Unit 5 Business Statistics and Research Methods PYQ’s 17th Oct 2020 Shift 1

The correct answer is – 2 Median – 3 Quartile Deviation = 2 Mean

In statistics, for a moderately skewed distribution, there exists a relation between mean, median, and mode. This mean median and mode relationship is known as the “empirical relationship”

Important Note

• Mean is the average of the data set which is calculated by adding all the data values together and dividing it by the total number of data sets.
• Median is the middle value among the observed set of values and is calculated by arranging the values in ascending order or in descending order and then choosing the middle value.
• The mode is the number from a data set that has the highest frequency and is calculated by counting the number of times each data value occurs.

Key Takeaways

In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median. Thus, the empirical relationship as Mean – Mode = 3 (Mean – Median).

If solved further, Mode = 3 Median – 2 Mean and 3 Median = 2 Mean + Mode

Hence 2 Median – 3 Quartile Deviation = 2 Mean is NOT true in a model distribution.

The correct answer is – 0.3077

Probability:

1. Probability is the study of experiments.
2. Experiments result in outcomes (also called simple events).
3. A set or collection of outcomes is called an event.
4. These concepts can be illustrated with a deck of cards.

1. Pulling a card from the deck is an experiment. Getting the ace of spades is an outcome (or simple event) (there are 52 possible outcomes for the experiment of drawing a card from a deck. Events include getting a spade (which event contains 13 outcomes) or getting a king (which event contains 4 outcomes).

Key Takeaways

Number Of King Cards = 4 (One each of Spades ♠, Hearts ♥, Clubs ♣, and Diamonds♦)

Number of Spade ♠ Cards = 13 (These include one King too, which we have already counted above)

So, the Number of King and Spade’s cards taken together = 16

Hence, probability of drawing a King or Spades card = 16/52 = 4/13 = 0.3077

Thus, option A is the correct answer.

The correct answer is – √npq

Binomial Distribution:

1. A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times.
2. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice).
3. For example, a coin toss has only two possible outcomes: heads or tails, and taking a test could have two possible outcomes: pass or fail.

Key Takeaways

The binomial distribution has the following properties:

1. The mean of the distribution (μ_x) is equal to np.
2. The variance (σ²_x) is npq.
3. The standard deviation (σ_x) is √npq.

Where,

• n: The number of trials in the binomial experiment.
• P: The probability of success on an individual trial.
• Q: The probability of failure on an individual trial. (This is equal to 1 – P.)

Thus, option 4 is the correct answer.

The correct answer is – SE_X̄ = σ√n √N − nN − 1

Standard Error:

1. The standard error of the mean is the standard deviation of the sampling distribution of the mean.
2. In other words, it is the standard deviation of a large number of sample means of the same sample size drawn from the same population.
3. The term standard error of the mean is commonly (though imprecisely) shortened to just standard error.
4. The “standard error of the mean” refers to the standard deviation of the distribution of sample means taken from a population.

1. The smaller the standard error, the more representative the sample will be of the overall population.

Key Takeaways

In sampling without replacement, the formula for the standard deviation of all sample means for samples of size n must be modified by including a finite population correction. The formula becomes:

σ_X̄ = σ√n √N − nN − 1 where N is the population size.

Thus, option 2 is the correct answer.

Additional Notes

When sampling with replacement the standard deviation of all sample means equals the standard deviation of the population divided by the square root of the sample size when sampling with replacement.

The correct answer is – Data array

Compression is the conversion of data to a format that requires less storage space.

Data Array:

1. The simple array is one of the simplest ways to present data.
2. It is an arrangement of given raw data in ascending or descending order.
3. In ascending order the scores are arranged in increasing order of their magnitude. For example, numbers 2,4,7,8,9,12, are arranged in ascending order.
4. In descending order the scores are arranged in decreasing order of their magnitude. For example, numbers 12, 9, 8, 7, 4,2, are arranged in descending order.
5. The simple array has several advantages as well as disadvantages over raw data.
6. Using a Simple array, we can easily point out the lowest and highest values in the data and the entire data can be easily divided into different sections.
7. Repetition of the values can be easily checked, and the distance between succeeding values in the data can be observed on the first look.

1. But sometimes a data array is not very helpful because it lists every observation in the array. It is cumbersome for displaying large quantities of data. Thus, it represents an uncompressed form of data.

Therefore, option 1 is the right answer.

Additional Notes

Frequency Distribution:

1. Frequency distributions summarize and compress data by grouping it into Return to Table of Contents classes and recording how many data points fall into each class.
2. That is, they show how many observations on a given variable have a particular attribute.
3. For example, a survey is taken from 50 people regarding their favorite color. The frequency distribution might indicate 15 people selected green, 12 blue, 6 red, 7 yellow, and 10 purple.
4. Converting these raw numbers into percentages would then provide an even more useful description of the data. The frequency distribution is the foundation of descriptive statistics.

Histogram:

1. It is one of the most popular methods for presenting continuous frequency distribution in a form of a graph.
2. In this type of distribution, the upper limit of a class is the lower limit of the following class.
3. The histogram consists of a series of rectangles, with its width equal to the class interval of the variable on the horizontal axis and the corresponding frequency on the vertical axis as its heights.

Ogive: The graph of a cumulative frequency distribution is known as a cumulative frequency curve or ogive. Since there are two types of cumulative frequency distribution e.g., “ less than” and “ more than” cumulative frequencies, we can have two types of ogives.

1. ‘Less than’ Ogive: In ‘less than’ ogive, the less than cumulative frequencies are plotted against the upper-class boundaries of the respective classes. It is an increasing curve having slopes upwards from left to right.
2. ‘More than’ Ogive: In more than ogive, the more than cumulative frequencies are plotted against the lower class boundaries of the respective classes. It is a decreasing curve and slopes downwards from left to right.

The frequency distribution, histogram, and ogive all represent a large amount of data with a much smaller diagram/graph or formula. Thus, they represent a compressed form of data.

The correct answer is – (B) only

The Regression Coefficient is the constant ‘b’ in the regression equation that tells about the change in the value of the dependent variable corresponding to the unit change in the independent variable.

The Regression Coefficient is also called a slope coefficient because it determines the slope of the line i.e. the change in the independent variable for the unit change in the independent variable.

Key Takeaways

Properties Of Regression Coefficient:

1. The correlation coefficient is the geometric mean of two regression coefficients. Symbolically, it can be expressed as:

Coefficient of correlation, r = √b_yx × b_xy

2. The value of the coefficient of correlation cannot exceed unity i.e. 1. Therefore, if one of the regression coefficients is greater than unity, the other must be less than unity.

3. The sign of both the regression coefficients will be the same, i.e. they will be either positive or negative. Thus, it is not possible that one regression coefficient is negative while the other is positive.

4. The coefficient of correlation will have the same sign as that of the regression coefficients, such as if the regression coefficients have a positive sign, then “r” will be positive and vice-versa.

5. The regression coefficients are independent of the change of origin, but not of the scale. By origin, we mean that there will be no effect on the regression coefficients if any constant is subtracted from the value of X and Y. By scale, we mean that if the value of X and Y is either multiplied or divided by some constant, then the regression coefficients will also change.

Thus, option 2 is the correct answer.

The correct answer is – (B), (C) only

The measure of central tendency: The measure of central tendency measures the average value of the sample. In descriptive statistics, there are two types of averages: the first are the mathematical averages and the second are the positional averages.

Key Takeaways

The mathematical averages are of three types: arithmetic mean, geometric mean, and harmonic mean:

1. The arithmetic mean is the most widely used measure for central tendency; it can be obtained by adding all the items of the series and dividing this total by the number of items.
2. In descriptive statistics, the geometric mean is defined as the nth root of the products of all the n values of the variable. In descriptive statistics, the geometric mean is used when the items in the series are very large.
3. The harmonic mean is defined as the reciprocal of the item. The harmonic mean is useful in finding the averages that involve speed, time, price, and ratio.

There are two types of positional average: the median and the mode.

1. The median is the average value of the series in which half the values are less than the median and half the values are greater than the median.
2. The mode, the second positional average, shows a higher frequency in the series.

Thus, option 1 is the correct answer.

The correct answer is – (A), (B), (C) only

Pearson’s correlation coefficient is the test statistics that measures the statistical relationship, or association, between two continuous variables. It is known as the best method of measuring the association between variables of interest because it is based on the method of covariance. It gives information about the magnitude of the association, or correlation, as well as the direction of the relationship.

Key Takeaways

PROPERTIES OF CORRELATION COEFFICIENT:

Property 1: Correlation coefficient lies between -1 and +1: Whenever we calculate the correlation coefficient by any one of the formulae its value always lies between -1 and +1.

Property 2: Correlation coefficient is independent of the change of origin and scale: Correlation coefficient is independent of the change of origin and scale, which means that if a quantity is subtracted and divided by another quantity (greater than zero) from original variables, i.e. U = (X – a)/h and V = (Y – b)/k then correlation coefficient between new variables U and V is same as the correlation coefficient between X and Y, i.e.Corr(x, y) = Corr(u, v).

Property 3: If X and Y are two independent variables then the correlation coefficient between X and Y is zero, i.e.Corr(x, y) = 0.

Statements A, B, and C are not true.

Thus, option 3 is the correct answer.

The correct answer is – (B), (D), (C), (E), (A)

Research is a process of systematic inquiry that entails collection of data; documentation of critical information; and analysis and interpretation of that data/information, in accordance with suitable methodologies set by specific professional fields and academic disciplines.

Key Takeaways

The following stages of research are in a proper sequence:

1. Budgeting is essential in the development of any major business project. Without a well-planned budget, projects can fall apart and be left incomplete. Having a budget allows you to establish the main objectives of a project.

1. Fieldwork refers to the collection of primary data from consumers or business-to-business customers through survey or observation. It is also desirable for a researcher to have sufficient knowledge of the research and data collection processes to be able to interact intelligently with a subcontractor.
2. Data collection is the process of gathering and measuring information on variables of interest, in an established systematic fashion that enables one to answer stated research questions, test hypotheses, and evaluate outcomes.
3. Research Outcomes seeks to understand the end results of particular health care practices and interventions. End results include effects that people experience and care about, such as a change in the ability to function.
4. Research writing is a reliable source to recount details about a conducted research and is most often considered to be a true testimony of all the work done to garner specificities of research.

Therefore, option 1 is the correct answer.

The correct answer is – A – IV, B – III, C – I, D – II

Key Takeaways

Therefore, it is clear from the above explanation that option 4) is the correct answer.