3. INFERENCE IN STATS
Statistical Inference – it is a the process of
drawing conclusions about a population based on a
sample information
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Dilshod Achilov
4. DISTRIBUTIONS
As sample size increases, histogram class
widths can be narrowed such that the
histogram eventually becomes a smooth
curve
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Dilshod Achilov
6. STEP 1
Specify the hypothesis to be tested and
the alternative that will be decided upon if
this is rejected
The hypothesis to be tested is referred to as
the Null Hypothesis (labelled H0)
The alternative hypothesis is labelled H1
For the earlier example this gives:
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mg500:
mg500:0
aH
H
7. STEP 1 (CONTINUED)
The Null Hypothesis is assumed to be true
unless the data clearly demonstrate
otherwise
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Dilshod Achilov
8. STEP 2
Specify a test statistic which will be used
to measure departure from
where is the value specified under the
Null Hypothesis, e.g. in the earlier
example.
For hypothesis tests on sample means the
test statistic is:
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00 : H
0
5000
n
s
x
t 0
Dilshod Achilov
9. STEP 2
The test statistic
is a ‘signal to noise ratio’, i.e. it measures how far
is from in terms of standard error units
The t distribution with df = n-1 describes the
distribution of the test statistics if the Null
Hypothesis is true
In the earlier example, the test statistic t has a t
distribution with df = 25
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n
s
x
t 0
x 0
Dilshod Achilov
10. STEP 3
= 0.05 gives cut-off values on the
sampling distribution of t called critical
values
values of the test statistic t lying beyond the
critical values lead to rejection of the null
hypothesis
For the earlier example the critical value for
a t distribution with df = 25 is 2.06
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11. 11
t distribution with df=25 showing critical region
0
0.1
0.2
0.3
0.4
Density
-4 -3 -2 -1 0 1 2 3 4
t
Overlay Y's
Y t distribution (df =25) Area t critical
Overlay Plot
critical values
critical region
0.025
0.025
12. STEP 4
Calculate the test statistic and see if it lies in the
critical region
For the example
t = -4.683 is < -2.06 so the hypothesis that the batch
potency is 500 mg/tablet is rejected
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683.4
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783.10
500096.490
t
13. P VALUE
The P value associated with a hypothesis test is the
probability of getting sample values as extreme or
more extreme than those actually observed,
assuming null hypothesis to be true
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15. TWO-TAIL AND ONE-TAIL TESTS
The test described in the previous example is
a two-tail test
The null hypothesis is rejected if either an
unusually large or unusually small value of the
test statistic is obtained, i.e. the rejection region
is divided between the two tails
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16. ONE-TAIL TESTS
Reject the null hypothesis only if the
observed value of the test statistic is
Too large
Too small
In both cases the critical region is
entirely in one tail so the tests are one-
tail tests
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