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STATISTICS & PROBABILITY
Hypothesis Testing
What is a Hypothesis?
I assume the mean GPA of this class
is 3.5!
• an assumption about
the population
parameter
• an educated guess
about the population
parameter
Hypotheses Testing: This is the process of making an inference or generalization on
population parameters based on the results of the study on samples.
Statistical Hypotheses: It is a guess or prediction made by the researcher
regarding the possible outcome of the study.
Reject?Accept?
Hypotheses Testing
is deciding between what is REALITY and what
is COINCIDENCE!
Null Hypothesis (Ho): is always hoped to be rejectedAlways contains “=“ sign
Alternative Hypothesis (Ha): •Challenges Ho•Never contains “=“ sign•Uses “< or > or “•It generally represents the idea which the researcher wants to prove.
Types of Statistical Hypotheses
The Null Hypothesis: Ho
Ha: The average GPA of this class is
a) higher than 3.5 (Ha: 3.5)
b) lower than 3.5 (Ha: 3.5)
c) not equal to 3.5 (Ha: 3.5)
The Alternative Hypothesis: Ha
Ex. Ho: The average GPA of this class is 3.5
H0: = 3.5
Types of Hypotheses Tests
1. One-tailed left directional test– this is used if Ha uses symbol
Rejection regionArea = 0.05
Acceptanceregion
= 0.05
Critical value is obtained
from the table
Types of Hypotheses Tests2. One-tailed right directional test
– this is used if Ha uses symbol
Rejection regionArea= 0.05
Acceptanceregion
Critical value is obtained
from the table = 0.05
Types of Hypotheses Tests
3. Two-tailed test: Non-directional – this is used if Ha uses symbol
Rejection region Rejection regionArea=.025 Area=.025
Acceptanceregion
= 0.05/2Critical value is
obtainedfrom the table
Level of Significance, and the Rejection Region
means the probability of being right is 95% , and the probability of being wrong is 5%. So what is = 0.01?
.
05.0
Acceptance Region
Rejection regionArea is 0.05
Level of Significance, and the Rejection Region
means the researcher is taking a 1% risk of being
wrong and a 99% risk of being right.So, what is = 0.05?
0.01
Acceptance Region
Rejection regionArea is 0.01
Level of Significance, and the Rejection Region
means the probability of committing Type I error is 5%.
Acceptanceregion
= 0.05/2= 0.025
Rejection region Rejection regionArea=.025 Area=.025
= 0.05, since it is 2-T, then
Level of Significance, and the Rejection Region
To summarize:
means the researcher is taking a 1% risk of being
wrong and a 99% risk of being right.So, what is = 0.05?
means the probability of being right is 95% and the probability of being wrong is 5%. So what is = 0.01?
means the probability of committing Type I error is 5%.
So what is = 0.01?
© 1984-1994 T/Maker Co.
Errors in Hypothesis Testing
Errors
Errors in Decisions
Errors in Conclusions
Type I ( error )Rejecting a true
Ho!
Ho: ERAP is not guilty
If the court convictsERAP, when in facthe is not guilty, the
court commitsType I error!
Type I is the same as the or the level of significance.
Errors in Hypothesis Testing
Errors
Errors in Decisions
Errors in Conclusions
Ho: ERAP is not guilty
If the court acquitsERAP, when in fact
he is guilty, thecourt commitsType II error!
Type II ( error )Accepting a false
Ho!
© 1984-1994 T/Maker Co.
Decisions made regarding Ho(Reject Ho/Do not reject Ho)
If we reject Ho, it means it is wrong!
If we do not reject Ho, it doesn’t mean it is correct,
we just don’t have enough evidence
to reject it!
Testing the Significance of Difference Between Means
Z-testn 30 is known
t-testn < 30
is unknown
F-test(ANOVA)
3 or more s
Testing the Significance of Difference Between Means“n is large or when n 30 & is known.”
• Hypothesized/population mean VS Sample mean and population standard deviation is known.
nx
Z
x - is the sample mean - is the population meann - is the sample size - is the population std. dev.
Z-testn 30
is known
Using PHStat: Go to… “One-Sample Tests; Z-Test for the Mean:Sigma Known”
• Sample mean 1 VS Sample mean 2 and population standard deviation is known.
Testing the Significance of Difference Between Means“n is large or when n 30 & is known.”
21
21
11nn
xxZ
1x - is the mean of sample 1
2x - is the mean of sample 2
21 & nn - are the sample sizes - is the population std. dev.
Z-testn 30 is known
• Sample mean 1 VS Sample mean 2 and 2 sample standard deviations are known.
Testing the Significance of Difference Between Means“n is large or when n 30 & is known.”
2
22
1
21
21
ns
ns
xxZ
1x - is the mean of sample 1
2x - is the mean of sample 2
21 & nn - are the sample sizes21 & ss - are the sample std. devs.
Z-testn 30 is unknown
Using Microsoft Excel: Go to…“Z-Test: Two-Sample For Means”
The Critical Value Approach in Testing the Significance of Difference
Between MeansThe 5-step solution
Step 1. Formulate Ho and Ha
Step 2. Set the level of significance , usually it is given in the problem.
Step 3. Formulate the decision rule (when to reject Ho); Find the critical value/P-value.
Step 4. Make your decision.
Step 5. Formulate your conclusion.
Approaches inHypothesis Testing
Critical valueapproach
p- valueapproach
Computed vs. Critical5-step solution1.Ho: ___________ Ha: ___________2. = ___; Cri-value= ______ 3. Decision rule: Reject Ho if 4. Decision:5. Conclusion:
valueCrivalueComp
p-value vs. 5-step solution1.Ho : ___________ Ha : ___________2. = ___; p- value=________ 3. Decision rule: Reject Ho if p- value 4. Decision:5. Conclusion:
Z=1.65
What Is Z Given = 0.05?
Finding Critical Values: One-Tailed
.4505.44951.6
.4394.43821.5
54Z = .05
.45
.05
Critical value
Critical Values: Z - Table
1.96 2.58Two-T
1.652.33One-T
0.05.01 Type
You will refer to this table to get the critical value of Zor the .tabularZ
CRITERION:
1. One-tailed test (right directional)
“Reject H0 if Zc ≥ Zt “
2. One-tailed test (left directional)
“Reject H0 if Zc ≤ Zt
3. Two-tailed test (Zc = +)
“Reject H0 if Zc ≥ Zt “
4. Two-tailed test (Zc = -)
‘Reject H0 if Zc ≤ Zt “
EXERCISES:
1. Past records showed that the average final examination grade of students in Statistics was 70 with standard deviation of 8.0. A random sample of 100 students was taken and found to have a mean final examination grade of 71.8. Is this an indication that the sample grade is better than the rest of the students? Test at 0.05 level of significance.
2. A certain type of battery is known to have a mean life of 60 hours. In random sample of 40 batteries, the mean life was found to be 58 hours with a standard deviation of 4.5 hours. Does it indicate that the mean lifetime of such battery has been reduced? Test at 0.01 level of significance.
3. The manager of a rent-a-car business wants to know whether the true average numbers of cars rented a day is 25 with a standard deviation of 6.9 rentals. A random sample of 30 days was taken and found to have an average of 22.8 rentals. Is there a significance between the mean and the sample mean? Test at 0.05 level of significance.
4. Advertisements claim that the average nicotine content of a certain kind of cigarette is 0.30 milligram. Suspecting that this figure is too low, a consumer protection service takes a random sample of 50 of these cigarette from different production slots and find that their nicotine content has a mean of 0.33 milligram with a standard deviation of 0.18 milligram. Use the 0.05 level of significance to test the null hypothesis µ = 0.30 against the alternative hypothesis µ < 0.30.
5. An experiment was planned to compare the mean time (in days) required to recover from common cold for person given a daily doze of 4 mgs. of vitamin C versus those who were not given a vitamin supplement. Suppose that 35 adults were randomly selected for each treatment category and that the mean recovery times and standard deviations for the 2 groups were as follows:
n X δ W/ vit. C 35 5.8 1.2 W/o vit. C 35 6.9 5.8 Suppose your research objective is to show that the use
of vit. C increases the mean time required to recover from common cold. Test using α = 0.05.