Charts and Graphs MCQs 4

The post is about Online Charts and Graphs MCQs with Answers. There are 20 multiple-choice questions from data visualizations (charts and graphs, such as histogram, frequency curve, cumulative frequency polygon, bar chart, pie chart, etc.) Let us start with the Online Charts and Graphs MCQs Test now.

Online Quiz Charts and Graphs MCQs

1. The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are below than upper edge of the box?

Charts and Graphs MCQs 4

 

 
 
 
 

2. When the sum of two or more categories equals 100, what chart type is ideally suited for displaying data?

 
 
 
 

3. Numerical methods and graphical methods are specialized procedures used in

 
 
 
 

4. The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are within the box?

Charts and Graphs MCQs 4

 

 
 
 
 

5. The following boxplots represent the entry test marks obtained by boys and girls. The position of the line within the box indicates —————-.

Charts and Graphs MCQs 4

 
 
 
 

6. The following boxplots represent the entry test marks obtained by boys and girls. The length of the graph represents —————–.

Charts and Graphs MCQs 4

 

 
 
 
 

7. What is a suitable way to display the relationship between two continuous variables?

 
 
 
 

8. A frequency curve with a right tail smaller than the left tail is called ————.

 
 
 
 

9. A histogram for an equal class interval is constructed by taking ————- on the x-axis and ————– on the y-axis.

 
 
 
 

10. Which of the graphs is useful to estimate the median and quantile of the data?

 
 
 
 

11. Which of the following is the suitable way to display the average income earned by men and women in a city?

 
 
 
 

12. If 84% of observations in a data set are less than $mean + SD$ then it indicates that data is

 
 
 
 

13. The type of rating scale that represents the response of respondents by marking at appropriate points is classified as

 
 
 
 

14. Which of the graphs is useful to identify the shape of the data?

 
 
 
 

15. The following boxplots represent the entry test marks obtained by boys and girls. The lowest marks obtained by one of the

Charts and Graphs MCQs 4

 
 
 
 

16. If 25% of observations in a data set are outside the interval ($Mean + 2SD$) then it indicates that data is

 
 
 
 

17. The following boxplots represent the entry test marks obtained by boys and girls. Data for marks of boys is ————– as compared to data for marks of girls.

Charts and Graphs MCQs 4

 

 
 
 
 

18. The following boxplots represent the entry test marks obtained by boys and girls. The length of the box represents ———-.

Charts and Graphs MCQs 4

 
 
 
 

19. The following boxplots represent the entry test marks obtained by boys and girls. The boys’ marks are on the average ————- girls’ marks.

Charts and Graphs MCQs 4

 
 
 
 

20. The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are above than lower edge of the box?

Charts and Graphs MCQs 4

 
 
 
 

Online Charts and Graphs MCQs with Answers

  • Which of the following is the suitable way to display the average income earned by men and women in a city?
  • What is a suitable way to display the relationship between two continuous variables?
  • When the sum of two or more categories equals 100, what chart type is ideally suited for displaying data?
  • Numerical methods and graphical methods are specialized procedures used in
  • The type of rating scale that represents the response of respondents by marking at appropriate points is classified as
  • A histogram for an equal class interval is constructed by taking ————- on the x-axis and ————– on the y-axis.
  • A frequency curve with a right tail smaller than the left tail is called ————.
  • If 25% of observations in a data set are outside the interval ($Mean + 2SD$) then it indicates that data is
  • If 84% of observations in a data set are less than $mean + SD$ then it indicates that data is
  • The following boxplots represent the entry test marks obtained by boys and girls. The lowest marks obtained by one of the
  • The following boxplots represent the entry test marks obtained by boys and girls. Data for marks of boys is ————– as compared to data for marks of girls.  
  • The following boxplots represent the entry test marks obtained by boys and girls. The boys’ marks are on the average ————- girls’ marks.
  • The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are below than upper edge of the box?  
  • The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are above than lower edge of the box?
  • The following boxplots represent the entry test marks obtained by boys and girls. What percent of the values are within the box?  
  • The following boxplots represent the entry test marks obtained by boys and girls. The length of the box represents ———-.
  • The following boxplots represent the entry test marks obtained by boys and girls. The length of the graph represents —————–.  
  • The following boxplots represent the entry test marks obtained by boys and girls. The position of the line within the box indicates —————-.
  • Which of the graphs is useful to estimate the median and quantile of the data?
  • Which of the graphs is useful to identify the shape of the data?

Graphs and charts are common methods to get a visual inspection of data. Graphs and charts are the graphical summaries of the data. Graphs represent diagrams of a mathematical or statistical function, while a chart is a graphical representation of the data. In the charts, the data is represented by symbols.

The important features of graphs and charts are (1) Title: the title of charts and graphs tells us what the subject of the chart or graph is, (2) Vertical Axis: the vertical axis tells us what is being measured in the chart and a graph, and (3) Horizontal Axis: the horizontal axis tells us the units of measurement represented.

There are various mathematical and statistical software that can be used to draw charts and graphs. For example, MS-Excel, Minitab, SPSS, SAS, STATA, Graph Maker, Matlab, Mathematica, R, Exlstat, Python, Maple, etc.

Note that

  • All graphs are charts, but not all charts are graphs.
  • Charts present information in a general way.
  • Graphs show the connections between pieces of data.
Online Charts and Graphs MCQs with Answers

R Frequently Asked Questions and Data Analysis

Efficiency of an Estimator

Introduction to Efficiency of an Estimator

The efficiency of an estimator is a measure of how well it estimates a population parameter compared to other estimators. It is possible to have more than one unbiased estimator of a parameter. We should have at least one additional criterion for choosing among the unbiased estimator of the parameter. Usually, unbiased estimators are compared in terms of their variances. Thus, the comparison of variances of estimators is described as a comparison of the efficiency of estimators.

Use of Efficiency

The efficiency of an estimator is often used to evaluate an estimator through the following concepts:

  • Bias: An estimator is unbiased if its expected value equals the true parameter value ($E[\hat{\theta}]=\theta$). The efficiency of an estimator can be influenced by bias; thus, unbiased estimators are often preferred.
  • Variance: Efficiency is commonly assessed by the variance of the estimator. An estimator having a lower variance is considered more efficient. The Cramér-Rao lower bound provides a theoretically lower limit for the variance of unbiased estimators.
  • Mean Squared Error (MSE): Efficiency can also be measured using MSE, which combines both variance and bias. MSE is given by: MSE = $Var(\hat{\theta}) + Bias (\hat{\theta})^2$. An estimator with a lower MSE is more efficient.
  • Relative Efficiency: The relative efficiency compares the efficiency of two estimators, often expressed as the ratio of their variances: Relative Efficiency = $\frac{Var(\hat{\theta}_2)}{Var(\hat{\theta}_1)}, where $\hat{\theta}_1$ is the estimator being compared, and $\hat{\theta}_2$ is a competitor.
Efficiency of an estimator

The efficiency of an estimator is stated in relative terms. If say two estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ are unbiased estimators of the same population parameter $\theta$ and the variance of $\hat{\theta}_1$ is less than the variance of $\hat{\theta}_2$ (that is, $Var(\hat{\theta}_1) < Var(\hat{\theta}_2)$ then $\hat{\theta}_1$ is relatively more efficient than $\hat{\theta}_2$. The ration is $E=\frac{Var(\hat{\theta}_2)}{var(\hat{\theta}_1)}$ is a measure of relative efficiency of $\hat{\theta}_1$ with respect to the $\hat{\theta}_2$. If $E>1$, $\hat{\theta}_1$ is said to be more efficient than $\hat{\theta}_2$.

If $\hat{\theta}$ is an unbiased estimator of $\theta$ and $Var(\hat{\theta})$ is minimum compared to any other unbiased estimator for $\theta$, then $\hat{\theta}$ is said to be a minimum variance unbiased estimator for $\theta$.

It is preferable to make efficient comparisons based on the MSE instead of its variance.

\begin{align*}
MSE(\hat{\theta}) & = E(\hat{\theta} – \theta)^2\\
&= E\left[(\hat{\theta} – E(\hat{\theta}) + E(\hat{\theta}) – \theta \right]\\
&= E\left[ \left(\hat{\theta} – E(\hat{\theta})\right) ^2 + \left(E(\hat{\theta})-\hat{\theta}\right)^2 + 2(\hat{\theta}-E(\hat{\theta}))(E(\hat{\theta}) -\theta)\right]\\
&= E[\hat{\theta} – E(\hat{\theta})]^2 + [E(\hat{\theta})-\theta]^2 \\
&= Var(\hat{\theta}) + (Bias)^2
\end{align*}

where $E[\hat{\theta}-E(\hat{\theta})] = E(\hat{\theta}) – E(\hat{\theta})=0$

Question about the Efficiency of an Estimator

Question: Let $X_1, X_2, \cdots, X_n$ be a random sample of size 3 from a population with mean $\mu$ and variance \sigma^2$. Consider the following estimators of mean $\mu$:

\begin{align*}
T_1 &= \frac{X_1+X_2+X_3}{2}\qquad Sample\,\, mean\\
T_2 &- \frac{X_1 + 2X_2 + X_3}{4} \qquad Weighted \,\, mean
\end{align*}

which estimator should be preferred?

Solution

First, we check the unbiasedness of $T_1$ and $T_2.

\begin{align*}
E(T_1) &= \frac{1}{3} E(X_1 + X_2 + X_3)=\mu\\
E(T_2) &= \frac{1}{4}E(X_1+2X_2 + X_4) = \mu
\end{align*}

Therefore, $T_1$ and $T_2$ are unbiased estimators of $\mu$.

For efficiency, let us check the variances of these estimators.

\begin{align*}
Var(T_1) &= Var\left(\frac{X_1 + X_2 + X_3}{3} \right)\\
&= \frac{1}{9} \left(Var(X_1) + Var(X_2) + Var(X_3)\right)\\
&= \frac{1}{9} (\sigma^2 + \sigma^2 + \sigma^2) = \frac{\sigma^2}{3}\\
Var(T_2) &= Var\left(\frac{X_1 + 2X_2 + X_3}{4}\right)\\
&= \frac{1}{16} \left(Var(X_1) + 4Var(X_2) + Var(X_3)\right)\\
&= \frac{1}{16}(\sigma^2 + 4\sigma^2 + \sigma^2) = \frac{3\sigma^2}{8}
\end{align*}

Since $\frac{1}{3} < \frac{3}{8}$, that is, $Var(T_1) < Var(T_2). The $T_1$ is better estimator of $\mu$ than $T_2$.

Reasons to Use Efficiency of an Estimator

  1. Optimal Use of Data: An efficient estimator makes the best possible use of the available data, providing more accurate estimates. This is particularly important in research, where the goal is often to make inferences or predictions based on sample data.
  2. Reducing Uncertainty: Efficiency reduces the variance of the estimators, leading to more precise estimates. This is essential in fields like medicine, economics, and engineering, where precise measurements can significantly impact decision-making and outcomes.
  3. Resource Allocation: In practical applications, using an efficient estimator can lead to savings in money, time, and resources. For example, if an estimator provides a more accurate estimate with less data, it can result in fewer resources needed for data collection.
  4. Comparative Evaluation: Comparisons between different estimators help researchers and practitioners choose the best method for their specific context. Understanding efficiency allows one to select estimators that yield reliable results.
  5. Statistical Power: Efficient estimators contribute to higher statistical power, which is the probability of correctly rejecting a false null hypothesis. This is particularly important in hypothesis testing and experimental design.
  6. Robustness: While efficiency relates mostly to variance and bias, efficient estimators are often more robust to violations of assumptions (e.g., normality) in some contexts, leading to more reliable conclusions.

In summary, the efficiency of an estimator is vital as it directly influences the accuracy, reliability, and practical utility of statistical analyses, ultimately affecting the quality of decision-making based on those analyses.

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Microsoft Excel Quiz 3

The post is about the Microsoft Excel Quiz. There are 20 multiple-choice questions about Microsoft Excel which serves as a valuable resource for enhancing skills and preparing for exams such as MS-CIT, entrance exams, and university assessments. The Microsoft Excel Quiz encompasses both fundamental and advanced aspects of MS Excel to assist professionals and students alike. Let us start with the Microsoft Excel Quiz Now.

Please go to Microsoft Excel Quiz 3 to view the test

Online Microsoft Excel Quiz with Answers

  • Which of the following would be considered a core capability of spreadsheets?
  • When you open an Excel workbook or spreadsheet, what kind of file is it?
  • On the Excel Home tab, which of the following groups are listed?
  • Which of the following key Data Analyst tasks is typically done last?
  • How are Excel worksheet cells referenced?
  • Which of the following Excel keyboard shortcuts could be used to find how many rows of data you have in a worksheet, assuming you have no empty rows in your data?
  • Which of the following is a valid way of editing existing data in a cell?
  • What is one of the key components of a typical formula?
  • In Excel for the web, how can you format data in cells to use a currency?
  • What are Excel cell references by default?
  • When creating formulas, what is a mixed reference?
  • How can you zoom to a specific area of data in an Excel spreadsheet?
  • What do you use the AutoFill feature for?
  • What character do you type first when you want to start writing a formula?
  • What is one of the functions found in the AutoSum drop-down list?
  • In Excel Desktop, what is one of the function categories on the Formulas tab in the Function Library group?
  • What is one of the ways to apply new data formats to the rest of a column?
  • What tools or features can we use to split a single column with two names in it into two separate columns with a name in each?
  • What do custom filters provide that AutoFilters don’t?
  • According to the video ‘Useful Functions for Data Analysis’, what is one of the most common functions a Data Analyst might use?
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Types of Hypothesis Tests in Statistics

Introduction to Types of Hypothesis Tests

In statistics, hypothesis tests are methods used to make inferences or draw conclusions about a population based on sample data. In this pose, we will discuss the Basic Types of Hypothesis Tests in Statistics. There are three basic types of hypothesis tests, namely (i) Left-Tailed Test, (ii) Right-Tailed Test, and (iii) Two-Tailed Test.

Note that I am not talking about Statistical tools used under specific conditions related to the data type and distribution. I am talking about the nature of the hypotheses being tested. Therefore, I will focus in this post on the area under the curve in the tails. In hypothesis testing, the distribution of the test’s rejection region can be characterized as either one-tailed or two-tailed. The one-tailed tests include both left- and right-tailed tests.

Hypothesis-Testing-Tails-Critical-Region

Left-Tailed Test

The left-tailed tests are used when the null hypothesis is being tested in a claim that the population parameter at least ($\ge$) a given value. Note that the alternative hypothesis then claims that the parameter is less than (<) the value. For example,

A tire manufacturer claims that their tires last on average more than 35000 miles. If one thinks that the claim is false, then one would write the claim as $H_o$, remembering to include the condition of equality. The hypothesis for this test would be: 
$$H_o:\mu\ge 35000$$
$$H_1: \mu<35000$$

One would hope that the sample data would allow the rejection of the null hypothesis, refuting the company’s claim.

The $H_o$ will be rejected in the case above if the sample mean is statistically significantly less than 35000. That is, if the sample mean is in the left-tail of the distribution of all sample means.

Right Tailed Test

The right-tailed test is used when the null hypothesis ($H_0$) being tested is a claim that the population parameter is at most ($\le$) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is greater than (>) the value.

Suppose, you worked for the tire company and wanted to gather evidence to support their claim then you will make the company's claim $H_1$ and remember that equality will not be included in the claim (H_o$). The hypothesis test will be

$$H_0:\mu \le 35000$$
$$H_1:\mu > 35000$$

If the sample data was able to support the rejection of $H_o$ this would be strong evidence to support the claim $H_1$ which is what the company believes to be true.

One should reject $H_o$ in this case if the sample mean was significantly more than 35000. That is, if the sample mean is in the right-tailed of the distribution of all sample means.

Two-Tailed Test

The two-tailed test is used when the null hypothesis ($H_o$ begins tested as a claim that the population parameter is equal to (=) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is not equal to ($\ne$) the value. For example, the Census Bureau claims that the percentage of Punjab’s area residents with a bachelor’s degree or higher is 24.4%. One may write the null and alternative hypotheses for this claim as:

$$H_o: P = 0.244$$
$$H_1: P \ne 0.244$$

In this case, one may reject $H_o$ if the sample percentage is either significantly more than 24.4% or significantly less than 24.4%. That is if the sample proportion was in either tail (both tails) of the distribution of all sample proportions.

Key Differences

  • Directionality: One-tailed tests look for evidence of an effect in one specific direction, while two-tailed tests consider effects in both directions.
  • Rejection Regions: In a one-tailed test, all of the rejection regions are in one tail of the distribution; in a two-tailed test, the rejection region is split between both tails.
Statistics and Data Analysis Types of Hypothesis Tests in Statistics

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