Design of Experiments Quiz 6

Online Quiz about Design of Experiments Quiz Questions with Answers. There are 20 MCQs in this DOE Quiz cover the basics of the design of experiments, hypothesis testing, basic principles, and single-factor experiments, fixed effect models, random effect models. Let us start with “Design of Experiments MCQs with Answer”. Let us start with the Design of Experiments Quiz Questions with Answers now.

Design of Experiments Quiz with Answers

Online design of experiments quiz with Answers

1. For one factor ANOVA, the model contains:

 
 
 
 

2. A researcher is interested in measuring the rate of production of five particular machines. The model will be a:

 
 
 
 

3. If the experimenter is interested in the variation among treatment means not the treatment means themselves. The model used is called:

 
 
 
 

4. If an interaction effect in a factorial design is significant the main effects of the factors involved in that interaction may be difficult to interpret.

 
 

5. For ANOVA we assume that treatments are applied to the experimental units:

 
 
 
 

6. One factor ANOVA means, there is only:

 
 
 
 

7. A factorial experiment can be run as an RCBD by assigning the runs from each replicate to separate blocks.

 
 

8. One of the ANOVA assumptions is that treatments have:

 
 
 
 

9. A fixed effect model is used when the effect of —————– is assumed to be fixed during the experiment.

 
 
 
 

10. To compare the IQ level of five students a series of tests is planned and IQ is computed based on their results. The model will be:

 
 
 
 

11. An interaction term in a factorial model with quantitative factors introduces curvature in the response surface representation of the results.

 
 

12. In a random effects model ————- are randomly chosen from a large population.

 
 
 
 

13. In a random effect model:

 
 
 
 

14. If the experiment were to be repeated and the same set of treatments would be included, we choose:

 
 
 
 

15. Single-factor ANOVA is also called:

 
 
 
 

16. In a fixed effect model:

 
 
 
 

17. If the treatments in a particular experiment are a random sample from a large population of similar treatments. we choose:

 
 
 
 

18. Factorial experiments cannot be used to detect the presence of interaction.

 
 

19. The treatment effect is associated with:

 
 
 
 

20. The experimenter is interested in treatment means only. The model used is called:

 
 
 
 

Design of Experiments Quiz with Answers

  • If an interaction effect in a factorial design is significant the main effects of the factors involved in that interaction may be difficult to interpret.
  • Factorial experiments cannot be used to detect the presence of interaction.
  • An interaction term in a factorial model with quantitative factors introduces curvature in the response surface representation of the results.
  • A factorial experiment can be run as an RCBD by assigning the runs from each replicate to separate blocks.
  • One of the ANOVA assumptions is that treatments have:
  • For ANOVA we assume that treatments are applied to the experimental units:
  • One factor ANOVA means, there is only:
  • For one factor ANOVA, the model contains:
  • Single-factor ANOVA is also called:
  • In a fixed effect model:
  • In a random effect model:
  • The treatment effect is associated with:
  • If the experiment were to be repeated and the same set of treatments would be included, we choose:
  • The experimenter is interested in treatment means only. The model used is called:
  • A fixed effect model is used when the effect of —————– is assumed to be fixed during the experiment.
  • A researcher is interested in measuring the rate of production of five particular machines. The model will be a:
  • To compare the IQ level of five students a series of tests is planned and IQ is computed based on their results. The model will be:
  • If the treatments in a particular experiment are a random sample from a large population of similar treatments. we choose:
  • If the experimenter is interested in the variation among treatment means not the treatment means themselves. The model used is called:
  • In a random effects model ————- are randomly chosen from a large population.

MCQs General Knowledge

Consistency: A Property of Good Estimator

Consistency refers to the property of an estimator that as the sample size increases, the estimator converges in probability to the true value of the parameter being estimated. In other words, a consistent estimator will yield results that become more accurate and stable as more data points are collected.

Characteristics of a Consistent Estimator

A consistent has some important characteristics:

  • Convergence: The estimator will produce values that get closer to the true parameter value with larger samples.
  • Reliability: Provides reassurance that the estimates will be valid as more data is accounted for.

Examples of Consistent Estimators

  1. Sample Mean ($\overline{x}$): The sample mean is a consistent estimator of the population mean ($\mu$). A larger sample from a population converges to the actual population mean, compared to a smaller smaller.
  2. Sample Proportion ($\hat{p}$): The sample proportion is also a consistent estimator of the true population proportion. As the number of observations increases, the sample proportion gets closer to the true population proportion.

Question: $\hat{\theta}$ is a consistent estimator of the parameter $\theta$ of a given population if

  1. $\hat{\theta}$ is unbiased, and
  2. $var(\hat{\theta}) \rightarrow 0$ when $n\rightarrow \infty$

Answer: Suppose $X$ is random with mean $\mu$ and variance $\sigma^2$. If $X_1,X_2,\cdots,X_n$ is a random sample from $X$ then

\begin{align*}
E(\overline{X}) &= \mu\\
Var(\overline{X}) & = \frac{\sigma^2}{n}
\end{align*}

That is $\overline{X}$ is unbiased and $\lim\limits_{n\rightarrow\infty} Var(\overline{X}) = \lim\limits_{n\rightarrow\infty} \frac{\sigma^2}{n} =0$

Question: Show that the sample mean $\overline{X}$ of a random sample of size $n$ from the density function $f(x; \theta) = \frac{1}{\theta} e^{-\frac{x}{\theta}}, \qquad 0<x<\infty$ is a consistent estimator of the parameter $\theta$.

Answer: First, we need to check that $E(\overline{x})=\theta$, that is, the sample mean $\overline{X}$ is unbiased.

\begin{align*}
E(X) &= \mu = \int x\cdot f(x; \theta) dx = \int\limits_{0}^{\infty}x\cdot \frac{1}{\theta} e^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta} \int\limits_{0}^{\infty} xe^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta} \left[ \Big| -\theta x e^{-\frac{x}{\theta}}dx\Big|_{0}^{\infty} + \theta \int\limits_{0}^{\infty} e^{-\frac{x}{\theta}}dx \right]\\
&= \frac{1}{\theta} \left[0+\theta(-\theta) e^{-\frac{x}{\theta}}\big|_0^{\infty} \right] = \theta\\
E(X^2) &= \int x^2 f(x; \theta)dx = \int\limits_{0}^{\infty}x^2 \frac{1}{\theta} e^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta}\left[ \Big| – x^2 \theta e^{-\frac{x}{\theta} }\Big|_{0}^{\infty} + \int\limits_0^\infty 2x\theta e^{-\frac{x}{\theta}}dx \right]\\
&= \frac{1}{\theta} \left[ 0 + 2\theta^2 \int\limits_0^\infty \frac{x}{\theta} e^{-\frac{x}{\theta}}dx\right]
\end{align*}

The expression is to be integrated into $E(X)$ which equals 0. Thus

\begin{align*}
E(X^2) &=\frac{1}{\theta} 2\theta^2\theta = 2\theta^2\\
Var(X) &=E(X^2) – [E(X)]^2 = 2\theta^2 – \theta^2 = \theta^2
and \quad Var(\overline{X}) &= \frac{\sigma^2}{n}\\
\lim\limits_{n\rightarrow \infty} \,\, Var(\overline{X}) &= \lim\limits_{n\rightarrow \infty} \frac{\sigma^2}{n} = 0
\end{align*}

Since $\overline{X}$ is unbiased and $Var(\overline{X})$ approaches 0 and $n\rightarrow \infty$, the $\overline{X}$ is a consistent estimator of $\theta$.

Importance of Consistency in Statistics

The following are a few key points about the importance of consistency in statistics:

Reliable Inferences: Consistent estimators ensure that as sample size increases, the estimates become closer and closer to the true population value/parameters. This helps researchers and statisticians to make sound inferences about a population based on sample data.

Foundation for Hypothesis Testing: Most of the statistical methods rely on consistent estimators. Consistency helps in validating the conclusions drawn from statistical tests, leading to confidence in decision-making.

Improved Accuracy: Since more data points are available due to the increase in sample size, the more consistently the estimates will converge to the true value. All this leads to more accurate statistical models, which can improve analysis and predictions.

Mitigating Sampling Error: Consistent estimators help to reduce the impact of random sampling error. As sample sizes increase, the variability in estimates tends to decrease, leading to more dependable conclusions.

Building Statistical Theory: Consistency is a fundamental concept in the development of statistical theory. It provides a rigorous foundation for designing and validating statistical methods and procedures.

Trust in Results: Consistency builds trust in the findings of statistical analyses. It is because the results are stable and reliable across different samples (due to large samples), therefore it is more likely to accept and act upon those results.

Framework for Model Development: In statistics and data science, developing models based on consistent estimators results in models with more accuracy.

Long-Term Decision Making: Consistency in data interpretation supports long-term planning, risk assessment, and resource allocation. It is required that businesses and organizations often make strategic decisions based on statistical analyses.

https://itfeature.com consistency a property of good estimator

R Frequently Asked Questions

MCQs Charts and Graphs Quiz 6

The post is about Online MCQs Charts and Graphs Quiz with Answers. There are 20 multiple-choice questions from Charts and Graphs (Data Visualizations, such as histogram, frequency curve, cumulative frequency polygon, bar chart, pie chart, heatmap, exploratory data analysis, etc.) Let us start with the Online MCQs Charts and Graphs Quiz Questions with Answers now.

Please go to MCQs Charts and Graphs Quiz 6 to view the test

Online MCQs Charts and Graphs Quiz Questions

  • What spreadsheet software is preferred when multiple users need to collaborate?
  • Which of the following is true about filled map charts?
  • What is the difference between Bar and Column charts?
  • Which chart uses nested rectangles?
  • Which two charts typically have categories arranged on the horizontal axis and values on the vertical axis?
  • When you add a filter to a pivot chart, how do you update the data in the source pivot table?
  • What is the capability of line charts?
  • How does a pivot chart differ from a standard chart in Excel?
  • What is the main difference between Area and Column charts?
  • What do treemaps use to represent hierarchical data categories?
  • Histograms can look like a bar chart, what’s the key difference?
  • What kind of data works best with filled map charts?
  • What are sparklines typically used for?
  • Which of the following is true about an area chart?
  • Which of the following is true of scatter charts?
  • In which situation is a bar graph preferred over a pie chart?
  • Suppose you want to visualize the results of a study. When assessing only one ordinal or nominal variable it is sufficient to use a (1) ———— When looking at the relationship between two of these ordinal or nominal variables you’d better use a (2) ———— When you’re assessing the correlation between two continuous variables it’s best to use a (3) ————- Fill in the right words on the dots.
  • Which chart is a type of correlation chart?
  • Which plot type is specifically employed to visualize the median and distribution within and across categories?
  • To visualize its distribution, binned data is often plotted in which of the following types of chart?
MCQs Charts and Graphs Quiz with Answers

Computer MCQs, R Language Frequently Asked Questions

MCQs Estimation Quiz 8

MCQs Estimation Quiz from Statistical Inference covers the topics of Estimation (Confidence Interval) and Bayes Factor for the preparation of exams and different statistical job tests in Government/ Semi-Government or Private Organization sectors. This test will also help get admission to different colleges and Universities. The online MCQS Estimation quiz will help the learner understand the related concepts and enhance their knowledge.

Please go to MCQs Estimation Quiz 8 to view the test

MCQs Estimation Quiz with Answers

  • An observed 95% confidence interval does not predict that 95% of the estimates from future studies will fall inside the observed interval.
  • Suppose a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The probability that the given study’s results are replicable is not equal to $1-p$.
  • Suppose that a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The value $p = 0.001$ does not directly confirm that the effect size was large.
  • Suppose that a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The p-value of a statistical test is the probability of the observed result or a more extreme result, assuming the null hypothesis is true.
  • The specific 95% confidence interval observed in a study has a 95% chance of containing the true effect size.
  • A Bayes Factor close to 1 (inconclusive evidence) means that the effect size is small.
  • To conclude that the difference between the two estimates is non-significant ($\alpha = 0.05$), the two 95% confidence intervals around the means do not overlap.
  • If two 95% confidence intervals around the means overlap, then the difference between the two estimates is necessarily non-significant ($\alpha = 0.05$).
  • Suppose that a research article indicates a value of p = .30 in the results section ($\alpha = 0.05$). The probability that the given study’s results are replicable is not equal to $1-p$.
  • Suppose that a research article indicates a value of $p = 0.30$ in the results section ($\alpha = 0.05$). You have absolutely proven the null hypothesis (that is, you have proven that there is no difference between the population means).
  • How are the three paths to statistical inference (frequentist, likelihood, Bayesian) related to each other?
  • Two researchers are investigating if people can see in the future. Person A believes there is no effect, which would mean that p-values are distributed as a —————-. B finds a test statistic at the very far end of the distribution, which means that —————-.
  • The probability of finding a significant result when there is no true effect is called ————– The probability of finding a significant result when there is a true effect, is called —————.
  • The likelihood ratio of the two hypotheses gives information about ————–, but not about —————-.
  • When a Bayesian t-test yields a $BF = 10$, it is ten times more likely that there is an effect than that there is no effect.
  • A Bayes Factor that provides strong evidence for the alternative model does not mean the alternative hypothesis is true.
  • When a Bayesian t-test yields a $BF = 0.1$, it is ten times more likely that there is no effect than that there is an effect.
  • Suppose that a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The p-value gives the probability of obtaining a significant result whenever a given experiment is replicated.
  • A Bayes Factor that provides strong evidence for the null model does not mean the null hypothesis is true.
  • Suppose, the Bayesian method is used to estimate a population mean of 10 with a 95% credible interval from 8 to 12, which means ————–. This interval depends on —————.
Online MCQs Estimation Quiz with Answers

Statistical inference is a branch of statistics in which we conclude (make some wise decisions) about the population parameter using sample information. Statistical inference can be further divided into the Estimation of the Population Parameters and the Hypothesis Testing.

Estimation is a way of finding the unknown value of the population parameter from the sample information by using an estimator (a statistical formula) to estimate the parameter. One can estimate the population parameter by using two approaches (I) Point Estimation and (ii) Interval Estimation.

R Programming Language