Test of hypothesis (z)
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Statistical Test of Hypothesis
STATISTICS
DATA
central tendency variability
distributions
graphicalrepresentation
HYPOTHESIS TESTING
http://www.RapidLearningCenter.com
is the study of
has
can be classifiedinto
are displayed with
is tested with
Statistical test of hypothesis
step by step process
used to verify a claim
using data
from a scientific study
Statistically significant a result that has been predicted as
unlikely to have occurred by chance alone, according to a significance
level.
Significance levell α (read as alpha)
a pre-determined threshold probability for a
test of hypothesis.
Definitions
hypothesis is a tentative explanation for an observation, phenomenon, or scientific problem that can be tested by further investigation.
In other words a scientific guess.
Two classifications
Null Hypothesis (HO) Alternative Hypothesis (HA)
status quo or default positionmaintained hypothesis or
research hypothesis
-no relationship between two measured phenomena
- that a treatment has no effect-no difference when compared
- relationship exist- treatment has effect
-significant difference when compared
already accepted accepted only when the null hypothesis is rejected
denoted by
= equality
denoted by
≠ only concerned that null hypothesis is not true.
>, < direction is important
ProcessProblem or Claim is posted
Null Hypothesis Alternative Hypothesis
are formulated
step 1
significance levelstep 2 is agreed upon
critical value* is determined
step 3 Appropriate Test* is identified and applied
step 4 Conclusionsare made by comparing test result and critical value"Null hypothesis is either accepted or rejected"
step 5 INTERPRETATION problem is solved or claim is verified
Interpretation of Significance level
Significance level Interpretation
α≤ .01 very strong evidence against the null hypothesis
.01< α ≤ .05 moderate evidence against the null hypothesis
.05< α ≤ .10 suggestive evidence against tthe null hypothesis
α> .10 little or no real evidence against tthe null hypothesis
Large-Sample Tests of Hypothesis
Population Mean vs Sample Mean
Population
samplesample
xHo :
xHa
xHa
xHa
:
:
:Assumptions:Large sample (n ≥ 30)Sample is randomly selected
Testing Population Mean
Example:
Test the hypothesis that weight loss in a new diet program exceeds 20 pounds during the first month.
Sample data : n = 36, mean = 21, s = 5, μ0 = 20, α = 0.05
Assumptions:Large sample (n ≥ 30)Sample is randomly selected n
s
xz 0
test statistics
Solution:
Step1 : H0 : μ = 20 (μ is not larger than 20)
Ha : μ > 20 (μ is larger than 20)Step2 : α = 0.05 zα (critical value) =1.645
20.1
)6(5
1
365
2021
0
z
z
z
ns
xz
Testing Population MeanStep3 : Z test since n ≥ 30
Step4: Decision: Accept Ho
Conclusion: At 5% significance level there is insufficient statistical evidence to concludethat weight loss in a new diet program exceeds 20 pounds per first month.
population 2 orsample 2
population 2 orsample 2
population 1 orsample 1
Test Concerning Two Means
Population vs Population or Sample vs Sample
population 1 orsample 1
21
21
:
:
xxHo
Ho
2121
2121
2121
:,:
:,:
:,:
xxHaHa
xxHaHa
xxHaHa
Comparing Two Population Means
Example: A random sample of 35 baby boys showed a mean birth weight of 7.4 lbs with a standard deviation 1.18 lbs while 40 baby girls showed a mean birth weight of 6.5 lbs with a standard deviation 1.5 lbs. Test if there are gender differences at 1% level of significance.
Assumptions:
1. Large samples ( n1 ≥ 30; n2 ≥ 30)2. Samples are randomly selected3. Samples are independent 2
22
1
21
21
nn
xxz
test statistics
Solution:
Step1 : H0 : μ1 = μ2 (no gender difference)
Ha : μ1 ≠ μ2 (there is gender difference) Step2 : α = 0.01 zα (critical value) =+/-2.58
Comparing Two Population MeansStep3 : z test ( n1 ≥ 30; n2 ≥ 30)
2.90420.0960
9.0
40(1.5)
35(1.18)
6.57.422
2
22
1
21
21
nn
xxz
Step4: Decision: Reject Ho
Conclusion:There is sufficient evidence to conclude that there is a significantdifference in the birthweight between boys and girls at 1% level of significance
Large-Sample Tests of Hypothesis
• Other tests– Testing a Population Proportion
– Comparing Two Population Proportions
are left as part of research
End of
First Part of the Discussions
Review Exercises
Review Exercises
1. Ambulatory Services Inc. claims that their average response time is within 30 minutes of receipt of call. The response time for a random sample of 64 cases were recorded, with a sample mean of 34 minutes and a standard deviation of 21 minutes.
(i) Is there sufficient evidence to conclude that the actual response time is
larger than what is claimed by Ambulatory Services Inc.? Use α = .05
2. A chemist from a university claimed that he has invented a new spray that will keep the flowers fresh longer. He based his claim on a test when he selected 500 blossoms of a single type of flower and placed into two groups. One group (consisting of 250 blossoms) was sprayed with his formulation and the other with no spray. For the treatment he found that the average wilting time was 7.2 days with a standard deviation of 1.2 days, while for the control group, 3.6 days with a standard deviation of 1.1 days. Do you agree with the claim of the chemist that the spray actually keeps the flowers fresh longer? Use α = .01
3. A pharmaceutical company claims that they have developed a new drug that will provide immediate relief for persons suffering from vertigo. VERTIPLUS is claimed to provide relief within 5 minutes. A clinical trial was undertaken to test this claim and out of 36 tests the mean relief time is recorded at 6.7 minutes with a standard deviation of 1.38 minutes. Is there sufficient evidence to uphold the claim? α = .01
.
THANK YOU!