Statistics and maths

1. UUnnlloocckkiinngg tthhee MMyysstteerriieess
ooff HHyyppootthheessiiss TTeessttiinngg
Muzamil Hussain
Arsalan Shaikh
Thanks to Sir
Mohammad Rafique

2. What’s this all about?
• Hypothesis
• An educated guess
• A claim or statement about a
property of a population
• The goal in Hypothesis Testing is to
analyze a sample in an attempt to
distinguish between population
characteristics that are likely to occur
and population characteristics that
are uunnlliikkeellyy to occur.

3. • Null Hypothesis
vs. Alternative
Hypothesis
• Type I vs. Type II
Error
" a vs. b
The Basics

4. Null Hypothesis vs. Alternative
Hypothesis
Null Hypothesis
• Statement about the
value of a population
parameter
• Represented by H0
• Always stated as an
Equality
Alternative Hypothesis
• Statement about the
value of a population
parameter that must be
true if the null
hypothesis is false
• Represented by H1
• Stated in on of three
forms
• >
• <
• ¹

5. Type I vs. Type II Error
Referring
to Ho, the
Null
Hypothesis
True False
Reject Type I
Error
O.K
Fail to
Reject
O.K. Type II
Error

6. Alpha vs. Beta
· a is the probability of Type I error
· b is the probability of Type II error
· The experimenters (you and I) have the
freedom to set the a-level for a
particular hypothesis test. That level is
called the level of significance for the
test. Changing a can (and often does)
affect the results of the test—whether
you reject or fail to reject H0.

7. Alpha vs. Beta, Part II
• It would be wonderful if we could force
both a and b to equal zero.
Unfortunately, these quantities have
an inverse relationship. As a
increases, b decreases and vice versa.
• The only way to decrease both a and b
is to increase the sample size. To
make both quantities equal zero, the
sample size would have to be infinite—
you would have to sample the entire
population.

8. Type I and Type II Errors
True State of Nature
We decide to
reject the
null hypothesis
We fail to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)
a
Correct
decision
Type II error
(rejecting a false
null hypothesis)
b
Correct
decision
Decision

9. Forming Conclusions
• Every hypothesis test ends with the
experimenters (you and I) either
• Rejecting the Null Hypothesis, or
• Failing to Reject the Null Hypothesis
• As strange as it may seem, you never
aacccceepptt the Null Hypothesis. The best
you can ever say about the Null
Hypothesis is that you don’t have
enough evidence, based on a sample,
to reject it!

10. Seven Steps to Hypothesis
Testing Happiness
(Traditional or Classical Method)

11. The Seven Steps…
1) Describe in words the population
characteristic about which
hypotheses are to be tested
2) State the null hypothesis, Ho
3) State the alternative hypothesis, H1
or Ha
4) Display the test statistic to be used

12. The Seven Steps…
5) Identify the rejection region
• Is it an upper, lower, or two-tailed
test?
• Determine the critical value
associated with a, the level of
significance of the test
5) Compute all the quantities in
the test statistic, and compute
the test statistic itself

13. The Seven Steps…
7) State the conclusion. That is,
decide whether to reject the null
hypothesis, Ho, or fail to reject the
null hypothesis. The conclusion
depends on the level of significance
of the test. Also, remember to state
your result in the context of the
specific problem.

14. Types of Hypothesis Tests
• Large Sample Tests, Population Mean
(known population standard deviation)
• Large Sample Tests, Population
Proportion (unknown population
standard deviation)
• Small Sample Tests, Mean of a Normal
Population

15. Test for mean one and
two sample test

16. • Our focus in this Presentation is
comparing the data from two different
samples
• For now, we will assume that these two
different samples are independent of
each other and come from two distinct
populations
Population 1:m 1 , s1
Sample 1: , s1
Population 2: m 2 , s2
Sample 2: , s2

17. Two-Sample Z test
We want to test the null hypothesis that the two
populations have different means
• H0: m1 = m2 or equivalently, m1 - m2 = 0
• Two-sided alternative hypothesis: m1 - m2 ¹ 0
• If we assume our population SDs s1 and s2 are known, we
can calculate a two-sample Z statistic:
• We can then calculate a p-value from this Z statistic using
the standard normal distribution
X1=mean1 x2=mean2 s1 s2 = standard deviation N1 N2 = sample 1 and sample 2

18. Two-Sample t test
We still want to test the null hypothesis that the two
populations have equal means (H0: m1 - m2 = 0)
• If s1 and s2 are unknown, then we need to use the sample
SDs s1 and s2 instead, which gives us the two-sample T
statistic:
• The p-value is calculated using the t distribution, but what
degrees of freedom do we use?
• df can be complicated and often is calculated by software
• Simpler and more conservative: set degrees of freedom equal to
the smaller of (n1-1) or (n2-1)

20. The End
Actually, it’s just the
beginning...