MCQs Basic Statistics with Answers 16

The post is about Online MCQs Basic Statistics with Answers. There are 20 multiple-choice questions about variables, data, data classification, data types, measurable and non-measurable characteristics, frequency distribution, tables, and attributes. Let us start with the MCQS Basic Statistics with Answers.

Online Multiple Choice Questions about Basic Statistics

1. Data classified by two characteristics at a time are called

 
 
 
 

2. How many classes are generally used for arranging data

 
 
 
 

3. Data classified by geographical regions is called

 
 
 
 

4. A single value which represents the whole set of data is

 
 
 
 

5. A Characteristic which cannot be measurable is called

 
 
 
 

6. Data arranged in ascending or descending order is called

 
 
 
 

7. $\sum\limits_{i=1}^n X_i=?$

 
 
 
 

8. The amount of milk given by a cow is a

 
 
 
 

9. Data classified by attributes are called

 
 
 
 

10. The number of accidents recorded yesterday in Multan is a

 
 
 
 

11. Data classified by the time of their occurrence is called

 
 
 
 

12. Measurements usually provide

 
 
 
 

13. Data in the population census reports are

 
 
 
 

14. The process of arranging data into rows and columns is called

 
 
 
 

15. Important and basic classification of data are

 
 
 
 

16. Hourly temperature recorded by the Weather Bureau represents

 
 
 
 

17. The number 115.9700 rounded off to the nearest tenth (one decimal place) is

 
 
 
 

18. Counting or enumerations usually provide

 
 
 
 

19. The Colour of hair is a

 
 
 
 

20. Data used by an agency that was originally collected by them are

 
 
 
 

Online MCQs Basic Statistics with Answers

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  • A Characteristic which cannot be measurable is called
  • The number 115.9700 rounded off to the nearest tenth (one decimal place) is
  • $\sum\limits_{i=1}^n X_i=?$
  • A single value which represents the whole set of data is
  • Important and basic classification of data are
  • Data classified by geographical regions is called
  • Data classified by attributes are called
  • Data classified by the time of their occurrence is called
  • Data classified by two characteristics at a time are called
  • Data used by an agency that was originally collected by them are
  • Data in the population census reports are
  • Measurements usually provide
  • Counting or enumerations usually provide
  • Hourly temperature recorded by the Weather Bureau represents
  • The amount of milk given by a cow is a
  • The number of accidents recorded yesterday in Multan is a
  • The colour of hair is a
  • Data arranged in ascending or descending order is called
  • The process of arranging data into rows and columns is called
  • How many classes are generally used for arranging data

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Five Number Summary Statistics

The five number summary statistics is a set of descriptive statistics that summarizes a data set under study. Five number summary statistics consists of five numerical values that divide the data set into four equal parts. The five number summary statistics are also known as quartiles five number summary.

Five Number Summary Statistics includes the following values:

  • Minimum Value: The smallest value in the data set.
  • First Quartile ($Q_1$): The value that separates the lowest 25% of the data from the remaining data sets.
  • Median ($Q_2$): The value that separates the lowest 50% from the highest 50% of the data.
  • Third Quartile ($Q_3$): The value that separates the lowest 75% of the data from the highest 25% of the data.
  • Maximum value: The largest value in the data set.

Visualization of Five Number Summary Statistics

A box plot can visually represent the five number summary statistics. The box plot displays the dataset’s range (Minimum and Maximum), the median ($Q_2$), and the quartiles ($Q_1$ and $Q_2$).

The Five number summary statistics is a useful way to quickly summarize: the central tendency, variability, and distribution of a data set.

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Interquartile Range

The interquartile range (IQR) is a measure of variability that is based on the five number summary of a dataset. It is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$) of a data set. The rectangle in the box plot represents the interquartile range. The box represents the middle 50% of the data (between $Q_1$ and $Q_3$), with a line inside the box marking the median ($Q_2$).

What is a Box Plot

A box plot is a graphical representation of the five number summary statistics. It is also known as a box-and-whisker plot. It is used to see the distribution of the data and to detect outliers graphically/visually.

five number summary statistics box plot

The relative positions of the quartiles and the median can provide clues about the shape of the distribution. For example, if the median is closer to $Q_1$, the distribution might be right-skewed. If the median is closer to $Q_3$, it might be left-skewed. If the median is roughly halfway between $Q_1$ and $Q_3$, the distribution might be roughly symmetric. The whiskers extend from the box to the minimum and maximum values, and sometimes outliers are plotted as individual points beyond the whiskers.

The five-number summary is a valuable tool for understanding the distribution of data and making comparisons between different datasets. It is often used in exploratory data analysis, quality control, and other statistical applications.

How to Compute the Five Number Summary Statistics:

  • First, arrange the data in ascending order.
  • Find the minimum and maximum values in the data set.
  • Find the median:
    • If the number of data points is odd, the median is the middlemost value in the sorted data.
    • If the number of data points is even, the median is the average of the two middlemost middle values of the sorted data.
  • Find $Q_1$ and $Q_3$:
    • $Q_1$ is the median of the lower half of the data (excluding the median if the number of data points is odd).
    • $Q_3$ is the median of the upper half of the data (excluding the median if the number of data points is odd).

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MCQs Sampling and Qualitative Research 13

The post is about MCQs Sampling and Qualitative Research. 20 multiple-choice questions cover the topic from sample, sampling and sampling distributions, and qualitative research. Let us start with the Online MCQs Sampling and Qualitative Research Quiz with Answers.

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Online MCQs Sampling and Qualitative Research

MCQs Sampling and Qualitative Research quiz
  • The types of probability sampling are
  • Three of the methods of unit selection in simple random sampling are
  • In a stratified sampling, the strata
  • A cluster sampling is when
  • In a multi-phase sampling
  • Spatial sampling is a sampling procedure in which
  • Accidental sampling is the sampling procedure
  • Most qualitative researchers
  • Theoretical sampling means that
  • Which of the following is NOT one of the criteria of qualitative sampling?
  • Concerning qualitative research, which of the following is NOT correct?
  • A researcher entered a large restaurant and briefly interviewed the oldest person sitting at every second table. This type of sampling is
  • A researcher interviewed the householder of two randomly selected houses in each of the streets of the Upper-Heights suburb of a new town. This sampling procedure is
  • In a study of attitudes to university policies, a researcher questioned 150 first-year students, 130 second-year students, and 100 third-year students. The sampling procedure used in this study was
  • In a study of attitudes to university policies, a researcher initially chose 150 first-year students, 130 second-year students, and 100 third-year students ($N_1=380$). Then, the researcher chose 25 male and 25 female students from each year group who were finally interviewed ($N_2=150$). The sampling procedure used in this study was
  • A researcher chose a sample by using a sampling frame and taking the person corresponding to the kth number in the list. This procedure is called
  • The author chose the respondents of his cohabitation study by interviewing a few available cohabiting couples and by obtaining the names of new couples from the previous respondents. This procedure is called
  • A researcher compiled a sample by interviewing the first two available respondents and by choosing further respondents according to the information collected from each additional respondent. This sampling procedure is called
  • Which of the following is an example of nonstatistical sampling?
  • What is sampling for groups with considerable variation within but similar to each other called?
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One Sample Hypothesis Test (t-test)

Introduction: One Sample Hypothesis Test

In this post, I will discuss One Sample Hypothesis Test (One Sample t-test). When testing a claim about the mean using sample data with a small number of observations (i.e., sample size), the appropriate t-distribution instead of the standard normal distribution should be used to determine the standardized test statistic, critical values, rejection region, and p-values.

Recall that if the sample of values drawn follows the normal distribution, the sample size (number of observations in the sample) is less than 30, and the population standard deviation is unknown, then the random variable

$$t=\frac{\overline{x} – \mu}{\frac{s}{\sqrt{n}}}$$

has the Student’s t-distribution with $n-1$ degrees of freedom.

The procedure of locating the rejection regions for a t-distribution hypotheses test is the same as for the normal distribution tests, however, the critical values will differ. To find the critical value(s) $t_0$ for a test, determine if the test is one-tailed or two-tailed and the significance level ($\alpha$). The critical values can be found in the t-distribution table by looking up the entry in the column giving the level of significance and the row showing the degrees of freedom.

Note that:

  • For a right-tailed test, $t_0$ is the positive value in the table
  • For a left-tailed test, $t_0$ is the negative of the value in the table
  • For a two-tailed test, there are two critical values $t_0$ both the value and its opposite.
One Sample Hypothesis Testing-Tails-Critical-Region

Assumptions of the One Sample Hypothesis Test (t-test)

  • Independence: Observations in the sample should be independent of each other.
  • Normality: The population from which the sample is drawn should be normally distributed. However, the t-test is relatively robust to violations of normality, especially for larger sample sizes.
  • Random Sampling: The sample should be a random sample from the population.

One Sample Hypothesis Test for Mean

Example 1: SAT Math scores are normally distributed. A sample of SAT Math scores for 16 students has an average score of 522.8 with a sample standard deviation of 154.5. Suppose, one wishes to support the claim that the average SAT Math score exceeds 500 using a level of significance of 0.05.

Solution

Step 1: The null and alternative hypotheses test in this case are

$H_0: \mu \le 500$ vs $H_1: \mu > 500$

From the alternative hypothesis, the test is right-hand-tailed with $\mu_0=500$.

Step 2: Level of Significance is 5% = 0.05

Step 3: Critical Value

Using the t-distribution table with one Tail, 5% level of significance, and $n-1=16-1=15$ degrees of freedom, the critical value is $t_0=1.753$. The rejection region is thus $t\ge 1.7535$.

Step 4: Test Statistics

The standardized test statistic is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{522.8 – 500}{\frac{154.5}{\sqrt{16}}}\\
&= \frac{22.8}{38.625} = 0.59
\end{align*}

Step 5: Interpretation of One Sample Mean Test

Since the standardized test statistic is not in the region of rejection, therefore, one should not reject $H_0$ and so the sample data is not sufficient to support the claim that the average exceeds 500 at the 0.05 level of significance.

Example 2: A biologist measures the weights of anesthetized female grizzly bears during winter. A sample of 14 bears is found to have an average weight of $\overline{X} = 376.6$lbs. with a sample standard deviation of $s=32.5$lbs. Is there sufficient evidence to support the claim that the average weight of all female bears in the area is less than 400 lbs? Use $\alpha=0.01$ level of significance.

Solution:

Step 1: The null and alternative hypotheses in this case

$H_0:\mu \ge 400$ vs $H_1:\mu < 400$

From the alternative hypothesis, the test is left-hand-tailed with $\mu_0=400$.

Step 2: Level of Significance is 1% = 0.01

Step 3: Critical Value

Using the Student’s t-distribution table with one tail, the level of significance $\alpha = 0.01$, and $n-1=14-1=13$ degrees of freedom, the critical value $t_0=-2.650$. The region of rejection is thus $t\le -2.650$.

Step 4: The Test Statistics is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{376.6 – 400}{\frac{32.5}{\sqrt{14}}}\\
&= \frac{-23.4}{8.686} = -2.694
\end{align*}

Since the standardized test statistic is in the rejection region, one should reject the null hypothesis ($H_0$), which supports the claim that the average bear weight is less than 400 lbs.

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