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RESEARCH METHODOLOGY INFERENCIAL STATISTICS BY ONDABU IBRAHIM TIRIMBA

RESEARCH METHODS LESSON 3

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RESEARCH METHODOLOGY

INFERENCIAL STATISTICS

BY

ONDABU IBRAHIM TIRIMBA

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

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).

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

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

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.

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.

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.

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

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

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

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.

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.

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%.

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.

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%”.

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.

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.

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.

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.

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.

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.

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

SignificanceSignificance• 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.

Type I and II ErrorsType 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.

Test of HypothesisSteps in Test of HypothesisThe 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.

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.

Statistical Tests

Statistical tests include:

• Linear Regression

• T-test

• ANOVA

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.

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.

Regression lineThe 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.

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.

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.

The Coefficient of Correlation

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

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.

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).

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

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.

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.

Formula

t = (X1 – µ)/sx

Where: X1= Sample mean

µ = Population mean

Sx = Standard error of the mean

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.

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.

 

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.

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.

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).

 

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.

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

 

 

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.

ANOVA or Analysis of VarianceExample: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.

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.

The ANOVA AssumptionsAlthough 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

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.

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: [email protected]