Hypothesis Testing
The Summary from class is outlined below. It outlines
our textbook, " Lind. (2005). Statistical techniques in business & economics
(11th ed). New York: McGrawHill, Chapter 9 and 10."
Hypothesis Testing is a statistical methodology
that seeks to prove or disprove if a certain belief about a series of data
is true. It is a procedure, based on sample evidence and probability
theory, used to determine whether the hypothesis is a reasonable statement and
should not be rejected, or is unreasonable and should be rejected.
A Hypothesis is a statement about the value of a population parameter
developed for the purpose of testing. The testing party begins by setting up a null hypothesis, which is typically
the original belief that is potentially incorrect given the new data; this
represents the statement being tested. The alternative hypothesis is the new
possibility that will potentially disprove the null hypothesis. If there is
sufficient evidence after testing to accept the alternative hypothesis, then it
will be considered true; if it is disproved, then the null hypothesis will be
accepted.
One
Sample Hypothesis Testing
Steps to Hypothesis Testing:
Step 1: State the Null and Alternative Hypothesis to Test:
Null Hypothesis, Ho, is a statement about the value of a population
parameter.
Alternative Hypothesis, H1, is a statement that is accepted if the sample
data provide evidence that the null hypothesis is false.
Step 2: Select the Level of Significance
The probability of rejecting the null hypothesis when it is actually true.
You could have two types of errors.
 Type I Error: Rejecting the null hypothesis when it is actually true.
 Type II Error: Accepting the null hypothesis when it is actually false.
Step 3: Identify the Test Statistic
Test statistic: A value, determined from sample information, used to
determine whether or not to reject the null hypothesis.
Critical value: The dividing point between the region where the null
hypothesis is rejected and the region where it is not rejected.
Step 4: Formulate the Decision Rule
Decide on type of test.
 OneTailed Tests of Significance: A test is onetailed when the
alternate hypothesis, H1 , states a direction (i.e, > or < a certain value).
 TwoTailed Tests of Significance: A test is twotailed when no direction
is specified in the alternate hypothesis H1 (i.e., = or not = to).
Testing for the Population Mean: Large Sample, Population Standard Deviation
Known
Testing for the Population Mean: Large Sample, Population
Standard Deviation Unknown.
In both cases as long as the sample size n 30, z can be
approximated
Testing for a Population Mean: Small Sample, Population Standard
Deviation Unknown
Step 5: Take a Sample,
Step 6: Analyze the Sample,
A pValue in Hypothesis Test is the probability, assuming that the null
hypothesis is true, of finding a value of the test statistic at least as extreme
as the computed value for the test.
 If the pValue is smaller than the significance level, H0 is rejected.
 If the pValue is larger than the significance level, H0 is not
rejected.
Computation of the pValue
 OneTailed Test: pValue = P{z absolute value of the computed test
statistic value}
 TwoTailed Test: pValue = 2P{z absolute value of the computed test
statistic value}
Step 7: Make a Decision (accept the null or reject the null)
Tests Concerning Proportion
A Proportion is the fraction or percentage that indicates the part of the
population or sample having a particular trait of interest.
p = (Number of Successes in the Sample/ Number Sampled)
Test Statistic for Testing a Single Population Proportion
The sample proportion is p and
B
is the population proportion.
Two Sample Tests of
Hypothesis
The purpose is to Test or Compare Two Sets of Data. We
wish to know whether the distribution of the differences in sample means
has a mean of 0.
2.1 One Variable, One or Two Populations/Treatments
2.1.1 Test of Averages/Means
2.1.2 Variances are Unknown but
assumed Equal, Two Treatment Means, Completely Randomized Design, Small Sample
Size (<30).
2.1.2.1 t
Test  Two Samples
2.1.2.2 t
Test  One Sample
2.1.3 Variances are Known, Sampling
From Independent Normal Distributions. If both samples contain at least 30
observations we use the z distribution as the test statistic.
2.1.3.1 Z
Test  Two Samples
2.1.3.2 Z
Test  One Sample
2.1.4 Asked For Other
2.1.4.1 Use
Confidence Interval
2.1.4.2 Use
PValue
2.2 Test of Variability
2.2.1 One Normal Population, Variance
Constant Known
2.2.1.1 c^2
Test  One Sample Test
2.2.2 Two normal Populations,
Independent Random Samples, Estimated Means and Variances Unknown for Both.
2.2.2.1 F
Test  More than one Sample
2.3 Significant Difference Between Means
2.3.1 md Test
2.3.2 Ask For Other
2.3.2.1 Use
Confidence Interval
2.4 One Variable, More Than Two Populations/Treatments
2.4.1 Single Factor Analysis of
Variance (ANOVA) Analysis of Fixed Effects Model to Test more than 2 means where
means are equal.
2.4.2 Equality of Variance
2.4.2.1
Bartiett's Test, to test Variances, Requires Normality.
2.4.3 Confidence Interval
2.4.3.1 to
Test  On differences in two means, Normal Distribution with Variances Unknown.
2.4.4 Sample Size
2.4.4.1 to
Test  Difference in Two Means Unknown Variances
2.4.4.2 Using
OC Curves
2.4.4.3 Given
SDV
2.4.5 KruskalWallis Test  Normality
assumption is unjustified, Good for test of differences in means  A
nonparametric Alternative
2.4.6 Randomized Complete Block
Design ANOVA
2.5 Comparing Pairs of Treatment Means after the ANOVA F
test where Ho is rejected. Test each Ho: mI = mj for all I=j.
2.5.1 Least Significant Difference
(LSD) Method very effective for detecting true differences in means if it is
applied only after F test in ANOVA where level of significant is at 5%.
2.5.2 Duncan's Multiple Range Test
2.5.3 NewmanKeuls Test  More
conservative than Duncan's Test in that the Type I error is smaller and the
method is less powerful.
2.5.4 Tukey's Test  Results are
somewhat more difficult to interpret than LSD or NewmanKeals Test. Tukey has a
type I error rate of a which is conservative (smaller rate) than either NewmanKeuls
or Duncan Test. (Therefor is less Powerful.)
2.6 Estimation of Model Parameters
2.6.1 Single Factor Model  Yij = m +
tI + Îij
2.6.2 Overall Mean  E(m) = Average
Y..
2.6.3 Treatment Effects  tI =
Average Yi.  Average Y.. (I=1,2,…..a)
2.6.4 Confidence Interval of Mean for
some 1a CI º [ Average Yi. ± ((ta/2,Na)*(MSE/n)^1/2)] = LL £ mI £ UL
2.6.5 Confidence Interval of Mean
Difference for some 1a CI º [(Average of Yi.  Average of Yj.) ± ((ta/2,Na)*(2*MSE/n)^1/2)]
= LL £ (mI mj) £ UL
