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PROBABILITY & STATISTICALINFERENCE LECTURE 6
MSc in Computing (Data Analytics)
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General Steps in Hypotheses testing
1. From the problem context, identify the parameter of interest.
2. State the null hypothesis, H0.
3. Specify an appropriate alternative hypothesis, H1.
4. Choose a significance level, .
5. Determine an appropriate test statistic.6. State the rejection region for the statistic.
7. Compute any necessary sample quantities, substitute these into theequation for the test statistic, and compute that value.
8. Decide whether or not H0should be rejected and report that in theproblem context.
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Type of questions that can be answered with Two sample
hypothesis tests
A manufacturing plant want to compare thedefective rate of items coming off two different
process lines.
Whether the test results of patients who received a
drug are better than test results of those who
received a placebo.
The question being answered is whether there is a
significant (or only random) difference in theaverage cycle time to deliver a pizza from Pizza
Company A vs. Pizza Company B.
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Difference in Means of Two Normal Distributions, Variances
Known
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Test Assumptions
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Example
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Example
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Example
The P-Valueis the exact significance level of a statistical test; thatis the probability of obtaining a value of the test statistic that
is at least as extreme as that when the null hypothesis is true
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Confidence Interval on a Difference in Means, Variances
Known
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Example
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Example
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Difference in Means of Two Normal Distributions,
Variances unknown
We wish to test:
The pooled estimator of 2:
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Difference in Means of Two Normal Distributions,
Variances unknown
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Example
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Example
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Example
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Confidence Interval on the Difference in Means, Variance
Unknown
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Example
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Example
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Example
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Practical Hypothesis Testing
1.
From the problem context, identify the parameter ofinterest.
2. State the null hypothesis, H0.
3. Specify an appropriate alternative hypothesis, H1.
4. Choose a significance level, .
5. Calculate the P-value using a software package of choice.
6. Decide whether or not H0should be rejected and report
that in the problem context. Reject H0when P-Value is lessthan .
(Golden rule: Reject H0for small )
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Some Reserach
Look up the correct formula for calculating thehypotheses test between two proportions
What are the assumptions for the test
Find an example of the research
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Answer to research
Large-sampletest on the difference in populationproportions
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Example
Example of large-sampletest on the difference inpopulation proportions
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Analysis of Variance
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Introduction
In the previous section we were concerned with theanalysis of data where we compared the sample
means.
Frequently data contains more that two samples,
they may compare several treatments.
In this lecture we introduce statistical analysis that
allows us compare the mean of more that two
samples. The method is called Analysis of Variance or AVOVA for short.
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Total Sum of Squares
Data set:
14, 12, 10, 6 ,4, 2
Group A:
6 ,4, 2
Group B:
14, 12, 10
Overall Mean : 8
Total Sum of Squares:
SST= (14-8)2 + (12-8)2 +
(10-8)2 + (6-8)2 + (4-8)2 +(2-8)2 =112
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Between Group Variation
Sum of Squares of the
Model:
SSm= na( - a)2 + nb( -
b)2
=3*(8-4)2 +3*(8-12)2
=96
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Within Group Variation
Sum of Squares of theError:
SSe=
= (14-12)2 + (12-12)2 +(10-12)2 + (6-4)2 + (4-
4)2 + (4-2)2 +
= 16
2
1 1
_ _
)(
k
i
n
j
jij xx
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Structure of the Data
Group Observation Total Mean
1 x11 x12 .......... x1n x1
2 x21 x22..........
x2n x2
.
.
.
..........
a xa1
xa2
.......... xan
xa
Total
1x
2x
ax
x
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ANOVA Table
Source Degrees of
Freedom
Sum Of Squares Mean
Square
F- Stat
Model a - 1 SSM /(a-1) MSM / MSE
Error n-aSSE /(n-a)
Total n-1SST /(n-1)2
1
)( xx
n
i
i
a
j
jj xxn
1
2)(
2
1 1
_ _
)(
a
i
n
j
jij xx
Where : n is the sample size and a is the number of
groups
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ANOVA TableOriginal Example
Source Degrees of
Freedom
Sum Of Squares Mean
Square
F- Stat
Model 2 - 1 = 1 96 96 24
Error 62 = 4 164
Total 61 = 5 112
Where : n is the sample size and a is the number of groups
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Model Assumptions
Independence of observations within and betweensamples
normality of sampling distribution
equal variance - This is also called the
homoscedasticity assumption
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The ANOVA Equation
We can describe the observations in the abovetable using the following equation:
nj
ai
Y ijiij ,......,2,1
,......,2,1
Where : n is the sample size and k is the number of groups
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ANOVA Hypotheses
We wish to test the hypotheses:
The analysis of variance partitions the total variability
into two parts.
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Example
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Graphical Display of Data
Figure 13-1 (a)Box plots of hardwood concentration data. (b) Display of
the model in Equation 13-1 for the completely randomized single-factor
experiment
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Example
We can use ANOVA to test the hypotheses thatdifferent hardwood concentrations do not affect the
mean tensile strength of the paper. The hypotheses
are:
The ANOVA table is below:
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Example
The p-value is less than 0.05 therefore the H0 canbe rejected and we can conclude that at least one
of the hardwood concentrations affects the mean
tensile strength of the paper.
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Test Model Assumptions
Use the Bartletts Test to test for homoscedasticityassumption
Bartlett's test (Snedecor and Cochran, 1983) is used
to test if ksamples have equal variances.
Bartlett's test is sensitive to departures from
normality. That is, if your samples come from non-
normal distributions, then Bartlett's test may simply
be testing for non-normality. The Levene test is analternative to the Bartlett test that is less sensitive to
departures from normality.
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Barlett Test for Equal Variance
The hypotheses for the Barlett test are as follows:
The barlett test statistic follows a chi-squared
distribution
Interpert the p-value like any other hypothese test
ji,paironleastatfor:H
...:H
22
i1
222
210
j
n
If the Assumption of Equal Variance is
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If the Assumption of Equal Variance is
not met
If the assumption for equal variance is not met usethe Welches ANOVA
Assignment for next week:
Investigate the difference between the standard
ANOVA and Welches ANOVA?
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Demo
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Confidence Interval about the mean
For 20% hardwood, the resulting confidence interval on the mean is
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Confidence Interval about on the difference of two treatments
For the hardwood concentration example,
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Multiple Comparisons Following the
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Multiple Comparisons Following the
ANOVA The least significant difference (LSD) is
If the sample sizes are different in each treatment:
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Example: Multi-comparison Test
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Example: Multi-comparison Test
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Demo
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Exercises