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7/30/2019 Statistical Test of Hypotheses
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Statistical Test of Hypotheses
Professor M. Kabir
Department of Statistics,Jahangirnagar University
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Hypothesis
A hypothesis may be defined is simply as
a statement about one or more
populations. The hypothesis is frequently concerned
with the parameters of the populations
about which the statement is made.
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Hypothesis
A hospital administrator may hypothesize thatan average length of stay of patient admittedto the hospital in five days;
A public health nurse may hypothesize that aparticular educational program will result inimproved communication between nurse andthe patient
A physician may hypothesize that a certaindrug will be effective in 90% of the cases forwhich it is used.
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Hypothesis
By means of hypothesis testing onedetermines whether or not suchstatements are compatible with available
data.
Types of Hypotheses
There are two types of hypotheses
- Research hypotheses
- Statistical hypotheses
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Hypothesis
The research hypothesis is the conjecture or
supposition that motivates the research.
Research hypotheses lead directly to statistical
hypotheses.
Statistical hypotheses are hypotheses are
stated in such a way that they may me
evaluated by appropriate statisticaltechniques.
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Hypothesis
Steps in Hypothesis Testing
Data
Assumptions
Hypotheses
Test Statistic
Distribution of Test Statistic
Decision Rule
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Hypothesis
There are two statistical hypotheses involved inhypotheses testing. These are null hypothesesand alternative hypotheses.
A null hypothesis specifies a hypothesized realvalue, or values for a parameter.
It is denoted by the symbol Ho. The nullhypothesis is sometimes referred to as a
hypothesis of no difference, since it is astatement of agreement with conditionspresumed to be true in the population ofinterest.
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Hypothesis
An alternative hypothesis specifies a real valueor range of values for a parameter that will beconsidered when the null hypothesis is
rejected.The alternative hypothesis is a statement of what
we will believe is true if our sample data causeus to reject the null hypothesis. Usually the
alternative hypothesis and research hypothesisare the same, and in fact the two terms areused interchangeably. We shall designatealternative hypothesis by the symbol Ha.
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Hypothesis
The test statistic is some statistic that may becomputed from the data of the sample. Thetest statistic serves as a decision maker, since
the decision to reject or not to reject the nullhypothesis depends on the magnitude of thetest statistic value
What is rejection region? The rejection region
consists of the set of values of a statistic forwhich the null hypothesis is rejected. Thevalues of the boundaries of the region arecalled the critical values.
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Hypothesis
What is type one error? A type I error occurs
when the null hypothesis is rejected when in
fact it is true. The significance level is the
probability of a type one error when the nullhypothesis is true.
What is type II error? A type II error occurs
when the null hypothesis is not rejected whenit is false
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Hypothesis
The power of a test is the probability of
rejecting the null hypothesis when it is false.
The probability of a type I error is denoted by
, and the probability of a type II error is by
.
The power is defined as
Power= 1probability of type II error Power = 1- .
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Hypothesis
Normal Test when population mean and
variance is known
n
xz
x
/
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Hypothesis
General Formula for Test Statistic
The following is a general formula for a
test statistic that will be applicable inmany of the hypothesis tests discussed
Test statistic = relevant statistic-
hypothesized parameter/ standard errorof the relevant statistic
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Hypothesis
Distribution of Test Statistic
It has been pointed out that the key to
statistical inference is the sampling
distribution.
The distribution of test statistic
for example follows the standard normal
distribution if the null hypothesis is true andthe assumptions are met.
n
xz
/
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Hypothesis
Decision rule:
The decision rule tells us to reject the null
hypothesis if the value of the test statistic that
we compute from our sample is one of the
values in the rejection and to reject the null
hypothesis if the computed value of the test
statistic is one of the values in the non-rejection region.
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Hypothesis
Significance level: The decision as to whichvalues go into the rejection region and whichones go into the non rejection region is made
on the basis of the desired level of significance,designated by .
The term level of significance reflects the factthat hypothesis tests are sometimes called
significance tests, and computed value of thetest statistic that falls in the rejection region issaid to be significant.
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Hypothesis
The level of significance, specifies the area under
the curve of the distribution of the test statistic that is
above the values on the horizontal axis constituting
the rejection region.Types of errors
The error committed when a true null hypothesis is
rejected is called type I error . The type II error is the
error committed when a false null hypothesis is notrejected. The probability of committing type II is
designated by
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Hypothesis
Whenever we reject a null hypothesis there is
always the concomitant risk of committing a
type I error, rejecting a true null hypothesis.
Whenever we fail to reject a null hypothesisthe risk of falling to reject a false null
hypothesis is always present.
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Hypothesis
Statistical Decision
The statistical decision consists of rejecting orof not rejecting the null hypothesis . It is
rejected if the computed value of the teststatistic falls in the rejection region, and is notrejected if the computed value of the teststatistic falls in the non-rejection region.
Conclusion: If Ho is rejected we conclude thatHa is true. If Ho is not rejected we concludethat Ho may be true.
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Hypothesis
The p value is the smallest value of for which the
null hypothesis can be rejected. For Z= -2.12 the p
value is 0.034.
The p value for a hypothesis testing is the probabilityof obtaining when Ho is true, a value of the test
statistic as extreme or more extreme than the one
actually computed.
If p value is less than or equal to , we reject thehypothesis. If p value is greater than , we do not
reject the hypothesis. We accept the hypothesis
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Hypothesis
Hypothesis Accept Ho Reject Ho
Accept Ho Correct Type II error
Reject Ho Type I error Correct
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Steps in Hypothesis Testing
Evaluate data
Review assumption
State hypothesis
Select rest statistics Determine distribution of test statistic
State decision rule
Calculate test statistic
Make statistical decision Do not reject Ho
Reject Ho
Conclude Ho may be true
Calculate Ha is true
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Hypothesis Testing: A Single
Population Mean
We consider the testing of a hypothesis about a
population mean fewer than three different
conditions
When sampling is from a normally distributed
population of values with known variance
When sampling is from a normally distributed
population of values with unknown variance
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Hypothesis Testing: A Single
Population MeanWhen sampling is from a normally distributed
population and the population variance isknown, the test statistic for testing Ho:
EX. If random sample of size 10 is drawn from anormal population with mean and variance
are respectively 27 and 20 respectively. Canwe conclude the mean age of this population isdifferent from 30 years?
n
x
z
/
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Hypothesis Testing: A Single
Population MeanEX. If random sample of size 10 is drawn from
a normal population with mean and varianceare respectively 27 and 20 respectively. Can
we conclude the mean age of this population isdifferent from 30 years?
Calculation of test statistic
We have z= (27-30)/ 1.4142 = - 2.12
We reject hypothesis. We conclude thatpopulation mean is different from 30 years.
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Hypothesis Testing: A Single
Population MeanTesting Ho by means of a confidence interval
The 95% confidence interval of population meanis given by
27 plus-minus 1.96 Square root of 20/1027+ 2.7718, 27-2.7718
The age lies between 29.77 to 24.23 years
Since the interval does not include 30, we say 30is not a candidate for the mean we areestimating and there fore population mean isnot equal =30 and Ho is rejected.
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Hypothesis Testing: A Single
Population Mean
In general, when testing null hypothesis by
means of a two sided confidence interval, we
reject Ho at the level of significance if the
hypothesized parameter is not contained withthe 100 ( 1-) percent confidence interval.
If the hypothesized parameter is contained
within the interval, Ho cannot be rejected at
the level of significance
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Hypothesis Testing: A Single
Population Mean
Sampling from a normally distributed
population: Population Variance is unknown
The test statistic for testing Ho: Population
mean= = o is
Statistic is
sample mean- pop population / s/square root of n
Example: Will we be able to conclude that themean BMI for the population is 35 .
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Hypothesis Testing: A Single
Population Mean
Can we reject the hypothesis that
population mean is equal to 35.
Body Mass Index ( BMI) measurementsfor 14 male subjects are given below.
Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14
BMI 23 25 21 37 39 21 23 24 32 57 23 26 32 45
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Hypothesis Testing: A Single
Population MeanData: The data consist of BMI measurements on
14 subjects as given above.
Assumptions: The 14 subjects constitute a simple
random sample from a population of similarsubjects. We assume that BMI measurementsin this population are approximately normallydistributed.
Hypotheses: Population mean 35Population is not equal to 35
Test statistic is with d.f is n-1,t= (30.5-35)/2.8434
ns
xt
/
ns
xt
/
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Hypothesis Testing: A Single
Population MeanThe calculated value of t = -1.58
With 13 degree of the value of t is2.16
Since computed value of t is less than the table
value. We accept the hypothesis. Based on thedata the mean population from which thesample drawn may be 35.
Hypothesis Testing: population standard deviation is
not known
If the population standard deviation is notknown , the usual practice is to use the samplestandard deviation as an estimate. The test
statistic for testing Ho= = o,
ns
xt
/
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Hypothesis Testing: A Single
Population Mean
then , which when Ho is true , is
distributed approximately as the
standard normal distribution if n is
large.ns
xt
/
ns
xt
/
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Hypothesis Testing: A Single
Population MeanEx. A study was conducted to describe the
menopausal status , menopausal symptoms,energy expenditure, and aerobic fitness of
healthy midlife women and to determine therelationship among these factors. The meanscore of maximum oxygen uptake for a sample242 was 33.3 with a standard deviation of12.14. The researcher wishes to know if, on thebasis of these data, one may conclude that themean score for a population of such women isgreater than 30
ns
xt
/
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Hypothesis Testing: A Single
Population Mean
Data Maximum oxygen uptake for 242women with mean = 33.3 and s = 12.14
Assumptions: The data constitute a simplerandom from a population of healthymidlife women similar to those in thesample.
Hypotheses Ho: grater than equal to 30Ha: is greater than 30
ns
xt
/
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Hypothesis Testing: A Single
Population Mean
Data Maximum oxygen uptake for 242women with mean = 33.3 and s = 12.14
Assumptions: The data constitute a simplerandom from a population of healthymidlife women similar to those in thesample.
Hypotheses Ho: grater than equal to 30Ha: is greater than 30
ns
xt
/
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Hypothesis Testing: A Single
Population Mean
Given the above test statistic
We have z= ( 33.3-30)/0.7804= 3.3/0.7804
= 4.23We reject the hypothesis since computed
value is greater than table value. We
conclude that the mean score for thesampled population is greater than 30.
ns
xt
/