This document provides an introduction to hypothesis testing. It discusses the components of a hypothesis test, including the null and alternative hypotheses, types of errors, and controlling errors. Specifically, it explains that the null hypothesis is the statement being tested, while the alternative is what would be true if the null is false. Type I error is rejecting the null when it is true, while Type II error is failing to reject a false null. The significance level and power help control these errors.
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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
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Chapter 3: Describing, Exploring, and Comparing Data
3.3: Measures of Relative Standing and Boxplots
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
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.3: Measures of Relative Standing and Boxplots
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.
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..
PUH 5302, Applied Biostatistics 1
Course Learning Outcomes for Unit III
Upon completion of this unit, students should be able to:
4. Recommend solutions to public health problems using biostatistical methods.
4.1 Compute and interpret probability for biostatistical analysis.
4.2 Draw conclusions about public health problems based on biostatistical methods.
5. Analyze public health information to interpret results of biostatistical analysis.
5.1 Analyze literature related to biostatistical analysis in the public health field.
5.2 Prepare an annotated bibliography that explores a topic related to public health issues.
Course/Unit
Learning Outcomes
Learning Activity
4.1
Unit Lesson
Chapter 5
Unit III Problem Solving
4.2
Unit Lesson
Chapter 5
Unit III Problem Solving
5.1
Chapter 5
Unit III Annotated Bibliography
5.2
Chapter 5
Unit III Annotated Bibliography
Reading Assignment
Chapter 5: The Role of Probability
Unit Lesson
Welcome to Unit III. In previous units, we discussed some fundamentals of biostatistics and their application
to solving public health problems. In Unit III, we will compute, interpret, and apply probability, especially in
relation to different populations.
Computing and Interpreting Probabilities
Probability means using a number (or numbers) to demonstrate how likely something is to occur. For
example, if a coin is tossed, the probability of getting a heads or tail is one out of two chances; that is ½.
Researchers have used probability studies to predict weather and other events and have been successful to
some extent. Public health professionals have used statistical methods to predict the chances of health-
related events, thereby providing arguments in favor of taking precautionary measures and warning the
general public on important health issues.
In biostatistics, we use both descriptive statistics and inferential statistics to address public health issues
within a population. In most cases, researchers are not able to study the entire population; they try to get a
sample from the population from which they can generalize their findings.
Descriptive Statistics
Aside from the use of probability sampling methods, there are other methods used for the computation and
interpretation of data; these are generally known as descriptive statistics. With descriptive statistics, we
UNIT III STUDY GUIDE
Probability
PUH 5302, Applied Biostatistics 2
UNIT x STUDY GUIDE
Title
normally compute the mean, mode, median, variance, and standard deviation. Information obtained using
such computation methods is used for descriptive purposes, as opposed to information obtained from
inferential statistics.
Let’s examine this example using the numbers 5, 10, 2, 4, 6, 10, 2, 3, and 2.
The mean is the sum of all the numbers ÷ the number of cases
= 37 ÷ 9
= 4.11
The median is the middle number after the numbers have been arranged in an ascending or descend ...
This lecture looks at:
- An explanation of each of the steps in the research process flowchart
- Types of data
- Generating and testing theories
- Measurement error
- Validity
- Reliability
Page 266LEARNING OBJECTIVES· Explain how researchers use inf.docxkarlhennesey
Page 266
LEARNING OBJECTIVES
· Explain how researchers use inferential statistics to evaluate sample data.
· Distinguish between the null hypothesis and the research hypothesis.
· Discuss probability in statistical inference, including the meaning of statistical significance.
· Describe the t test and explain the difference between one-tailed and two-tailed tests.
· Describe the F test, including systematic variance and error variance.
· Describe what a confidence interval tells you about your data.
· Distinguish between Type I and Type II errors.
· Discuss the factors that influence the probability of a Type II error.
· Discuss the reasons a researcher may obtain nonsignificant results.
· Define power of a statistical test.
· Describe the criteria for selecting an appropriate statistical test.
Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES. In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.
SAMPLES AND POPULATIONS
Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements about populations. Would the results hold up if the experiment were conducted repeatedly, each time with a new sample?
In the hypothetical experiment described in Chapter 12 (see Table 12.1), mean aggression scores were obtained in model and no-model conditions. These means are different: Children who observe an aggressive model subsequently behave more aggressively than children who do not see the model. Inferential statistics are used to determine whether the results match what would happen if we were to conduct the experiment again and again with multiple samples. In essence, we are asking whether we can infer that the difference in the sample means shown in Table 12.1 reflects a true difference in the population means.
Recall our discussion of this issue in Chapter 7 on the topic of survey data. A sample of people in your state might tell you that 57% prefer the Democratic candidate for an office and that 43% favor the Republican candidate. The report then says that these results are accurate to within 3 percentage points, with a 95% confidence level. This means that the researchers are very (95%) confident that, if they were able to study the entire population rather than a sample, the actual percentage who preferred th ...
hypothesis-Meaning need for hypothesis qualities of good hypothesis type of hypothesis null and alternative hypothesis sources of hypothesis formulation of hypothesis, hypothesis testing
Clinical trials are the gold standard of evidence-based medicine. Properly designed clinical trials can lead to chance findings and potentially lead to erroneous conclusions.
Importantly, clinical trials can also be badly designed on purpose to increase the risk of false or chance findings leading to support misleading claims. Such techniques are frequently used by bad researchers and charlatans to substantiate their claims with biased clinical trials. It is therefore important to be weary of the limitations of clinical trials and understand how causal inference should be approach. In that presentation, I discuss the situations under which the risk of erroneous conclusions from clinical trials is increased and I discuss ways to identify and prevent bad clinical research.
The views expressed and presented in that presentation are my own views and may not represent the views of the National Institute for Health and Care Excellence.
Check out this brief paper if you want to know more about P value calculations. There is a misconception that a very small p value means the difference between groups is highly relevant. Looking at the p value alone deviates our attention from the effect size. Consider an experiment in which 10 subjects receive a placebo, and another 10 receive an experimental diuretic. After 8 h, the average urine output in the placebo group is 769 mL, versus 814 mL in the diuretic group—a difference of 45 mL. How do we know if that difference means the drug works and is not just a result of chance? Read on and let me know if you have any questions...
Hypothesis is a hunch the researcher or research team has. Basically a hypothesis is nothing more or less than a hunch to solve your research problem.
A good hypothesis by adding predictions on the how or why. So use sentences that include variations. If one cannot assess the predictions by observation or by experience, the hypothesis classes as not yet useful, and must wait for others who might come afterward to make possible the needed observations. For example, a new technology or theory might make the necessary experiments feasible.
Hypothesis testing and estimation are used to reach conclusions about a population by examining a sample of that population.
Hypothesis testing is widely used in medicine, dentistry, health care, biology and other fields as a means to draw conclusions about the nature of populations
Hypothesis Testing. Inferential Statistics pt. 2John Labrador
A hypothesis test is a statistical test that is used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. A hypothesis test examines two opposing hypotheses about a population: the null hypothesis and the alternative hypothesis.
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Session 9 intro_of_topics_in_hypothesis_testing
1. Training on Teaching
Basic Statistics for
Tertiary Level Teachers
Summer 2008
Note: Most of the Slides were prepared by Prof.
Josefina Almeda of the School of Statistics, UP
Diliman
Introduction to Topics
in Hypothesis Testing
2. TEACHING BASIC STATISTICS ….
Session 7.2
Two Areas of Inferential Statistics
Estimation
Point Estimation
Interval Estimation
Hypothesis Testing
3. TEACHING BASIC STATISTICS ….
Session 7.3
Research Problem: How effective is
Minoxidil in treating male pattern baldness?
Specific Objectives:
1. To estimate the population proportion of patients
who will show new hair growth after being
treated with Minoxidil.
2. To determine whether treatment using Minoxidil
is better than the existing treatment that is
known to stimulate hair growth among 40% of
patients with male pattern baldness.
Question: How do we achieve these objectives
using inferential statistics?
This can be answered
by ESTIMATION
This can be answered by
HYPOTHESIS TESTING
4. TEACHING BASIC STATISTICS ….
Session 7.4
What is Hypothesis Testing?
Hypothesis testing is an area of
statistical inference in which one
evaluates a conjecture about some
characteristic of the parent population
based upon the information contained
in the random sample.
Usually the conjecture concerns one
of the unknown parameters of the
population.
6. TEACHING BASIC STATISTICS ….
Session 7.6
Example of Hypothesis
The mean body temperature for patients
admitted to elective surgery is not equal to
37.0oC.
Note: The parameter of interest here is
which is the mean body
temperature for patients admitted to
elective surgery.
7. TEACHING BASIC STATISTICS ….
Session 7.7
Example of a Hypothesis
The proportion of registered voters in
Quezon City favoring Candidate A
exceeds 0.60.
Note: The parameter of interest here is p
which is the proportion of registered
voters in Quezon City favoring Candidate
A.
8. TEACHING BASIC STATISTICS ….
Session 7.8
Things to keep in mind
Analyze a sample in an attempt to
distinguish between results that can
easily occur and results that are
highly unlikely
We can explain the occurrence of highly
unlikely results by saying either that a rare
event has indeed occurred or that things
aren’t as they are assumed to be.
9. TEACHING BASIC STATISTICS ….
Session 7.9
Example to illustrate the basic
approach in testing hypothesis
A product called “Gender Choice”
claims that couples “increase their
chances of having a boy up to 85%, a
girl up to 80%.” Suppose you conduct
an experiment that includes 100
couples who want to have baby girls,
and they all follow the Gender Choice
to have a baby girl.
10. TEACHING BASIC STATISTICS ….
Session 7.10
Con’t of Example
Using common sense and no real
formal statistical methods, what
should you conclude about the
effectiveness of Gender Choice if
the 100 babies include
a. 52 girls
b. 97 girls
11. TEACHING BASIC STATISTICS ….
Session 7.11
Solution to Example:
a. We normally expect around 50 girls
in 100 births. The result of 52 girls is
close but higher than 50, so we should
conclude that Gender Choice is
effective. If the 100 couples used no
special methods of gender selection,
the result of 52 girls could easily occur
by chance.
12. TEACHING BASIC STATISTICS ….
Session 7.12
Solution to Example
b. The result of 97 girls in 100 births is
extremely unlikely to occur by chance. It
could be explained in one of two ways:
Either an extremely rare event has
occurred by chance, or Gender Choice is
effective because of the extremely low
probability of getting 97 girls by chance,
the more likely explanation is that the
product is effective.
13. TEACHING BASIC STATISTICS ….
Session 7.13
Explanation of Example
We should conclude that the product is
effective only if we get significantly more
girls that we would expect under normal
circumstances.
Although the outcomes of 52 girls and 97
girls are both “above average,” the result of
52 girls is not significant, whereas 97 girls
does constitute a significant result.
15. TEACHING BASIC STATISTICS ….
Session 7.15
Null Hypothesis
denoted by Ho
the statement being tested
it represents what the experimenter doubts to
be true
must contain the condition of equality and
must be written with the symbol =, , or
When actually conducting the
test, we operate under the
assumption that the
parameter equals some
specific value.
16. TEACHING BASIC STATISTICS ….
Session 7.16
For the mean, the null hypothesis will be
stated in one of these three possible forms:
Ho: = some value
Ho: some value
Ho: some value
Note: the value of can be obtained
from previous studies or from
knowledge of the population
17. TEACHING BASIC STATISTICS ….
Session 7.17
Example of Null Hypothesis
The null hypothesis corresponding to the
common belief that the mean body
temperature is 37oC is expressed as
Ho:
We test the null hypothesis directly in
the sense that we assume it is true and
reach a conclusion to either reject Ho or
fail to reject Ho.
Co
37
18. TEACHING BASIC STATISTICS ….
Session 7.18
Alternative Hypothesis
denoted by Ha
Is the statement that must be true if the null
hypothesis is false
the operational statement of the theory that
the experimenter believes to be true and
wishes to prove
Is sometimes referred to as the research
hypothesis
19. TEACHING BASIC STATISTICS ….
Session 7.19
For the mean, the alternative hypothesis will be
stated in only one of three possible forms:
Ha: some value
Ha: > some value
Ha: < some value
Note: Ha is the opposite of Ho. For example,
if Ho is given as = 37.0, then it follows
that the alternative hypothesis is given
by Ha: 37.0.
20. TEACHING BASIC STATISTICS ….
Session 7.20
Note About Using or in Ho:
Even though we sometimes express Ho
with the symbol or as in Ho: 37.0
or Ho: 37.0, we conduct the test by
assuming that = 37.0 is true.
We must have a single fixed value for
so that we can work with a single
distribution having a specific mean.
21. TEACHING BASIC STATISTICS ….
Session 7.21
Note About Stating Your Own
Hypotheses:
If you are conducting a research study
and you want to use a hypothesis test
to support your claim, the claim must
be stated in such a way that it
becomes the alternative hypothesis, so
it cannot contain the condition of
equality.
22. TEACHING BASIC STATISTICS ….
Session 7.22
If you believe that your brand of
refrigerator lasts longer than the mean
of 14 years for other brands, state the
claim that > 14, where is the
mean life of your refrigerators.
Ho: = 14 vs. Ha: > 14
Example in Stating your Hypothesis
23. TEACHING BASIC STATISTICS ….
Session 7.23
Some Notes:
In this context of trying to support the goal of
the research, the alternative hypothesis is
sometimes referred to as the research
hypothesis.
Also in this context, the null hypothesis is
assumed true for the purpose of conducting
the hypothesis test, but it is hoped that the
conclusion will be rejection of the null
hypothesis so that the research hypothesis is
supported.
24. TEACHING BASIC STATISTICS ….
Session 7.24
Suppose that the government is deciding
whether or not to approve the manufacturing of a
new drug. A drug is to be tested to find out if it
can dissolve cholesterol deposits in the heart’s
arteries. A major cause of heart diseases is the
hardening of the arteries caused by the
accumulation of cholesterol. The Bureau of
Food and Drug (BFaD) will not allow the
marketing of the drug unless there is strong
evidence that it is effective.
Example of a Research Problem
25. TEACHING BASIC STATISTICS ….
Session 7.25
Con’t of Research Problem
A random sample of 98 middle-aged men has been
selected for the experiment. Each man is given a
standard daily dosage of the drug for 2 consecutive
weeks. Their cholesterol levels are measured at the
beginning and at the end of the test. The interest is to
determine if the intake of the drug lead to reduced
cholesterol levels; that is, a hypothesis test will have to
be performed to determine if the drug is effective or not.
Based on the results of the experiment, the director of
BFaD will decide whether to release the drug to the
public or postpone its release and request for more
research.
26. TEACHING BASIC STATISTICS ….
Session 7.26
Con’t of Research Problem
To perform a statistical hypothesis test, we must
firstly identify the parameter of interest, and have
some educated guess about the true value of the
parameter. In the case of the BFaD example, the
possible states of the drug’s effectiveness are
referred to as hypotheses. Because the director
wants only to know whether it is effective or not,
either of the following hypotheses applies.
The drug is ineffective.
The drug is effective.
27. TEACHING BASIC STATISTICS ….
Session 7.27
Con’t of Research Problem
To measure the effectiveness of the drug for each
middle-aged man, we can look at the percent change
in cholesterol levels experienced by all middle-aged
men who took the drug before and after they took the
drug.
We summarize effectiveness in terms of the
population mean.
Let be the population mean of the percent change in
cholesterol levels. BFaD decides to classify the drug
as effective only if, on the average, it reduces the
cholesterol levels by more than 30% ( 30%).
28. TEACHING BASIC STATISTICS ….
Session 7.28
Stating the Null Hypothesis
The null hypothesis represents no
practical change in cholesterol levels
before and after the drug use. In terms of
, we say
Ho: 30%
This means that the cholesterol level is reduced
by 30% or less.
29. TEACHING BASIC STATISTICS ….
Session 7.29
Stating the Alternative Hypothesis
Ha: 30%
Whenever sample results fail to support the
null hypothesis, the conclusion we accept is
the alternative hypothesis. In our illustration,
if results from the sample of percentage
change in cholesterol levels fail to support Ho,
then the director concludes Ha and says that
the drug is effective.
30. TEACHING BASIC STATISTICS ….
Session 7.30
What is a Test of Significance?
• A test of significance is a problem of
deciding between the null and the alternative
hypotheses on the basis of the information
contained in a random sample.
• The goal will be to reject Ho in favor of Ha,
because the alternative is the hypothesis that
the researcher believes to be true. If we are
successful in rejecting Ho, we then declare
the results to be “significant”.
31. TEACHING BASIC STATISTICS ….
Session 7.31
Note About Testing the Validity of
Someone Else’s Claim
Sometimes we test the validity of someone else’s claim,
such as the claim of the Coca Cola Bottling Company that
“the mean amount of Coke in cans is at least 355 ml,”
which becomes the null hypothesis of Ho: 355
In this context of testing the validity of someone
else’s claim, their original claim sometimes
becomes the null hypothesis (because it contains
equality), and it sometimes becomes the
alternative hypothesis (because it does not
contain the equality).
33. TEACHING BASIC STATISTICS ….
Session 7.33
Type I Error
The mistake of rejecting the null hypothesis
when it is true.
It is not a miscalculation or procedural
misstep; it is an actual error that can occur
when a rare event happens by chance.
The probability of rejecting the null
hypothesis when it is true is called the
significance level ( ).
The value of is typically predetermined,
and very common choices are = 0.05
and = 0.01.
34. TEACHING BASIC STATISTICS ….
Session 7.34
Examples of Type I Error
1. The mistake of rejecting the null
hypothesis that the mean body
temperature is 37.0 when that mean
is really 37.0.
2. BFaD allows the release of an
ineffective medicine
35. TEACHING BASIC STATISTICS ….
Session 7.35
Type II Error
The mistake of failing to reject the
null hypothesis when it is false.
The symbol (beta) is used to
represent the probability of a type II
error.
36. TEACHING BASIC STATISTICS ….
Session 7.36
Examples of Type II Errors
1. The mistake of failing to reject the
null hypothesis ( = 37.0) when it
is actually false (that is, the mean
is not 37.0).
2. BFaD does not allow the release
of an effective drug.
37. TEACHING BASIC STATISTICS ….
Session 7.37
Summary of Possible Decisions in
Hypothesis Testing
True Situation
Decision
The null
hypothesis
is true.
The null
hypothesi
s is false.
We decide
to reject the
null
hypothesis.
TYPE I error
(rejecting a
true null
hypothesis)
CORRECT
decision
We fail to
reject the
null
hypothesis.
CORRECT
decision
TYPE II error
(failing to
reject
a false null
hypothesis)
TRIAL: The accused did not do it.
TRUTH
VERDICT INNOCEN
T
GUILTY
GUILTY Type 1
Error
Correct
INNOCEN
T
Correct Type 2
Error
ANALOGY
38. TEACHING BASIC STATISTICS ….
Session 7.38
The experimenter is free to determine . If the
test leads to the rejection of Ho, the researcher
can then conclude that there is sufficient evidence
supporting Ha at level of significance.
Usually, is unknown because it’s hard to
calculate it. The common solution to this difficulty
is to “withhold judgment” if the test leads to the
failure to reject Ho.
and are inversely related. For a fixed
sample size n, as decreases increases and
vice-versa.
Controlling Type I and Type II
Errors
39. TEACHING BASIC STATISTICS ….
Session 7.39
In almost all statistical tests, both and
can be reduced by increasing the sample
size.
Controlling Type I and Type II
Errors
Because of the inverse relationship of
and , setting a very small should also
be avoided if the researcher cannot afford
a very large risk of committing a Type II
error.
40. TEACHING BASIC STATISTICS ….
Session 7.40
Common Choices
of
Consequences of
Type I error
0.01 or smaller
0.05
0.10
very serious
moderately serious
not too serious
The choice of usually depends on the
consequences associated with making a
Type I error.
Controlling Type I and Type II
Errors
41. TEACHING BASIC STATISTICS ….
Session 7.41
Controlling Type I and Type II
Errors
The usual practice in research and industry
is to determine in advance the values of
and n, so the value of is determined.
Depending on the seriousness of a type I
error, try to use the largest that you can
tolerate.
For type I errors with more serious
consequences, select smaller values of .
Then choose a sample size n as large as is
reasonable, based on considerations of
time, cost, and other such relevant factors.
42. TEACHING BASIC STATISTICS ….
Session 7.42
Example to illustrate Type I and
Type II Errors
Consider M&Ms (produced by Mars,
Inc.) and Bufferin brand aspirin tablets
(produced by Bristol-Myers Products).
The M&M package contains 1498
candies. The mean weight of the
individual candies should be at least
0.9085 g., because the M&M package
is labeled as containing 1361 g.
43. TEACHING BASIC STATISTICS ….
Session 7.43
The Bufferin package is labeled as holding 30
tablets, each of which contains 325 mg of
aspirin.
Because M&Ms are candies used for
enjoyment whereas Bufferin tablets are drugs
used for treatment of health problems, we are
dealing with two very different levels of
seriousness.
Example to illustrate Type I and
Type II Errors
44. TEACHING BASIC STATISTICS ….
Session 7.44
If the M&Ms don’t have a population
mean weight of 0.9085 g, the
consequences are not very serious,
but if the Bufferin tablets don’t have a
mean of 32.5 mg of aspirin, the
consequences could be very serious.
Example to illustrate Type I and
Type II Errors
45. TEACHING BASIC STATISTICS ….
Session 7.45
If the M&Ms have a mean that is too
large, Mars will lose some money but
consumers will not complain.
In contrast, if the Bufferin tablets have
too much aspirin, Bristol-Myers could
be faced with consumer lawsuits.
Example to illustrate Type I and
Type II Errors
46. TEACHING BASIC STATISTICS ….
Session 7.46
Consequently, in testing the claim that
= 0.9085 g for M&Ms, we might choose
= 0.05 and a sample size of n = 100.
In testing the claim of = 325 mg for
Bufferin tablets, we might choose = 0.01
and a sample size of n = 500.
The smaller significance level and large
sample size n are chosen because of the more
serious consequences associated with the
commercial drug.
Example to illustrate Type I and
Type II Errors
47. TEACHING BASIC STATISTICS ….
Session 7.47
The test statistic should tend to take on certain
values when Ho is true and different values when
Ha is true.
The decision to reject Ho depends on the value of
the test statistic
• A decision rule based on the value of the test
statistic:
Reject Ho if the computed value of the
test statistic falls in the region of rejection.
The Test Statistic - a statistic computed from the
sample data that is especially sensitive to the differences
between Ho and Ha
48. TEACHING BASIC STATISTICS ….
Session 7.48
Factors that Determine the Region of
Rejection
the behavior of the test statistic if the null hypotheses
were true
the alternative hypothesis: the location of the region
of rejection depends on the form of Ha
level of significance ( ): the smaller is,
the smaller the region of rejection
Region of Rejection or Critical Region-the set
of all values of the test statistic which will lead to the
rejection of Ho
49. TEACHING BASIC STATISTICS ….
Session 7.49
Critical Value/s
the value or values that separate the
critical region from the values of the
test statistic that would not lead to
rejection of the null hypothesis.
It depends on the nature of the null
hypothesis, the relevant sampling
distribution, and the level of
significance.
50. TEACHING BASIC STATISTICS ….
Session 7.50
Types of Tests
Two-tailed Test. If we are primarily concerned with
deciding whether the true value of a population
parameter is different from a specified value, then
the test should be two-tailed. For the case of the
mean, we say Ha: 0.
Left-tailed Test. If we are primarily concerned with
deciding whether the true value of a parameter is
less than a specified value, then the test should be
left-tailed. For the case of the proportion, we say
Ha: P P0.
Right-tailed Test. If we are primarily concerned
with deciding whether the true value of a parameter
is greater than a specified value, then we should
use the right-tailed test. For the case of the standard
deviation, we say Ha: 0.
51. TEACHING BASIC STATISTICS ….
Session 7.51
Level of Significance
and the Rejection Region
Ho: 30
Ha: < 30
0
0
0
Ho: 30
Ha: > 30
Ho: 30
Ha: 30
/2
Critical
Value(s)
Rejection
Regions
52. TEACHING BASIC STATISTICS ….
Session 7.52
The p-value - the smallest level of
significance at which Ho will be rejected
based on the information contained in the
sample
An Alternative Form of Decision Rule
(based on the p-value)
Reject Ho if the p-value is less than or
equal to the level of significance ( ).
53. TEACHING BASIC STATISTICS ….
Session 7.53
If the level of significance =0.05,
p-value Decision
0.01 Reject Ho.
0.05 Reject Ho.
0.10 Do not reject Ho
Example of Making Decisions
Using the p-value
54. TEACHING BASIC STATISTICS ….
Session 7.54
Conclusions in Hypothesis Testing
1. Fail to reject the null hypothesis Ho.
2. Reject the null hypothesis Ho.
Notes:
Some texts say “accept the null hypothesis”
instead of “fail to reject the null hypothesis.”
Whether we use the term accept or fail to
reject, we should recognize that we are not
proving the null hypothesis; we are merely
saying that the sample evidence is not strong
enough to warrant rejection of the null
hypothesis.
55. TEACHING BASIC STATISTICS ….
Session 7.55
Wording of Final Conclusion
Start
Does
the
original
claim contain
the condition
of
equality
No (Original claim
does not contain
equality and
becomes Ha)
Do you reject
Ho?
“The sample data
support the claim
that….(original
claim).”
Yes
(Reject Ho)
No
(Fail to
Reject Ho)
“There is no sufficient
sample evidence to
support the claim
that….(original claim).”
(This is the
only case in
which the
original claim
is
supported.)
56. TEACHING BASIC STATISTICS ….
Session 7.56
Wording of Final Conclusion
Start
Does
the
original
claim contain
the condition
of
equality
Yes (Original claim
contains equality
and becomes Ho)
Do you reject
Ho?
“There is sufficient
evidence to
warrant rejection of
the claim
that….(original
claim).”
Yes
(Reject Ho)
No
(Fail to Reject
Ho)
“There is no sufficient
sample evidence to
warrant rejection of the
claim that….(original
claim).”
(This is the
only case in
which the
original claim
is rejected.)
57. TEACHING BASIC STATISTICS ….
Session 7.57
Example in Making Final
Conclusion
If you want to justify the claim that the
mean body temperature is different
from 37.0oC, then make the claim that
37.0. This claim will be an
alternative hypothesis that will be
supported if you reject the null
hypothesis of Ho: = 37.0.
58. TEACHING BASIC STATISTICS ….
Session 7.58
Example in Making Final
Conclusion
If, on the other hand, you claim that
the mean body temperature is
37.0oC, that is = 37.0, you will
either reject or fail to reject the
claim; in either case, you will not
support the original claim.
59. TEACHING BASIC STATISTICS ….
Session 7.59
A Summary of the Steps in Hypothesis Testing
Determine the objectives of the experiment
(responsibility of the experimenter).
1. State the null and alternative hypotheses.
2. Decide on a level of significance, .
Determine the testing procedure and methods of
analysis (responsibility of the statistician).
3. Decide on the type of data to be collected and
choose an appropriate test statistic and testing
procedure.
60. TEACHING BASIC STATISTICS ….
Session 7.60
4. State the decision rule.
5. Collect the data and compute for the value of
the test statistic using the sample data.
6. a) If decision rule is based on region of
rejection: Check if the test statistic falls in the
region of rejection. If yes, reject Ho.
b) If decision rule is based on p-value:
Determine the p-value. If the p-value is less
than or equal to , reject Ho.
7. Interpret results.
Con’t of Steps in Hypothesis
Testing