Design of Experiments Objective Questions 3

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MCQs about the Design of Experiments Objective Questions for the preparation of PPSC, FPSC Lecturer Statistics job, and BS, M.Phil, Ph.D. Statistics Degree Programs. Let us start with the Design of Experiments Objective Questions quiz.

Multiple Choice Questions about Design of Experiments.

1. Planning an experiment to obtain appropriate data and drawing inferences out of the data for any problem under investigation is known as _______.

 
 
 
 

2. What will be the F ratio if an experiment is performed in CRD with 3 replications to compare four treatments? The treatment mean sum of squares = 96, error mean sum of squares = 12.

 
 
 
 

3. Analysis of variance is a statistical method of comparing the – – – – – – – – of several populations.

 
 
 
 

4. An experiment is performed in CRD with 3 replications to compare four treatments, then what will be the degrees of freedom for error?

 
 
 
 

5. In systematic designs, the treatments are applied to the experimental units by some

 
 
 
 

6. Which one provide the estimate of experimental error in the design of the experiment

 
 
 
 

7. An experiment is performed in CRD with 3 replications to compare four treatments. Then, the total experimental units will be

 
 
 
 

8. To test the hypothesis about one population variance, the test statistic will be – – – – – – -.

 
 
 
 

9. Manova stands for

 
 
 
 

10. The scientific method for the construction of a statistical layout plan for an experiment is – – – – – -?

 
 
 
 

11. The experiment is performed on

 
 
 
 

12. What will be a degree of freedom (df) for column-wise blocking in $5 \times 5$ Latin Square Design?

 
 
 
 

13. Under one-way variability in environmental conditions, the appropriate design for experimenting will be – – – – – – – -.

 
 
 
 

14. An experiment is performed in CRE with 3 replications to compare four treatments. The treatment sum of squares = 8. If the error sum of square = 12 then what will be the total sum of square

 
 
 
 

15. For a $7 \times 7$ Latin Square Design, there will be ___________ observations.

 
 
 
 

16. One of the purposes of experimental design is to

 
 
 
 

17. If the experimental material is not homogenous and there is one source of variation in the experiment then we use

 
 
 
 

18. There are – – – – – – – – basic principles of sound statistical design.

 
 
 
 

19. The random process of assigning treatments to the experimental units can be done by

 
 
 
 

20. An experiment is performed in CRD with 3 replications to compare four treatments. The treatment sum of squares = 9. If the error sum of squares = 12 then what will be the mean sum of squares for treatment?

 
 
 
 

Planning an experiment to obtain appropriate data and drawing inferences from the data concerning any problem under investigation is known as the design and analysis of the experiments.

Design of Experiments Objective Questions

An experiment deliberately imposes a treatment on a group of objects or subjects to observe the response. The experimental unit is the basic entity or unit on which the experiment is performed. It is an object to which the treatment is applied and the variable under investigation is measured and analyzed.

Single-Factor Design: In a single-factor experiment only a single factor varies while all others are constant. The CRD, RCBD, and LSD are examples of single-factor designs.

Multi-Factor Design: Multi-factor designs are also known as factorial experiments. When several factors are investigated simultaneously in a single experiment, such experiments are known as factorial experiments.

Systematic Designs: In systematic designs treatments are applied to the experimental units by some systematic pattern, i.e., by the choice of the experimenter. For example, the experimenter wishes to test three treatments and he decides to have four repetitions of each treatment.

Randomized Designs: In randomized designs, as the treatments are applied randomly, therefore the conclusions drawn are supported by statistical tests. In this way, inferences are applicable in a wider range and the random process minimizes the systematic error. The analysis of variance techniques is also suitable for randomized designs only.

The purpose of the Design of Experiments is:

  • Get maximum information for minimum expenditure in the minimum possible time.
  • Helps to reduce the experimental error.
  • To ignore spurious effects, if any.
  • To evaluate and examine the outcomes critically and logically.

Design of Experiments Objective Questions

  • Which one provides the estimate of experimental error in the design of the experiment
  • There are – – – – – – – – basic principles of sound statistical design.
  • If the experimental material is not homogenous and there is one source of variation in the experiment then we use
  • An experiment is performed in CRD with 3 replications to compare four treatments. Then, the total experimental units will be
  • An experiment is performed in CRD with 3 replications to compare four treatments, then what will be the degrees of freedom for error?
  • An experiment is performed in CRE with 3 replications to compare four treatments. The treatment sum of squares = 8. If the error sum of square = 12 then what will be the total sum of square?
  • An experiment is performed in CRD with 3 replications to compare four treatments. The treatment sum of squares = 9. If the error sum of squares = 12 then what will be the mean sum of squares for treatment?
  • What will be the F ratio if an experiment is performed in CRD with 3 replications to compare four treatments? The treatment mean sum of squares = 96, error mean sum of squares = 12.
  • Analysis of variance is a statistical method of comparing the – – – – – – – – of several populations.
  • To test the hypothesis about one population variance, the test statistic will be – – – – – – -.
  • Under one-way variability in environmental conditions, the appropriate design for experimenting will be – – – – – – – -.
  • The scientific method for the construction of a statistical layout plan for an experiment is – – – – – -?
  • Manova stands for
  • For a $7 \times 7$ Latin Square Design, there will be ___________ observations.
  • The experiment is performed on
  • The random process of assigning treatments to the experimental units can be done by
  • In systematic designs, the treatments are applied to the experimental units by some
  • Planning an experiment to obtain appropriate data and drawing inferences out of the data for any problem under investigation is known as ___________
  • One of the purposes of experimental design is to
  • What will be a degree of freedom (df) for column-wise blocking in $5 \times 5$ Latin Square Design?
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Classification of Randomized Designs (2023)

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Randomized designs are a type of experimental design where randomization process is used to assign the units (like people or objects) to different treatment groups. The randomization process helps to control for bias and ensures that any observed differences between the groups are likely due to the treatment itself, rather than some other factors.

Randomized Designs

In randomized designs, the treatments are applied randomly, therefore the conclusions drawn are supported by statistical tests. The classification of randomized designs for single-factor are:

Example: A market gardener wants to test three types of peas, $A$, $B$, and $C$, on his land. He divides a square plot into nine equal squares, three to be planted with each type of pea. The problem he then faces is which square to plant with which type.

Classification of Randomized Designs

One method is a Completely Randomized Design (CRD) which might,

123
CAC
BAA
BBC
Allocation of Different Types of Peas Randomly to plots

This would be all right if all the plots were equally desirable. If however, there were prevailing north wind so that the northernmost plots were exposed, he might decide to use, a Randomize Complete Block Design (RCBD).

Randomized Complete Block Design, where each of the types $A$, $B$, and $C$ is planted once in each west-east block.

123
ABC
ACB
CBA
Allocation of Different Types of Pease in each West-East Block

If the gardener also felt that the soil to the east was rather better than that to the west, he would use, a Latin Square Design (LSD).

A Latin Square design, where each type of pea is planted once in each row (west-east), and once in each column (north-south).

Block 1Block 2Block 3
ABC
BCA
CAB
Allocation of Different Types of Pease planted once in each row (West-East), and once in each column (North-South)

For Randomized Designs, Note that

  • Completely Randomized Design (CRD) is a statistical experimental design where the treatments are assigned completely at random so that each treatment unit has the same chance (equal chance) of receiving any one treatment.
  • In CRD any difference among experimental units receiving the same treatment is considered as an experimental error.
  • CRD is applicable only when the experimental material is homogeneous (eg., homogeneous soil conditions in the field).
  • Since soil is heterogeneous in the field, the CRD is not a preferable method in field experiments. Therefore, CRD generally applies to the lab experimental conditions, as in labs, the environmental conditions can be easily controlled.
  • The concept of “local control” is not used in CRD.
  • CRD is best suited for experiments with a small number of treatments.
Design of Experients

The best design for a study will depend on the specific research question and the factors that one needs to control for. By incorporating randomization, you can control for extraneous variables that might influence the outcome and improve the validity of the findings.

However, the choice of the randomized design depends on the specific research question(s) being asked. It is important to consider the strengths and weaknesses of each design before making a decision.

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Chi-Square Distribution ($\chi^2$) Made Easy

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The Chi-square distribution is a continuous probability distribution that is used in many hypothesis tests. The Chi-Square statistic always results in a positive value.

A Chi-Square variate (with $v$ degrees of freedom (df)) is the sum of $v$ independent, squared standard normal variates ($\sum\limits_{i=1}^v z_i^2$). It is denoted by $\chi^2_v$. The variance $s^2$ from a sample of normally distributed observations is distributed as $\chi^2$ with $v$ (the df) as a parameter referred to as df of the calculated variance. Symbolically,

$$\frac{v\cdot s^2}{\sigma^2} \sim \chi^2_v$$

Chi Square Distribution Table

The variance $s^2$ for $n$ observations from a $N(\mu, \sigma^2)$, the df is equal to $v=n-1$. The Chi-Square distribution is also used for the contingency (analysis of frequency) tables as an approximation to the distribution of complex statistics. All the families of Chi-Square distribution are specified by their degrees of freedom.

Chi-Square Family of Distributions

Chi-Square Distribution Case of the Gamma Distribution

The Chi-Square distribution is a particular case of the Gamma Distribution, the pdf is

$$P_{\chi^2}(x) = [2^{v/2}\Gamma(v/2)]^{-1} \chi^{(v-2)/2}e^{-x/2}, \quad x\ge 0$$

where $\Gamma(x)$ is the Gamma Distribution.

Normal Approximation to $\chi^2$

Method 1: The PDF and df of Chi-Square can be approximated by the normal distribution. For large $v$ df, the first two moments $z=\frac{(X-v)}{\sqrt{2v}}$, $X\sim \chi^2$.

Method 2: Fisher approximation (compensates the skewness of $X$)

$$\sqrt{2X} – \sqrt{2v-1} \sim N(0, 1)$$

Method 3: Approximation by Wilson and Hilferty is quite accurate. Defining $A=\frac{2}{9v}$, we have

$$\frac{\sqrt[3]{(X/v)}-1+A}{\sqrt{A}}\sim N(0, 1)$$

For the determination of percentage points

$$\chi^2_{v[P]}=v[z_P\sqrt{A}+1-A]^3$$

Generating Pseudo Random Variates

Following the schema allows the generation of random variates from $\chi^2_v$ distribution with $v>2$ df. It requires to generate serially random variates from the standard uniform $U(0,1)$ distribution.

Let $n=v$ degrees of freedom

\begin{align*}
C1 &= 1 + \sqrt{2/e} \approx 1.8577638850\\
C2 &= \sqrt{n/2}\\
C3 &= \frac{3n^2-2}{3n(n-2)}\\
C4 &= \frac{4}{n-2}\\
C5 &= n-2\\
\end{align*}

FAQs about Chi-Square Distribution

  1. What is the use of Chi-Square Probability Distribution?
  2. By which parameter is the family of $\chi^2$Distribution is specified?
  3. How Pseudo Random variates be used to generate a Chi-Square distribution?
  4. What is a normal approximation to Chi-Square?
  5. For $v>100$, the Chi-Square percentiles may be approximated by?

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