00144667)(.116( −
= 21.7
Since χ2
= 21.7 < χ2
.01,15 = 30.5779, the decision is to fail to reject the null
hypothesis.
9.31 H0: σ2
= 199,996,164 α = .10 α/2 = .05 n = 13 df =13 - 1 = 12
Ha: σ2
≠ 199,996,164 s2
= 832,089,743.7
χ2
.05,12 = 21.0261 χ2
.95,12 = 5.22603
χ2
=
164,996,199
)7.743,089,832)(113( −
= 49.93
Since χ2
= 49.93 > χ2
.05,12 = 21.0261, the decision is to reject the null
hypothesis. The variance has changed.
9.32 H0: σ2
= .04 α = .01 n = 7 df = 7 – 1 = 6 s = .34 s2
= .1156
Ha: σ2
> .04
χ2
.01,6 = 16.8119
χ2
=
04.
)1156)(.17( −
= 17.34
Since χ2
= 17.34 > χ2
.01,6 = 16.8119, the decision is to reject the null hypothesis
19. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 19
9.33 Ho: µ = 100
Ha: µ < 100
n = 48 µ = 99 σ = 14
a) α = .10 z.10 = -1.28
zc =
n
xc
σ
µ−
-1.28 =
48
14
100−cx
x c = 97.4
z =
n
xc
σ
µ−
=
48
14
994.97 −
= -0.79
from Table A.5, area for z = -0.79 is .2852
β = .2852 + .5000 = .7852
b) α = .05 z.05 = -1.645
zc =
n
xc
σ
µ−
-1.645 =
48
14
100−cx
x c = 96.68
z =
n
xc
σ
µ−
=
48
14
9968.96 −
= -1.15
20. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 20
from Table A.5, area for z = -1.15 is .3749
β = .3749 + .5000 = .8749
c) α = .01 z.01 = -2.33
zc =
n
xc
σ
µ−
-2.33 =
48
14
100−cx
x c = 95.29
z =
n
xc
σ
µ−
=
48
14
9929.95 −
= -1.84
from Table A.5, area for z = -1.84 is .4671
β = .4671 + .5000 = .9671
d) As gets smaller (other variables remaining constant), beta gets larger.
Decreasing the probability of committing a Type I error increases the probability
of committing a Type II error if other variables are held constant.
9.34 α = .05 µ = 100 n = 48 σ = 14
a) µa = 98.5 zc = -1.645
zc =
n
xc
σ
µ−
-1.645 =
48
14
100−cx
21. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 21
x c = 96.68
z =
n
xc
σ
µ−
=
48
14
9968.96 −
= -0.90
from Table A.5, area for z = -0.90 is .3159
β = .3159 + .5000 = .8159
b) µa = 98 zc = -1.645
x c = 96.68
zc =
n
xc
σ
µ−
=
48
14
9868.96 −
= -0.65
from Table A.5, area for z = -0.65 is .2422
β = .2422 + .5000 = .7422
c) µa = 97 z.05 = -1.645
x c = 96.68
z =
n
xc
σ
µ−
=
48
14
9768.96 −
= -0.16
from Table A.5, area for z = -0.16 is .0636
β = .0636 + .5000 = .5636
d) µa = 96 z.05 = -1.645
x c = 97.4
22. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 22
z =
n
xc
σ
µ−
=
48
14
9668.96 −
= 0.34
from Table A.5, area for z = 0.34 is .1331
β = .5000 - .1331 = .3669
e) As the alternative value get farther from the null hypothesized value, the
probability of committing a Type II error reduces. (All other variables being held
constant).
9.35 Ho: µ = 50
Ha: µ ≠ 50
µa = 53 n = 35 σ = 7 α = .01
Since this is two-tailed, α/2 = .005 z.005 = ±2.575
zc =
n
xc
σ
µ−
±2.575 =
35
7
50−cx
x c = 50 ± 3.05
46.95 and 53.05
z =
n
xc
σ
µ−
=
35
7
5305.53 −
= 0.04
from Table A.5 for z = 0.04, area = .0160
Other end:
23. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 23
z =
n
xc
σ
µ−
=
35
7
539.46 −
= -5.11
Area associated with z = -5.11 is .5000
β = .5000 + .0160 = .5160
9.36 a) Ho: p = .65
Ha: p < .65
n = 360 α = .05 pa = .60 z.05 = -1.645
zc =
n
qp
ppc
⋅
−ˆ
-1.645 =
360
)35)(.65(.
65.ˆ −cp
pˆ c = .65 - .041 = .609
z =
n
qp
ppc
⋅
−ˆ
=
360
)40)(.60(.
60.609. −
= -0.35
from Table A.5, area for z = -0.35 is .1368
β = .5000 - .1368 = .3632
b) pa = .55 z.05 = -1.645
pˆ c = .609
z =
n
QP
Ppc
⋅
−ˆ
=
360
)45)(.55(.
55.609. −
= -2.25
from Table A.5, area for z = -2.25 is .4878
24. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 24
β = .5000 - .4878 = .0122
c) pa = .50 z.05 = -1.645
pˆ c = .609
z =
n
qp
ppc
⋅
−ˆ
=
360
)50)(.50(.
50.609. −
= -4.14
from Table A.5, the area for z = -4.14 is .5000
β = .5000 - .5000 = .0000
9.37 n = 58 x = 45.1 σ = 8.7 α = .05 α/2 = .025
H0: µ = 44
Ha: µ ≠ 44 z.025 = ± 1.96
z =
58
7.8
441.45 −
= 0.96
Since z = 0.96 < zc = 1.96, the decision is to fail to reject the null hypothesis.
+ 1.96 =
58
7.8
44−cx
± 2.239 = x c - 44
x c = 46.239 and 41.761
For 45 years:
z =
58
7.8
4529.46 −
= 1.08
from Table A.5, the area for z = 1.08 is .3599
25. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 25
β = .5000 + .3599 = .8599
Power = 1 - β = 1 - .8599 = .1401
For 46 years:
z =
58
7.8
46239.46 −
= 0.21
From Table A.5, the area for z = 0.21 is .0832
β = .5000 + .0832 = .5832
Power = 1 - β = 1 - .5832 = .4168
For 47 years:
z =
58
7.8
479.46 −
= -0.67
From Table A.5, the area for z = -0.67 is .2486
β = .5000 - .2486 = .2514
Power = 1 - β = 1 - .2514 = .7486
For 48 years:
z =
58
7.8
48248.46 −
= 1.54
From Table A.5, the area for z = 1.54 is .4382
β = .5000 - .4382 = .0618
Power = 1 - β = 1 - .0618 = .9382
26. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 26
9.38 H0: p= .71
Ha: p < .71
n = 463 x = 324 pˆ =
463
324
= .6998 α = .10
z.10 = -1.28
z =
463
)29)(.71(.
71.6998.ˆ −
=
⋅
−
n
qp
pp
= -0.48
Since the observed z = -0.48 > z.10 = -1.28, the decision is to fail to reject the null
hypothesis.
27. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 27
Type II error:
Solving for the critical proportion, pˆ c:
zc =
n
qp
ppc
⋅
−ˆ
-1.28 =
463
)29)(.71(.
71.ˆ −cp
pˆ = .683
For pa = .69
z =
463
)31)(.69(.
69.683. −
= -0.33
From Table A.5, the area for z = -0.33 is .1293
The probability of committing a Type II error = .1293 + .5000 = .6293
For pa = .66
z =
463
)34)(.66(.
66.683. −
= 1.04
From Table A.5, the area for z = 1.04 is .3508
The probability of committing a Type II error = .5000 - .3508 = .1492
For pa = .60
z =
493
)40)(.60(.
60.683. −
= 4.61
From Table A.5, the area for z = 4.61 is .5000
The probability of committing a Type II error = .5000 - .5000 = .0000
28. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 28
9.39
1) Ho: µ = 36
Ha: µ ≠ 36
2) z =
n
x
σ
µ−
3) α = .01
4) two-tailed test, α/2 = .005, z.005 = + 2.575
If the observed value of z is greater than 2.575 or less than -2.575, the decision
will be to reject the null hypothesis.
5) n = 63, x = 38.4, σ = 5.93
6) z =
n
x
σ
µ−
=
63
93.5
364.38 −
= 3.21
7) Since the observed value of z = 3.21 is greater than z.005 = 2.575, the decision is
to reject the null hypothesis.
8) The mean is likely to be greater than 36.
9.40 1) Ho: µ = 7.82
Ha: µ < 7.82
2) The test statistic is
t =
n
s
x µ−
3) α = .05
4) df = n - 1 = 16, t.05,16 = -1.746. If the observed value of t is less than -1.746, then
the decision will be to reject the null hypothesis.
5) n = 17 x = 7.01 s = 1.69
29. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 29
6) t =
n
s
x µ−
=
17
69.1
82.701.7 −
= -1.98
7) Since the observed t = -1.98 is less than the table value of t = -1.746, the decision
is to reject the null hypothesis.
8) The population mean is significantly less than 7.82.
9.41
a. 1) Ho: p = .28
Ha: p > .28
2) z =
n
qp
pp
⋅
−ˆ
3) α = .10
4) This is a one-tailed test, z.10 = 1.28. If the observed value of z is greater than
1.28, the decision will be to reject the null hypothesis.
5) n = 783 x = 230
783
230
ˆ =p = .2937
6) z =
783
)72)(.28(.
28.2937. −
= 0.85
7) Since z = 0.85 is less than z.10 = 1.28, the decision is to fail to reject the null
hypothesis.
8) There is not enough evidence to declare that p is not .28.
b. 1) Ho: p = .61
Ha: p ≠ .61
2) z =
n
qp
pp
⋅
−ˆ
30. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 30
3) α = .05
4) This is a two-tailed test, z.025 = + 1.96. If the observed value of z is greater than
1.96 or less than -1.96, then the decision will be to reject the null hypothesis.
5) n = 401 pˆ = .56
6) z =
401
)39)(.61(.
61.56. −
= -2.05
7) Since z = -2.05 is less than z.025 = -1.96, the decision is to reject the null
hypothesis.
8) The population proportion is not likely to be .61.
9.42 1) H0: σ2
= 15.4
Ha: σ2
> 15.4
2) χ2
= 2
2
)1(
σ
sn −
3) α = .01
4) n = 18, df = 17, one-tailed test
χ2
.01,17 = 33.4087
5) s2
= 29.6
6) χ2
= 2
2
)1(
σ
sn −
=
4.15
)6.29)(17(
= 32.675
7) Since the observed χ2
= 32.675 is less than 33.4087, the decision is to fail to
reject the null hypothesis.
8) The population variance is not significantly more than 15.4.
9.43 a) H0: µ = 130
Ha: µ > 130
n = 75 σ = 12 α = .01 z.01 = 2.33 µa = 135
31. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 31
Solving for x c:
zc =
n
xc
σ
µ−
2.33 =
75
12
130−cx
x c = 133.23
z =
75
12
13523.133 −
= -1.28
from table A.5, area for z = -1.28 is .3997
β = .5000 - .3997 = .1003
b) H0: p = .44
Ha: p < .44
n = 1095 α = .05 pa = .42 z.05 = -1.645
zc =
n
qp
ppc
⋅
−ˆ
-1.645 =
1095
)56)(.44(.
44.ˆ −cp
cpˆ = .4153
z =
1095
)58)(.42(.
42.4153. −
= -0.32
from table A.5, area for z = -0.32 is .1255
β = .5000 + .1255 = .6255
32. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 32
9.44 H0: p = .32
Ha: p > .32
n = 80 α = .01 pˆ = .39
z.01 = 2.33
z =
80
)68)(.32(.
32.39.ˆ −
=
⋅
−
n
qp
pp
= 1.34
Since the observed z = 1.34 < z.01 = 2.33, the decision is to fail to reject the null
hypothesis.
9.45 x = 3.45 n = 64 σ2
= 1.31 α = .05
Ho: µ = 3.3
Ha: µ ≠ 3.3
For two-tail, α/2 = .025 zc = ±1.96
z =
n
x
σ
µ−
=
64
31.1
3.345.3 −
= 1.05
Since the observed z = 1.05 < zc = 1.96, the decision is to Fail to reject the null
hypothesis.
9.46 n = 210 x = 93 α = .10
210
93
ˆ ==
n
x
p = .443
Ho: p = .57
Ha: p< .57
For one-tail, α = .10 zc = -1.28
z =
210
)43)(.57(.
57.443.ˆ −
=
⋅
−
n
qp
pp
= -3.72
33. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 33
Since the observed z = -3.72 < zc = -1.28, the decision is to reject the null
hypothesis.
9.47 H0: σ2
= 16 n = 12 σ = .05 df = 12 - 1 = 11
Ha: σ2
> 16
s = 0.4987864 ft. = 5.98544 in.
χ2
.05,11 = 19.6751
χ2
=
16
)98544.5)(112( 2
−
= 24.63
Since χ2
= 24.63 > χ2
.05,11 = 19.6751, the decision is to reject the null
hypothesis.
9.48 H0: µ = 8.4 α = .01 α/2 = .005 n = 7 df = 7 – 1 = 6 s = 1.3
Ha: µ ≠ 8.4
x = 5.6 t.005,6 = + 3.707
t =
7
3.1
4.86.5 −
= -5.70
Since the observed t = - 5.70 < t.005,6 = -3.707, the decision is to reject the null
hypothesis.
9.49 x = $26,650 n = 100 σ = $12,000
a) Ho: µ = $25,000
Ha: µ > $25,000 α = .05
For one-tail, α = .05 z.05 = 1.645
z =
n
x
σ
µ−
=
100
000,12
000,25650,26 −
= 1.38
34. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 34
Since the observed z = 1.38 < z.05 = 1.645, the decision is to fail to reject the null
hypothesis.
b) µa = $30,000 zc = 1.645
Solving for x c:
zc =
n
xc
σ
µ−
1.645 =
100
000,12
)000,25( −cx
x c = 25,000 + 1,974 = 26,974
z =
100
000,12
000,30974,26 −
= -2.52
from Table A.5, the area for z = -2.52 is .4941
β = .5000 - .4941 = .0059
9.50 H0: σ2
= 4 n = 8 s = 7.80 α = .10 df = 8 – 1 = 7
Ha: σσσσ2
> 4
χ2
.10,7 = 12.017
χ2
=
4
)80.7)(18( 2
−
= 106.47
Since observed χ2
= 106.47 > χ2
.10,7 = 12.017, the decision is to reject the null
hypothesis.
9.51 H0: p = .46
Ha: p > .46
n = 125 x = 66 α = .05
125
66
ˆ ==
n
x
p = .528
35. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 35
Using a one-tailed test, z.05 = 1.645
z =
125
)54)(.46(.
46.528.ˆ −
=
⋅
−
n
qp
pp
= 1.53
Since the observed value of z = 1.53 < z.05 = 1.645, the decision is to fail to reject
the null hypothesis.
Solving for cpˆ :
zc =
n
qp
ppc
⋅
−ˆ
1.645 =
125
)54)(.46(.
46.ˆ −cp
cpˆ = .533
z =
125
)50)(.50(.
50.533.ˆ −
=
⋅
−
n
qp
pp
aa
ac
= 0.74
from Table A.5, the area for z = 0.74 is .2704
β = .5000 + .2704 = .7704
9.52 n = 16 x = 175 s = 14.28286 df = 16 - 1 = 15 α = .05
H0: µ = 185
Ha: µ < 185
t.05,15 = - 1.753
t =
n
s
x µ−
=
16
28286.14
185175 −
= -2.80
Since observed t = - 2.80 < t.05,15 = - 1.753, the decision is to reject the null
hypothesis.
36. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 36
9.53 H0: p = .16
Ha: p > .16
n = 428 x = 84 α = .01
428
84
ˆ ==
n
x
p = .1963
For a one-tailed test, z.01 = 2.33
z =
428
)84)(.16(.
16.1963.ˆ −
=
⋅
−
n
qp
pp
= 2.05
Since the observed z = 2.05 < z.01 = 2.33, the decision is to fail to reject the null
hypothesis.
The probability of committing a Type I error is .01.
Solving for cpˆ :
zc =
n
qp
ppc
⋅
−ˆ
2.33 =
428
)84)(.16(.
16.ˆ. −cp
cpˆ = .2013
z =
428
)79)(.21(.
21.2013.ˆ −
=
⋅
−
n
qp
pp
aa
ac
= -0.44
from Table A.5, the area for z = -0.44 is .1700
β = .5000 - .1700 = .3300
9.54 Ho: µ = $15
Ha: µ > $15
x = $19.34 n = 35 σ = $4.52 α = .10
For one-tail and α = .10 zc = 1.28
37. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 37
z =
n
x
σ
µ−
=
35
52.4
1534.19 −
= 5.68
Since the observed z = 5.68 > zc = 1.28, the decision is to reject the null
hypothesis.
9.55 H0: σ2
= 16 n = 22 df = 22 –1 = 21 s = 6 α = .05
Ha: σ2
> 16
χ2
.05,21 = 32.6705
χ2
=
16
)6)(122( 2
−
= 47.25
Since the observed χ2
= 47.25 > χ2
.05,21 = 32.6705, the decision is to reject the
null hypothesis.
9.56 H0: µ = 2.5 x = 3.4 s = 0.6 α = .01 n = 9 df = 9 – 1 = 8
Ha: µ > 2.5
t.01,8 = 2.896
t =
n
s
x µ−
=
9
6.0
5.24.3 −
= 4.50
Since the observed t = 4.50 > t.01,8 = 2.896, the decision is to reject the null
hypothesis.
9.57 a) Ho: µ = 23.58
Ha: µ ≠ 23.58
n = 95 x = 22.83 σ = 5.11 α = .05
Since this is a two-tailed test and using α/2 = .025: z.025 = + 1.96
38. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 38
z =
n
x
σ
µ−
=
95
11.5
58.2383.22 −
= -1.43
Since the observed z = -1.43 > z.025 = -1.96, the decision is to fail to reject the
null hypothesis.
b) zc =
n
xc
σ
µ−
+ 1.96 =
95
11.5
58.23−cx
cx = 23.58 + 1.03
cx = 22.55, 24.61
for Ha: µ = 22.30
z =
n
x ac
σ
µ−
=
95
11.5
30.2255.22 −
= 0.48
z =
n
x ac
σ
µ−
=
95
11.5
30.2261.24 −
= 4.41
from Table A.5, the areas for z = 0.48 and z = 4.41 are .1844 and .5000
β = .5000 - .1844 = .3156
The upper tail has no effect on β.
39. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 39
9.58 n = 12 x = 12.333 s2
= 10.424
H0: σ2
= 2.5
Ha: σ2
≠ 2.5
α = .05 df = 11 two-tailed test, α/2 = .025
χ2
.025,11 = 21.92
χ2
..975,11 = 3.81575
If the observed χ2
is greater than 21.92 or less than 3.81575, the decision is to
reject the null hypothesis.
χ2
= 2
2
)1(
σ
sn −
=
5.2
)424.10(11
= 45.866
Since the observed χ2
= 45.866 is greater than χ2
.025,11 = 21.92, the decision is to
reject the null hypothesis. The population variance is significantly more than
2.5.
9.59 The sample size is 22. x is 3.967 s = 0.866 df = 21
The test statistic is:
t =
n
s
x µ−
The observed t = -2.34. The p-value is .015.
The results are statistical significant at α = .05.
The decision is to reject the null hypothesis.
9.60 H0: p = .25
Ha: p ≠ .25
This is a two-tailed test with α = .05. n = 384.
Since the p-value = .045 < α = .05, the decision is to reject the null hypothesis.
The sample proportion, pˆ = .205729 which is less than the hypothesized p = .25.
40. Chapter 9: Statistical Inference: Hypothesis Testing for Single Populations 40
One conclusion is that the population proportion is lower than .25.
9.61 H0: µ = 2.51
Ha: µ > 2.51
This is a one-tailed test. The sample mean is 2.555 which is more than the
hypothesized value. The observed t value is 1.51 with an associated
p-value of .072 for a one-tailed test. Because the p-value is greater than
α = .05, the decision is to fail to reject the null hypothesis.
There is not enough evidence to conclude that beef prices are higher.
9.62 H0: µ = 2747
Ha: µ < 2747
This is a one-tailed test. Sixty-seven households were included in this study.
The sample average amount spent on home-improvement projects was 2,349.
Since z = -2.09 < z.05 = -1.645, the decision is to reject the null hypothesis at
α = .05. This is underscored by the p-value of .018 which is less than α = .05.
However, the p-value of .018 also indicates that we would not reject the null
hypothesis at α = .01.