Download Chapter 8 Hypothesis Testing

Chapter 8
Hypothesis Testing
“Could these observations really have
occurred by chance?”
Shannon Sprott
GEOG 3000
2/3/2010
Definition
• A statistical hypothesis test is a method of making statistical
decisions using experimental data. Hypothesis testing is one
of the most important tools of application of statistics to real
life problems.
• There are five ingredients to any statistical test :
(a) Null Hypothesis (H0)
(b) Alternate Hypothesis (HA)
(c) Test Statistic
(d) Rejection/Critical Region (Level of Significance)
(e) Conclusion
Steps to Hypothesis Testing
• Null Hypothesis (H0) : It is a hypothesis which states that there is
no difference between the procedures and is denoted by H0. Always
the null hypothesis is tested.
• Alternative Hypothesis (HA) : It is a hypothesis which states that
there is a difference between the procedures and is denoted by HA.
Steps to Hypothesis Testing cont…
• Test Statistic : It is the random variable X whose value is tested to
arrive at a decision.
• Rejection Region : It is the part of the sample space (critical region)
where the null hypothesis H0 is rejected. The size of this region, is
determined by the probability (a) of the sample point falling in the
critical region when H0 is true. a is also known as the level of
significance, the probability of the value of the random variable
falling in the critical region.
• Conclusion : If the test statistic falls in the rejection/critical region,
H0 is rejected, else H0 is accepted.
Null/Alternate hypothesis
Let’s say we have a hat with two kinds of numbers in it: some of the
numbers are drawn from a standard normal distribution (i.e. 2 = 1) with
mean μ = 0, and some of the numbers are drawn from a standard normal
distribution with unknown mean.
Now let’s say we take a number out of the hat. There are two hypotheses
that are possible:
• H0: the null hypothesis. The number is from a standard normal
distribution with μ = 0.
• HA: the alternative hypothesis. The number is not from a
standard normal distribution with μ = 0.
Numbers drawn from two different standard normal
distributions are thrown into hat.
In any testing situation, two kinds
of error could occur:
• Type I (false positive). We reject the null hypothesis when it’s
actually true.
• Type II (false negative). We accept the null hypothesis when
it’s actually false.
The probability of committing a Type I error is typically
denoted α , and the probability of a Type II error is denoted β.
• α: the probability of making a Type I error (false positive).
• β: the probability of making a Type II error (false negative).
α is often called a significance level or sensitivity. Typically, we
try to fix an accepted level, α of Type I error, and go on to find
ways of minimizing the level of Type II error, β.
P - Value
• Probability statement which answers the question: If the null hypothesis
were true, than what is the probability of observing a test statistic at least
as extreme as the one observed.
• The lower the p-value, the less likely the result, assuming the null
hypothesis, the more "significant" the result, in the sense of statistical
significance. One often rejects a null hypothesis if the p-value is less than
0.05 or 0.01, corresponding to a 5% or 1% chance respectively of an
outcome at least that extreme, given the null hypothesis.
Large Sample
Significance Test for Proportions
Step 1
HO : p = po
HA : p > po, p < po, p ≠ po
Step 2
Test statistic is z test
Zobs = (p-hat - p) / √(p(1-p)/√n)
Step 3
P-Value will depend on which alternate
hypothesis is relevant:
Right Handed
Left Handed
Two Sided
Step 4
Set Significance Value ex. (α = .01)
Small Sample
Test for Population Mean
Step 1
HO: µ ≥ mean
HA: µ < mean
Step 2
Test statistic is t test
T = (x - μ) / SE (x)
Hypothesis Testing Equations
• One Sample z-test
– The test statistic is a z-score (z) defined by the following
equation: z = (p - P) / σ
• Two Sample z-test
– The test statistic is a z-score (z) defined by the following
equation. z = (p1 - p2) / SE
• One sample t-test
– The test statistic is a t-score (t) defined by the following
equation. t = (x - μ) / SE
Equations Cont…….
• Two sample t-test
– The test statistic is a t-score (t) defined by the following
equation. t = [ (x1 - x2) - d ] / SE
• Matched Pairs t-test
– The test statistic is a t-score (t) defined by the following
equation. t = [ (x1 - x2) - D ] / SE = (d - D) / SE
• Chi-Squared goodness of Fit Test
– The test statistic is a chi-square random variable (Χ2)
defined by the following equation. Χ2 = Σ [ (Oi - Ei)2 / Ei ]
References
• http://www.cee.vt.edu/ewr/environmental/teach/smprimer/hypotest/ht.
html
• Hypothesis Testing, Vincent A. Voelz:
http://www.stanford.edu/~vvoelz/lectures/hypotesting.pdf
• Johnson, R. A. and Bhattacharya, G. K, 1992 Statistics : Principles and
Methods. 2nd Edition. John Wiley and Sons.
• The Little Handbook of Statistical Practice; Gerard E. Dallal, Ph.D
• Elementary Statistics for Geographers; James E. Burt, Gerald M. Barber,
David L. Rigby. 2009
• http://stattrek.com/Lesson1/Formulas.aspx?Tutorial=Stat