Mathematical Statistics: Inferential Statistics Assignment, EED501

This assignment provides a comprehensive overview of inferential statistics, covering various types of variables, measurement methods, and hypothesis testing. It begins by differentiating between categorical and numerical variables, further classifying numerical variables into discrete and continuous types. The assignment then delves into measurement methods, including nominal, ordinal, interval, and ratio measurements, detailing how each is used to measure different types of variables. A significant portion of the assignment is dedicated to hypothesis testing, outlining the stages involved: formulating null and alternative hypotheses, setting the level of significance, computing test statistics, making a decision, and stating a conclusion. The document also explores different types of hypotheses, including simple and composite, as well as one-tailed and two-tailed tests. The assignment references key statistical concepts and provides examples to illustrate the application of these concepts.

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Mathematical statistics
Inferential statistics
Variables are usually of different kinds. There are categorical variables and numerical variables
(Cho and Abe, 2013). The numerical variables can be classified as discrete and continuous
variables. Different types of variables are measured using different methods. This implies that to
perform a statistical test, the method used for measuring categorical variables cannot be very
convenient in measuring continuous variables. Mostly categorical variables take on values that
are called dummy variables to represent them. Dummy variables mean the values that have been
coded by the user and assigned uniquely to the categorical variables, they are just meant to be
variable identifiers. Since numerical variables will only take on values that continuous or discrete
there will be no other measurement method. Since the categorical variable is a measurement
scale whose variable contains a set of categories different scale of measurement is used to
enhance statistical computation of the variable. The measurement methods that are used are the
nominal, ordinal, interval, and ratio measurements. These methods have been elaborated further
in the section below.
A nominal method is a measurement method used when the categorical variable contains
uncorded categories (Bianch et al. 2011). This means that the variable has not been ordered. For
instance, if you consider religion a person can be a Muslim, a Christian, a Hindu, or a pagan.
Such data there is no specific method in which the religion can be arranged say from the most
superior or less superior. This indicates that a person in any religion doesn’t feel that his/her
religion is better than that of another person since there is no order in which it can be measured.
Another example of a nominal measurement is gender. Gender cant is measured using any
quantitative values. For gender, you are either male or female and this no way a person can be

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assigned gender its just natural that you are either male or female. Since such kind of data cannot
give desirable information when computing, its usually coded dummy variable. The dummy
variables are usually coded using n-1 rule. The n-1 rule states that for any nominal variable you
are going to have dummy variables that are less than the nominal variables by one (Taranis et al,
2011). After coding the variable with dummy numbers, they are converted into factors a data
structure in which they are treated as variables with the levels that correspond to the number of a
dummy variable. For illustration if you have a nominal variable called gender it will be assigned
dummy variables 0 or 1, these variables will be assigned to the levels male and female.
Assigning a dummy variable to either male or female doesn’t have any order that is required.
The variable will have two levels, and statistical computing can be conveniently carried out
using various statistical software’s.
Ordinal measurement. This is a type of measurement that is used for ordered categorical
variables (Malhotra et al, 2012). This means that the categorical variable has a certain order
which has to be followed either in ascending or descending order. Such measurement is used for
measuring variables such as students performance. A student can score in the test as follows,
excellent, good, fair or poor. This order in which the variables have been arranged means that the
higher a students perform in a test the best he/she is and the converse is also true. Another
example of an ordinal measurement is the time of the day, it’s either morning, afternoon, or
night. This implies that you can’t move from morning to evening without going through the
afternoon. Ordinal measurement means that a certain order has to be followed that brings in
certain superiority of the highly ranked order.
Interval measurement. This measurement involves assigning values of variables into certain
intervals. The intervals can be of equal width or they can be unequal. Those intervals represent a

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certain level of achievement. Examples of such measurement include the students score in a test,
that can be assigned as follows, 0-39, 40-49, 50-59, 60-69, 70-100. This interval indicates that if
you score within a certain interval you will be awarded a grade that corresponds to that interval.
Finally, there is the ratio measurement. Under this measurement, the variables are measured
using a ratio scale. This measurement method is quantitative in nature. In this case, the presence
of zero indicates that a value is missing. Ratio measurement is similar to interval measurement
except that the true zero has been defined (Javanmard and Montanari, 2014). An example of this
measurement method is the use of a Likert scale. the Likert scale contains values between 0-10.
In this case, the values are given intervals 0-2, 3-5, 6-7, and 8-10. These intervals are then
labeled strongly agree, agree, disagree, and strongly disagree respectively. Most ratio
measurement involves using scales that have already been defined like the Kelvin scale that is
used in most physical sciences and the Fahrenheit scale which is used for measuring temperature.
Each measurement method is used appropriately according to the given data. Ordinal and
nominal variables are more convenient in measuring categorical variables while the interval and
ratio measurement are more convenient in measuring numerical data.
Hypothesis testing is a very significant part of inferential statistics. Inferential statistics also
called inductive statistics involves drawing a conclusion about the population based on sample
data. A statistical hypothesis is a statement about the probability distribution of a random
variable (Walker and Nowacki, 2011). This implies that statistical hypothesis depends on the
distribution of a random variable of the study. The process of hypothesis testing includes the
following stages;
1. Formulate the null hypothesis and the alternative. The null hypothesis is the researcher's
claim which is normally tested. The alternative hypothesis is the hypothesis that is tested

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against the null hypothesis and accepted when the null hypothesis is false. In statistics,
the null hypothesis is denoted by H0 and the alternative hypothesis is denoted by Hα or
(H1)
2. Set the level of significance (α) of the test and the sampling distribution. The level of
significance of the test refers to the probability of committing type 1 error, that is,
rejecting the null hypothesis when it's true (Dryden, 2014). In this step, you also
determine the critical region. In which the critical region refers to the rejection region
3. Compute the test statistics from the sample data. The test statistics is a measure that
results in the rejection or acceptance of the null hypothesis.
4. Make a decision. The decision is done by comparing the test statistics with the critical
values from the critical region. Usually, if the test statistic is greater than the critical
value the null hypothesis is rejected otherwise the null hypothesis is not rejected
5. State the conclusion. After making the decision the conclusion usually is to reject or fail
to reject the null hypothesis at the level of significance that was initially stated.
In hypothesis testing the decisions largely determined by the test statistics and the test, the
statistic is determined by the sample size. As the sample size increases the more accurate it will
be. Also, the level of significance determines in making the decision, most survey research
makes use of a 5% level of significance which is considered optimal (Wackerly et al, 2014).
However, the choice of level of significance depends on the researcher depending on how certain
they want to be about the results. For instance, if you were to perform the following hypothesis
test at 0.05 level of significance;
H0; there is no association between the response and the explanatory variable
Versus

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Hα: there is an association between the response and the explanatory variable.
If the sampling distribution is normal, then the test statistics is a standard normal distribution. If
the test statistic is greater than the critical value, then the null hypothesis is rejected at 5% level
of significance and this lead to the conclusion that the there is an association between the
response variable and the explanatory variable (Cramér, 2016).
In hypothesis testing, there are two types of hypothesis. They include the composite and simple
hypothesis. For simple hypothesis, the probability distribution of the random variable is
completely specified while for composite hypothesis the probability distribution is not specified
(Diamantopoulos et al, 2012). The composite hypothesis can either be a two-tailed test or a one-
tailed test. The one-tailed test implies that the alternative hypothesis can be a one-tailed upper
test or one-tailed lower test. For the two-tailed test, the alternative hypothesis is usually not equal
to the null hypothesis value. For the one-tailed upper test, the alternative hypothesis is usually of
greater than or equal to type while one-tailed lower test the alternative hypothesis is usually of
less than or equal to type. Suppose that a test is performed about the mean of a random variable,
the following are the possible alternative hypothesis that can be carried out;
Two-tailed test
H0: μ= μ0 versus Hα: μ≠ μ0
One-tailed upper test
H0: μ= μ0 versus Hα: μ≥ μ0
One-tailed lower test
H0: μ= μ0 versus Hα: μ≤ μ0

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Where μ0 the null hypothesis that is being tested.
References
Bianchi, F., Nicassio, F., Marzi, M., Belloni, E., Dall'Olio, V., Bernard, L., ... & Di Fiore, P. P.
(2011). A serum circulating miRNA diagnostic test to identify asymptomatic high‐risk
individuals with early stage lung cancer. EMBO molecular medicine, 3(8), 495-503.
Cho, H. C., & Abe, S. (2013). Is two-tailed testing for directional research hypotheses tests
legitimate?. Journal of Business Research, 66(9), 1261-1266.
Cramér, H. (2016). Mathematical methods of statistics (PMS-9) (Vol. 9). Princeton university
press.
Diamantopoulos, A., Sarstedt, M., Fuchs, C., Wilczynski, P., & Kaiser, S. (2012). Guidelines for
choosing between multi-item and single-item scales for construct measurement: a
predictive validity perspective. Journal of the Academy of Marketing Science, 40(3), 434-
449.
Dryden, I. L. (2014). Shape analysis. Wiley StatsRef: Statistics Reference Online.
Javanmard, A., & Montanari, A. (2014). Confidence intervals and hypothesis testing for high-
dimensional regression. The Journal of Machine Learning Research, 15(1), 2869-2909.
Malhotra, N. K., Mukhopadhyay, S., Liu, X., & Dash, S. (2012). One, few or many?: An
integrated framework for identifying the items in measurement scales. International
Journal of Market Research, 54(6), 835-862.

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Taranis, L., Touyz, S., & Meyer, C. (2011). Disordered eating and exercise: development and
preliminary validation of the compulsive exercise test (CET). European Eating Disorders
Review, 19(3), 256-268.
Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with
applications. Cengage Learning.
Walker, E., & Nowacki, A. S. (2011). Understanding equivalence and noninferiority
testing. Journal of general internal medicine, 26(2), 192-196.