A one factor design (also known as a one-way ANOVA) is a statistical method used to determine if there are significant differences between the means of multiple groups. In this design, there is one independent variable (factor) with multiple levels or categories.
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Suppose $y_{ij}$ is the response is the $i$th treatment for the $j$th experimental unit, where $i=1,2,\cdots, I$. The statistical model for a completely randomized one-factor design that leads to a One-Way ANOVA is
$$y_{ij} = \mu_i + e_{ij}$$
where $\mu_i$ is the unknown (population) mean for all potential responses to the $i$th treatment, and $e_{ij}$ is the error (deviation of the response from population mean).
The responses within and across treatments are assumed to be independent and normally distributed random variables with constant variance.
One Factor Design’s Statistical Model
Let $\mu = \frac{1}{I} \sum \limits_{i} \mu_i$ be the grand mean or average of the population means. Let $\alpha_i=\mu_i-\mu$ be the $i$th group treatment effect. The treatment effects are constrained to add to zero ($\alpha_1+\alpha_2+\cdots+\alpha_I=0$) and measure the difference between the treatment population means and the grand mean.
Therefore the one way ANOVA model is $$y{ij} = \mu + \alpha_i + e_{ij}$$
Comparing means of multiple groups: When one wants to determine if there are significant differences in the mean of a dependent variable across different groups or levels of a factor.
Exploring the effect of a categorical variable: When one wants to investigate how a categorical variable influences a continuous outcome.
Assumptions of One-Factor ANOVA
Normality: The data within each group should be normally distributed.
Homogeneity of variance (Equality of Variances): The variances of the populations from which the samples are drawn should be equal.
Independence: The observations within each group should be independent of each other.
When to Use One Factor Design
When one wants to compare the means of multiple groups.
When the independent variable has at least three levels.
When the dependent variable is continuous (e.g., numerical).
Note that
If The Null hypothesis is rejected, one can perform post-hoc tests (for example, Tukey’s HSD, Bonferroni) to determine which specific groups differ significantly from each other.
Remember: While one-factor designs are useful for comparing multiple groups, they cannot establish causation.
Statistics is used as a tool to make appropriate decisions in the face of uncertainty. We all apply statistical concepts in our daily life either we are educated or uneducated. Therefore the importance of Statistics cannot be ignored.
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The information collected in the form of data (observation) from any field/discipline will almost always involve some sort of variability or uncertainty, so this subject has applications in almost all fields of research. The researchers use statistics in the analysis, interpretation, and communication of their research findings.
Some examples of the questions which statistics might help to answer with appropriate data are:
How much better yield of wheat do we get if we use a new fertilizer as opposed to a commonly used fertilizer?
Does the company’s sales are likely to increase in the next year as compared to the previous?
What dose of insecticide is used successfully to monitor an insect population?
Statistical techniques being powerful tools for analyzing numerical data are used in almost every branch of learning. Statistics plays a vital role in every field of human activity. Statistics has an important role in determining the existing position of per capita income, unemployment, population growth rate, housing, schooling medical facilities, etc in a country. Now statistics holds a central position in almost every field like Industry, Commerce, Biological and Physical sciences, Genetics, Agronomy, Anthropometry, Astronomy, Geology, Psychology, Sociology, etc are the main areas where statistical techniques have been developed and are being used increasingly.
Statistics has its application in almost every field where research is carried out and findings are reported. Application of statistics (by keeping in view the importance of statistics) in different fields as follows:
Social Sciences
In social sciences, one of the major objectives is to establish a relationship that exists between certain variables. This end is achieved through postulating hypothesis and their testing by using different statistical techniques. Most of the areas of our economy can be studied by econometric models because these help in forecasting and forecasts are important for future planning.
Plant Sciences
The most important aspect of statistics in plant sciences is its role in the efficient planning of experiments and drawing valid conclusions. A technique in statistics known as “Design of Experiments” helps introduce new varieties. Optimum plot sizes can be worked out for different crops like wheat, cotton, sugarcane, and others under different environmental conditions using statistical techniques.
Physical Sciences
The application of statistics in physical sciences is widely accepted. The researchers use these methods in the analysis, interpretation, and communication of their research findings, linear and nonlinear regression models are used to establish cause and effect relationships between different variables, and also these days computers have facilitated experimentation and it is possible to simulate the process rather than experimentation.
Medical Sciences
The interest may be in the effectiveness of new drugs, the effect of environmental factors, heritability, standardization of various records, and other related problems. Statistics come to the rescue. It helps to plan the next investigation to get trustworthy information from the limited resources. It also helps to analyze the current data and integrate the information with that previously existing.
How statistics is used by banks, insurance companies, Business and economic planning and administration, Accounting and controlling of events, Construction Companies, Politicians
Banks
Banks are a very important economic part of a country. They do their work on the guess that all the depositors do not take their money on the same day. Bankers use probability theory to approximate the deposits and claims to take out their money.
Insurance Companies
Insurance companies play an important role in increasing economic progress. These companies collect payment from the people. They approximate the death rate, accident rate, and average expected life of people from the life tables. The payment per month is decided on these rates.
Business
Business planning for the future is very important such as the price, quality, quantity, demand, etc of a particular product. Businessmen can make correct decisions about the location of the business, marketing of the products, financial resources, etc. Statistics helps a businessman to plan production according to the taste of the customers, the quality of the products can also be checked more efficiently by using statistical methods
The relationship between supply and demand is a very important topic of everyday life. The changes in prices and demands are studied by index numbers. The relation between supply and demand is determined by correlation and regression.
Economic Planning
Economic planning for the future is a very important problem for economists. For example (i) opening of new educational institutions such as schools, and colleges, revision of pay scales of employees, construction of new hospitals, and preparation of government budgets, etc. all these require estimates at some future time which is called forecasting which is done by regression analysis and the different sources of earning, planning of projects, forecasting of economic trends are administered by the use of various statistical techniques.
Accounting and Controlling of Events
All the events in the world are recorded, for example, births, deaths, imports, exports, and crops grown by the farmer etc. These are recorded as statistical data and analyzed to make better policies for the betterment of the nation.
Administrator
An administrator whether in the public or private sector leans on statistical data to provide a factual basis for appropriate decisions.
Politician
A politician uses statistical advantageously to lend support and credence to his argument while elucidating the problems he handles.
Construction Companies
All kinds of construction companies start and run their programs after making judgments about the total cost of the project (job, work). To guess the expenditure a very important statistical technique of estimation is used.
Biology
In biology correlation and regression are used for analysis of hereditary relations. To classify the organization into different classes according to their age, height, weight, hair color, eyebrow color, etc. the rules of classification are tabulation of statistics are used.
The post is about Multiple Choice Questions Related to Sampling and Sampling Techniques Quiz. There are 20 MCQs about Sampling Techniques covering the topics related, to stratified sampling, cluster sampling, and simple random sampling. Let us start with the Sampling Techniques Quiz.
Online Multiple Choice Questions about Sampling and Sampling Techniques with Answers
MCQs Sampling Techniques Quiz with Answers
Sample allocation plan that provides the most precision, given a fixed sample size is
In stratified random sampling with strata weights 0.35, 0.55, and 0.10, and standard deviations 16, 23, and 19, and sample sizes 70, 110, and 20, the variance of the sample mean estimator is?
A group consists of 300 people and we are interviewing all members of a given group called
When the procedure of selecting the elements from the population is not based on probability is known as
Problem of non-Response
If the respondents do not provide the required information to the researcher, then it is known as
If larger units have more probability of their inclusion in the sample, the sampling is known as
The cluster sampling method differs from the stratified sampling method in that
One similarity between the stratified sampling method and the cluster sampling method is that
In a cluster random sampling method
In a stratified random sampling method
Suppose that all the units in the population are divided into ten mutually exclusive groups. The word “mutually exclusive” means that
When the population is badly affected, which type of sampling is appropriate?
A randomly selected sample of 1000 college students was asked whether they had ever used the drug Ecstasy. Sixteen percent (16% or 0.16) of the 1000 students surveyed said they had which one of the following statements about the number 0.16 is correct?
Which ONE of the following is the main problem with using non-probability sampling techniques?
Which of the following is not a characteristic of quota sampling?
Before completing a survey, an individual acknowledges reading information about how and why the data they provide will be used. What is this concept called?
————- is a set of elements taken from a larger population according to certain rules.
The mean of clusters average is the biased estimator of a population mean when
Which of the following would usually require the smallest sample size because of efficiency?
A statistical hypothesis is a claim about a population parameter. For example,
The mean height of males is less than 65 inches tall
The percentage of people favoring a bullet train is about 59%
The daily average expense for a college student is more than Rs. 250
At least 5% of Pakistan earn more than Rs 2,500,000 per year
A statistical method is used to determine if there is enough evidence in sample data to support a claim about a population.
The claimed hypotheses are written in certain statistical and concise forms. For example, the above statements about population can be written as
$H_0: \mu < 65$
$H_0: \pi = 0.59$
$H_0:\mu > 250$
$H_0: \pi \ge 0.05$
If someone is interested in knowing that above stated statistical hypotheses are either true or false, one needs to conduct a hypothesis test. To test a statistical hypothesis, one needs to follow the following basic procedure, to fulfill the requirements.
Draw a random sample from the population of interest (for example, the height of males)
Determine if the results from the sample data are consistent or not with the hypothesis under study.
If the collected sample data is (significantly) different from the claimed hypothesis, then reject the hypothesis as being false. However, if the sampled data is not significantly different, one would not reject the hypothesis.
Statistical Hypotheses Example
Example: Suppose a battery manufacturer claims that the average life of their batteries is at least 300 minutes.
To test this hypothesis, we follow the procedure as
Select a sample of say $n=100$ batteries. The sample of batteries is tested and the mean life of sampled batteries was found to be $\overline{x} = 294$ minutes with a sample standard deviation of $s=204 minutes.
We need to test whether “is this data sufficiently different from the manufacturer’s claim to justify rejecting the claim as false”?
Since the sample drawn is large enough, the Central Limit Theorem allows us to conclude that the distribution of sample means $\overline{x}$ is approximately normal.
If the manufacturer’s claim is correct, then $\mu_{\overline{x}} = \mu \ge 300$ and we will assume that $\mu_{\overline{x}} = \mu = 300$.
The Z-score will be $$Z = \frac{\overline{x} – \mu_{\overline{x}}}{\frac{s}{\sqrt{n}}}=\frac{294-300}{\frac{20}{\sqrt{100}}} = -3.0$$
Search the Probability value from Standard Normal Table, as $P(\overline{x} \le 294)=0.0013$
Decision about Hypothesis
Now one of the following must be true
The assumption that $\mu = 300$ is incorrect
The sample drawn has a so small mean that only 13 in 10,000 samples have a mean as low.
The probability of the second statement being true is quite small (0.0013). Thus there is strong evidence to believe that the first statement is true, and hence the manufacturer overstated the mean life of their batteries.
