Hypothesis testing
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Transcript of Hypothesis testing
Hypothesis TestingThe Basics
Advanced StatisticsSRSTHS
Mrs. Ma. Cristina C. Pegollo
What is Hypothesis? in statistics, is a claim or statement
about a property of a population an educated guess about the population
parameter
What is hypothesis testing?
This is the process of making an inference or generalization on population parameters based on the results of the study on samples.
What is statistical hypothesis?
It is a guess or prediction made by the researcher regarding the possible outcome of the study.
Fundamentals of
Hypothesis Testing
Central Limit Theorem
If n (the sample size) is large, the theoretical sampling distribution of the mean can be approximated closely with a normal distribution.
If researchers increase the samples to a considerable number, the shape of the distribution approximates a
normal curve.
µx = 98.6
Central Limit TheoremThe Expected Distribution of Sample Means
Assuming that = 98.6
Likely sample means
z = - 1.96
x = 98.48
or
z = 1.96
x = 98.72
or
µx = 98.6
Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means
Assuming that = 98.6
Likely sample means
Components of aFormal Hypothesis Test
Null Hypothesis: H0 This is the statement that is under investigation or
being tested.
It is always hoped to be rejected.
Usually the null hypothesis represents a statement
of “no effect”, “no difference”, or , put another way,
“things haven’t changed.”
Must contain condition of equality
=, , or
Reject H0 or fail to reject H0
Alternative Hypothesis: H1
This is the statement you will adopt in the situation in which the evidence (data) is so strong that you reject H0
‘opposite’ of H0
, <, >
generally represents the idea which the researcher wants to prove.
Example1: Null and Alternate Hypothesis
Null HypothesisHo: The average grade of this class is 88.
(Ho:
Alternative HypothesisH1: The average grade of this class is
a. Higher than 88 (Ho:
b. Lower than 88 (Ho:
c. Not equal to 88 (Ho:
Example 2The researcher wants to know if
online learning has increased the average grade of Einstein students from 80%
Ho : Online learning has not increased the average grade of Einstein studentsHo : ; Online learning has increased the average grade of Einstein students
Example 3The management of Starstruck canteen
wants to know if the services in the newly renovated canteen are different from the services before the renovation took place.Ho : The services in the newly renovated Starstruck canteen have not changed (meaning, the services now are just the same as before.)H1 : The services in the newly renovated Starstruck canteen have changed (meaning, the services now are different as before.)
Example 4The mathematics coordinator wants to
know if the number of failures in all math courses is lower than 25% of the total enrollment.
Ho : p The number of failures in all math courses is 25%
H1 : p< 25 The number of failures in all math courses is lower than 25%
Example 5
A car manufacturer advertises that its new subcompact models get 45 miles per gallon(mpg). Let be the mean of the mileage distribution for these cars. You assume that the manufacturer will not underrate the car, but you suspect that the mileage might be overrated.
Ho : µ (Car manufacturer’s claim)
H1 : µ
Example 6
A company manufactures ball bearings for precision machines. The average diameter of a certain type of ball bearing should be 6.0mm. To check that the average diameter is correct , the company formulates a statistical test.
Ho : µ (Car manufacturer’s claim)
H1 : µ
Note about Forming Your Own Claims (Hypotheses)
If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.
Note about Testing the Validity of Someone Else’s Claim
Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality).
Exercises:Formulate the null and alternative hypotheses of the following research problems1. A manager wants to know if the average length of
time for board meetings is 3 hours.2. The researchers want to know if the proportion of car
accidents in Balete Drive has increased from 5% of the total car accidents recorded in Quezon City in a day.
3. A teacher wants to know if there is a significant difference in the academic performance between Pasteur and Linnaeus students.
4. The registrar wants to know if the average encoding time is lower than 30 minutes.
5. The Discipline officer wants to know if the new policy on smoking has reduced the number of smokers this year than the previous year.
Test Statistic
a value computed from the sample data that is used in making the decision about the
rejection of the null hypothesis
Test Statistic
a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis
For large samples, testing claims about population means
z = x - µxn
Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
CriticalRegion
Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
CriticalRegion
Critical Region
Set of all values of the test statistic that would cause a rejection of the null hypothesis
CriticalRegions
Significance Level denoted by the probability that the test
statistic will fall in the critical region when the null hypothesis is actually true.
common choices are 0.05, 0.01, and 0.10
Critical ValueValue or values that separate the critical region
(where we reject the null hypothesis) from the values of the test statistics that do not lead
to a rejection of the null hypothesis
Critical ValueValue or values that separate the critical region
(where we reject the null hypothesis) from the values of the test statistics that do not lead
to a rejection of the null hypothesis
Critical Value( z score )
Critical ValueValue or values that separate the critical region
(where we reject the null hypothesis) from the values of the test statistics that do not lead
to a rejection of the null hypothesis
Critical Value( z score )
Fail to reject H0Reject H0
Two-tailed,Right-tailed,Left-tailed Tests
The tails in a distribution are the extreme regions bounded
by critical values.
Two-tailed TestH0: µ = 100
H1: µ 100
Two-tailed TestH0: µ = 100
H1: µ 100 is divided equally between the two tails of the critical
region
Two-tailed TestH0: µ = 100
H1: µ 100
Means less than or greater than
is divided equally between the two tails of the critical
region
Two-tailed TestH0: µ = 100
H1: µ 100
Means less than or greater than
100
Values that differ significantly from 100
is divided equally between the two tails of the critical
region
Fail to reject H0Reject H0 Reject H0
Right-tailed TestH0: µ 100
H1: µ > 100
Right-tailed TestH0: µ 100
H1: µ > 100
Points Right
Right-tailed TestH0: µ 100
H1: µ > 100
Values that differ significantly
from 100100
Points Right
Fail to reject H0 Reject H0
Left-tailed Test
H0: µ 100
H1: µ < 100
Left-tailed Test
H0: µ 100
H1: µ < 100
Points Left
Left-tailed Test
H0: µ 100
H1: µ < 100
100
Values that differ significantly
from 100
Points Left
Fail to reject H0Reject H0
Conclusions in Hypothesis Testing
always test the null hypothesis
1. Reject the H0
2. Fail to reject the H0
need to formulate correct wording of final conclusion
See Figure 7-4
FIGURE 7-4 Wording of Final Conclusion
Does theoriginal claim contain
the condition ofequality
Doyou rejectH0?.
Yes
(Original claim
contains equality
and becomes H0)
No(Original claimdoes not containequality and
becomes H1)
Yes
(Reject H0)
“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“The sample datasupports the claim that . . . (original claim).”
“There is not sufficientevidence to support the claimthat. . . (original claim).”
Doyou reject
H0?
Yes
(Reject H0)
No(Fail to
reject H0)
No(Fail to
reject H0)
(This is theonly case inwhich theoriginal claimis rejected).
(This is theonly case inwhich theoriginal claimis supported).
Start
Accept versus Fail to Reject some texts use “accept the null hypothesis
we are not proving the null hypothesis
sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect)
If you reject Ho, iit means it is wrong!
If you fail to reject Ho , it doesn’t mean it is correct – you simply do not have enough evidence to reject it!
Type I ErrorThe mistake of rejecting the null hypothesis
when it is true.
(alpha) is used to represent the probability of a type I error
Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6
Type II Errorthe mistake of failing to reject the null
hypothesis when it is false.
ß (beta) is used to represent the probability of a type II error
Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6
Table 7-2 Type I and Type II Errors
True State of Nature
We decide to
reject the
null hypothesis
We fail to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)
Type II error
(rejecting a false
null hypothesis)
Correct
decision
Correct
decision
Decision
Controlling Type I and Type II ErrorsFor any fixed , an increase in the sample
size n will cause a decrease in
For any fixed sample size n , a decrease in will cause an increase in . Conversely, an increase in will cause a decrease in .
To decrease both and , increase the sample size.
Approaches in Hypothesis Testing
1. Critical Value Approach
2. P-value Approach
5-step solution1. Ho: Ha:2. ; Critical Value=_____3. Decision rule: Reject Ho if 4. Decision:5. Conclusion:
Example:The average score in the final examination in College Algebra at ABC University is known to be 80 with a standard deviation of 10. A random sample of 39 students was taken from this year’s batch and it was found that they have a mean score of 84. Test at 0.05 level of significance.