LESSON 3

1. RESEARCH METHODOLOGY
INFERENCIAL STATISTICS
BY
ONDABU IBRAHIM TIRIMBA

2. Overview
 In this lesson, we will briefly cover a few main concepts
used in inferential Statistics, such as estimating a
population parameter, hypothesis testing, T-tests, linear
regression and Analysis of Variance (ANOVA).
 After completing this section you should be able to do
the following:
 Recognize common inferential statistical tests
 Identify and compute basic point estimates of population
parameters
 Describe the basics of hypothesis testing
 Understand and identify the use of regression modeling

3. Introduction
• Inferential Statistics’ are mathematical
tools that permits the researcher to
generalize to a population of individuals
based upon information obtained from a
limited number of research participants
(the sample).

4. Example
• For instance, consider an experiment where sales
were increased by 25% following a media
advertisement on 10 products compared to sales of 10
products which were not advertised. Inferential
Statistics allows us to decide if the increased sales are
due to chance or from the effect of advertising.
• There are primarily two ways to use inferential
statistics:
• Parameter Estimation
• Test of Hypothesis

5. Parameter Estimation
• A Parameter is any of the factors that
limits the way in which something can
be done.
• Parameter estimation falls into two
Categories:
• Point estimation
• Confidence interval (CI) estimation

6. Point Estimation
• Point estimation: The Estimate or Prediction
of a population parameter is often referred to as
a Point estimate.
• That is to say, the estimate is a single value
based on a sample, a statistic, which is then
used to estimate the corresponding value in the
population (a parameter).
• The average (mean = a parameter) of our
sample can be used as an estimator of the
population mean.

7. Sampling Error
• Sampling Error: the difference between the
population value of interest (e.g. mean), and
the sample value. Our sample value is often
referred to as an estimate of our population
value.
• If the sample is randomly drawn from the
population, then sampling error will be random
and will be distributed normally.

8. Confidence Interval (CI)
• Confidence Interval: Is a range of numbers which
are calculated so that the true populations mean
lies within this range with a particular degree of
certainty.
• The certainty in which a population mean lies
within the range is typically expressed as 95%
confidence interval, or a 99% confidence interval.
As you add more certainty the width of the interval
will increase.
• A confidence interval gives an estimated range of
values which is likely to include an unknown
population parameter, the estimated range being
calculated from a given set of sample data.

9. Confidence Interval cont.
• Confidence interval for the mean is given by
formula:
CI = ¯x ± Zα s/√n ¯x = mean
Zα = constant for 95% CI
= 1.96 and 2.56 for 99%
¯x – 1.96xs/√n < µ < ¯x + 1.96xs/√n

10. Confidence Interval cont.
So if for the selected sample the sample size is
36 (= n) with mean of 5 (= ¯x) and standard
deviation of 2 (= s) then the 95% confidence
interval (CI) of the population mean is given
by:
4.35=5–1.96 x 2/√36 < µ < 5+1.96 x 2/√36=5.65
Since, 1.96 x 2/√36 = ± 0.65
Thus, CI ranges between 4.35< µ < 5.65

11. Confidence Interval cont.
• So the 95% confidence interval for the mean
using this formula is between 4.35 and 5.65.
Notice, that if we select another random sample
of size 36, its mean and standard deviation
would be different so we would obtain a
different confidence interval.
Exercise: Use the same data given above to
calculate the 99% confidence interval of the
population mean

12. Confidence Interval cont.
• If independent samples are taken repeatedly
from the same population, and a confidence
interval calculated for each sample, then a
certain percentage (confidence level) of the
intervals will include the unknown population
parameter.
• Confidence intervals are usually calculated
so that this percentage is 95%, but we can
produce 90%, 99%, 99.9% (or whatever)
confidence intervals for the unknown
parameter.

13. Confidence Interval cont.
• The width of the confidence interval gives us
some idea about how uncertain we are about
unknown parameter.
• A very wide interval may indicate that more
data should be collected before anything very
definite can be said about the parameter.
• Confidence intervals are more informative than
the simple results of hypothesis tests (where
we decide “reject Ho” or “don’t reject Ho”) since
they provide a range of plausible values for the
unknown parameter.

14. Confidence Interval cont.
• Confidence limits are the lower and the upper
boundaries/values of a confidence interval, that
is, the values which define the range of a
confidence interval.
• The upper and lower bounds of a 95%
confidence interval are the 95% confidence
limits. Such limits may be taken for other
confidence levels, for example, 90%, 99%,
99.9%.

15. Hypothesis Testing
• The second type of inferential statistics is
hypothesis testing. This is sometimes called
statistical testing as well.
• In point estimation and in constructing
confidence interval, we had no expectations
about the values we calculated, whereas in
hypothesis testing we have formed some
expectation about the population parameter.

16. HYPOTHESIS TESTING cont.
Example
• Our hypothesis is that “tree mortality after a particular
forest fire will be greater than 60%”, in other words average
tree mortality > 60%.
• Once our notion of the population parameter has been
developed, we can write two contradictory hypotheses:
 The first is research (or alternative) hypothesis, which in
our case is that “the mean tree mortality > 60%”.
 The second hypothesis is called the null hypothesis, and is
the opposite of our research hypothesis. In our example, the
null hypothesis would be stated as “the mean tree mortality is
less than or equal to 60%”.

17. Hypothesis Testing cont.
Basic Concepts in Test of Hypothesis
• Def.: A Hypothesis is a tentative explanation
for an observation, phenomenon, or scientific
problem that can be tested by further
investigation.

18. Null and Alternative Hypothesis
• Null Hypothesis: The null hypothesis, (Ho),
represents a theory that has been put forward, either
because it is believed to be true or because it is to be
used as a basis for argument, but has not been
proved.
• For example, in a clinical trial of a new drug, the null
hypothesis might be that “the new drug is no better,
on average, than the current drug”.
We would write
Ho: there is no difference between the two drugs on
average.

19. Null and Alternative Hypothesis
• Alternative Hypothesis: The alternative
hypothesis, H1, is a statement of what a
statistical hypothesis test is set to establish.
• For example, in a clinical trial of a new drug,
the alternative hypothesis might be that “the
new drug has a different effect, on average,
compared to that of the current drug,
We would write:
• H1: the two drugs have different effects, on
average.

20. Null and Alternative Hypothesis
• The alternative hypothesis might also be
that the new drug is better, on average,
than the current drug.
In this case we would write:
• H1: the new drug is better than the current
drug, on average.

21. Null and Alternative Hypothesis
• We give special consideration to the null hypothesis. This
is due to the fact that the null hypothesis relates to the
statement of being tested, whereas the alternative
hypothesis relates to the statement to be accepted if /when
the null is rejected.
• The final conclusion once the test has been carried out is
always given in terms of the null hypothesis. We either
reject Ho in favor of H1 or do not reject Ho.
We never conclude, Reject H1 or even Accept H1.
• We conclude “Do not reject Ho”, this does not necessarily
mean that the null hypothesis is true, it only suggests that
there is not sufficient evidence against Ho in favor of H1.
Rejecting the null hypothesis then, suggests that the
alternative hypothesis may be true.

22. One and Two Tailed Tests
One Tailed Tests (T-Test)
Example
• Our hypothesis is that tree mortality after a
particular forest fire will be greater that 60%.
In other words average tree mortality > 60%.
• In this example, it is a one-tailed test.
Here we were simply considering the idea
that the population mean was larger than
some number. So we would reject the null
hypothesis if we had large values of tree
mortality.

23. Two Tailed Tests cont.
A two-tailed test is used when a research hypothesis is
stated as the following:
Example
• “Tree mortality following fire will be equal to 60%”,
whereas
• our null hypothesis would read “tree mortality
following fire is not equal to 60%”.
• Under this scenario, we could reject our research
hypothesis if tree mortality was much larger than 60
or much smaller than 60.
• This is a two-tailed test

24. Significance
Significance
• The probability of an outcome given the
null hypothesis is a p-value.
• A low probability value indicates rejection
of the null hypothesis.
• Typically: reject Ho if p-value ≤ 0.05 (for a
95% levels of significance test) or 0.01 (for
a 99% levels of significance test).
Statistically, significant means the effect is
not due to chance.

25. Type I and II Errors
Type I and II Errors
• We define a type I error as the event of rejecting
the null hypothesis when the null hypothesis was
true. The probability of a type I error (a) is called
the significance level.
• We define type II error (with probability b) as the
event of failing to reject the null hypothesis when
the null hypothesis was false.
• The type I risk is the chance of deciding that a
significant effect is present when it isn’t.
• The type II risk is the chance of not detecting a
significance effect when one exists.

26. Test of Hypothesis
Steps in Test of Hypothesis
The usual process of hypothesis testing
consists of four steps:
• Formulate the null hypothesis Ho
(commonly, that the observations are the
result of pure chance)
• and the alternative hypothesis H1
(commonly, that the observations show a
real effect combined with a component of
chance variation).
• Identify a test statistic that can be used to
assess the truth of the null hypothesis.

27. Test of Hypothesis cont.
• Compute the P-value, which is the probability
that a test statistic at least as significant as
the one observed would be obtained
assuming that the null hypothesis were true.
The smaller the P-value, the stronger the
evidence against the null hypothesis.
• Compare the P-value to an acceptable
significance value α (sometimes called an
alpha value). If P≤ α, that the observed
effect is statistically significant, the null
hypothesis is ruled out, and the alternative
hypothesis is valid.

28. Statistical Tests
Statistical tests include:
• Linear Regression
• T-test
• ANOVA

29. Regression Models and
Correlation
• The use of regression models is very common, and
serves a very specific point to us as managers.
• Regression models allow us to predict the outcome of
one variable from another variable.
• When two variables are related, it is possible to
predict a persons score on one variable from their
score on he second variable with better than chance
accuracy.
• This section describes how these predictions are
made and what can be learned about the relationship
between the variables by developing a prediction
equation.

30. Regression Models and Correlation
• It will be assumed that the relationship
between the two variables is linear.
• Given that the relationship is linear,
the prediction problem becomes one of
findings the straight line that best fits
the data.
• Since the terms “regression” and
“prediction” are synonymous, this line
is called the regression line.

31. Regression line
The mathematical form of the regression line
predicting Y from X is:
Y = Bo + B1X
• Where:
- X is the variable represented on the X-
axis (Independent variable)
- B1 is the slope of the line,
- Bo is the Y-intercept and
- Y consist of the predicted (dependent variable)
values of Y for the various values of X.

32. The Coefficient of Correlation
• The correlation between two variables reflects the
degree to which the variables are related. The most
common measure of correlation is the Pearson
Product Moment Correlation (called Pearson’s
correlation in short).
• When measured in a population, the Pearson
Product Moment correlation is designated by the
Greek letter rho (p).
• When computed in a sample, it is designated by the
letter r and is sometimes called “Pearson’s r”.
• Pearson’s correlation reflects the degree of linear
relationship between two variables. It ranges from
+1 to -1.

33. The Coefficient of Correlation
• A correlation of +1 means that there is a
perfect positive linear relationship.
• A positive relationship shows high scores
on the X axis that are associated with high
scores on the Y-axis.
• A correlation of -1 means that there is a perfect
negative linear relationship between variables.
• A negative relationship shows high scores on
the X-axis that are associated with low scores on
the Y-axis.

34. The Coefficient of Correlation
• A correlation of 0 means there is no linear
relationship between the two variables.

35. Coefficient
of Determination
• The coefficient of determination r2 gives the proportion
of the variance (fluctuation) of one variable that is
predictable from the other variables.
• It is a measure that allows us to determine how certain
one can be in making predictions from a certain
model/graph.
• The coefficient of determination is a measure of how
well the regression line represents the data.
• If the regression line passes exactly through every
point on the scatter plot, it would be able to explain all
of the variation.

36. Coefficient
of Determination
• The further the line is away from the points, the
lesser it is able to explain.
For example, if r = 0.922, then r2 = 0.850, which
means that 85% of the total variation in Y can
be explained by the linear relationship between
X and Y. The other 15% of the total variation in
Y remains unexplained (or is by chance).

37. T-test
• The T-test gives an indication of the separateness of two sets
of measurements, and is thus used to check whether two sets
of measures are essentially different.
• In many situations, we will want to compare two populations
parameters. To compare these two populations, we can
compare the differences between the two sample means.
• T-test looks for significant difference in means between two
samples or between a population and a sample.
There are 3 types of T-tests;
- One sample T-test
- Independent 2 samples T-test
- Paired sample T-test

38. One Sample T-test
• One sample t-test: is a statistical procedure that is
used to know the mean differences between the
sample and the known value of the population
mean.
• In one sample t-test, we know the population mean.
We draw a random sample from the population and
then compare the sample mean with the population
mean and make a statistical decision as to whether
or not the sample mean is different from the
population.

39. Assumptions in One Sample
t-test
• In one sample t-test, dependent variables should be
normally distributed.
• In one sample t-test, samples drawn from the
population should be random.
• In one sample t-test, cases of the samples should be
independent
• The data is measurement data-interval/ratio
• In one sample t-test, we should know the
population mean.

40. Formula
t = (X1 – µ)/sx
Where: X1= Sample mean
µ = Population mean
Sx = Standard error of the mean

41. Independent t-test
Independent t-test: the independent-
measures t-test (or independent t-test) is
used when measures from the two samples
being compared do not come in matched
pairs. It is used when groups are
independent.

42. Related Formula
t = x1 – x2/√{s2 (1/n1 + 1/n2)}
For an independent 2 sample t-test, it is
important to know if the 2 samples have
similar variances as we interpret data. The
requirement for variance homogeneity test
may be measure with Levine’s test. Results
for this can be given in SPSS along with the t-
test results.

43. Assumption in 2 sample
independence T-test
1.0 Normality: Assumes that the population distributions are
normal. The t-test is quite robust over moderate violations
of this assumption. It is especially robust if a two tailed test
is used and if the sample sizes are not especially small.
Check for normality by creating a histogram.
2.0 Independent Observations: The
observations within each treatment condition
must be independent.

44. Assumption in 2 sample independence
t-test cont.
3.0 Equal Variances: Assume that the
population distributions have the same
variance. This assumption is quite important
(If it is violated, it makes the test’s averaging
of the 2 variances meaningless).
If it is violated, then use a modification of the t-
test procedures as needed. See
“Understanding the Output” in this section for
how to check this with Levenes Test for Equality
of Variances.

45. Paired Sample T test
The matched-pair t-test (or paired t-test or
paired samples t-test or dependent t-test) is
used when the data from the two groups can
be presented in pairs,
For example where the same people are
being measured in before-and-after
comparison or when the group is given two
different tests at different times (e.g
pleasantness of two different types of
chocolate).

46. Assumptions in paired
sample t-test
1. The first assumption in the paired sample t-test is that only the
matched pair can be used to perform the paired sample t-test.
2. In the paired sample t-test, normal distributions are
assumed.
3. Variance in paired sample t-test: in a paired sample t-
test, it is assumed that the variance of two sample is
same.
4. The data is measurement data-interval/ratio
5. Independence of observation in paired sample t-test:
in a paired sample t-test, observations must be
independent of each other.

47. Formula:
t = d/ √ s2
/n
Where:
d bar is the mean difference between two
samples;
s2
is the sample variance,
n is sample size and
t is a paired sample t-test with n-1 degree of
freedom

48. ANOVA or Analysis of
Variance
So far we have discussed comparing the means of two
populations to each other and comparing the population
mean to another number. However, we often want to
compare many populations to each other.

49. ANOVA or Analysis of Variance
Example:
We may want to compare regeneration rates for three
different tree species in northern Idaho. We would begin by
taking samples from each population and then calculate the
means from the three samples and make an inference about
the population means from this.
It is common since these three mean regeneration rates
would all be different numbers however, this does not mean
that there is a difference between the population means for
the three tree types.
To answer that question we can use a statistical test called
an analysis of variance or ANOVA. This test is widely used in
natural resources, and you are bound to come across it when
reading scientific literature.

50. The ANOVA Assumptions
The use of an ANOVA assumes that:
• All the populations are normally distributed (follow a bell
shaped curve)
• All the population variances are equal,
• And all the samples were taken independently of each other
and are randomly collected from their population.
Generally, our null hypothesis when conducting an ANOVA is
that all the population means are equal and our research
(alternative) hypothesis will be that at least one of the
population means is not equal.

51. The ANOVA Assumptions
Although an ANOVA is widely used and it does
indicate that a population mean is different
than others, it does not tell us which one is
different from the others.
Analysis of variance tests the null hypothesis
that all the population means are equal:
Formula:
Ho: µ1 = µ2 = µ3……. = µa
You can read more from Text books

52. The ANOVA cont.
• By comparing two estimates of variance (………) recall that ……….. is the
variance within each of the “a” treatment populations.) one estimate (called
the mean square error or MSE for short) is based on the variances within
the samples. The MSE is an estimate of ………. Whether or not the null
hypothesis is true. The second estimate (mean square between or MSB for
short) is based on the variance of the sample means. The MSB is only an
estimate of ………. If the null hypothesis is true. If the null hypothesis is
false then MSB estimates something larger than ……… The logic by which
analysis of variance tests the null hypothesis is as follows: if the null
hypothesis is true, then MSE and MSB should be about the same since they
are both estimates of the same quantity (…): however, if the null hypothesis
is false then MSB can be expected to be larger than MSE since MSB is
estimating a quantity larger than ……..
• Therefore, if MSB is sufficiently larger than MSE, the null hypothesis can be
rejected. If MSB is not sufficiently larger than MSE then the null hypothesis
cannot be rejected. How much larger is sufficiently larger.

53. END
•Questions
•Next Class
•Assignments
•AOB
Prof. Joseph M. Keriko
Principal, JKUAT - Nairobi Campus
Professor of Organic Chemistry and
EIA/EA Leader Expert
P.O. Box 39125 – 00623 Nairobi
Tel. 0722-915026
Email: kerikojm@yahoo.co.uk