Solved 2024: FYUG Even Semester Exam, 2024<br>STATISTICS (2nd Semester)<br>Statistical Methods and Probability

Course No.: STADSM-151T
Full Marks: 70 | Time: 3 Hours
Instructions: Answer any ten from Section-A and five from Section-B. (Full paper solved here).

SECTION-A (2 x 10 = 20 Marks)

1. Define 'statistics' and mention some limitations.

2 Marks

Definition: Statistics is the science concerned with collecting, organizing, analyzing, interpreting, and presenting data

.

Limitations:

It does not study qualitative phenomena directly (like honesty or beauty) unless they are quantified.

It deals with aggregates (groups) rather than individual items.

Statistical results are true only on average.

2. Define qualitative and quantitative data with examples.

2 Marks

Qualitative Data: Data describing attributes or characteristics that cannot be measured numerically. Example: Eye color, Gender.

Quantitative Data: Data that can be expressed numerically and measured. Example: Height (cm), Income (dollars).

3. What are discrete and continuous data? Give examples.

2 Marks

Discrete Data: Data that takes only specific, isolated values (usually integers). Example: Number of children in a family.

Continuous Data: Data that can take any value within a given range. Example: Temperature of a room.

4. Define mode and give its formula for continuous distribution.

2 Marks

Mode: The value that appears most frequently in a data set

.

Formula:

Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] * h Where: l = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of preceding class, f2 = frequency of succeeding class, h = class width.

5. What are measures of dispersion? Name them.

2 Marks

Definition: Measures that describe the spread or variability of data around a central value

.

Measures: Range, Quartile Deviation, Mean Deviation, and Standard Deviation

.

6. What are raw and central moments of the rth order?

2 Marks

Raw Moment (mu'_r): The expected value of the rth power of the variable X about the origin. Formula: mu'_r = E(X^r).

Central Moment (mu_r): The expected value of the rth power of the deviation of X from its mean. Formula: mu_r = E[(X - Mean)^r].

7. Write properties of Karl Pearson's correlation coefficient (r).

2 Marks

The value of r lies between -1 and +1 inclusive (-1 <= r <= 1).

It is independent of change of origin and scale.

It is a pure number (dimensionless).

8. Prove: If bXY > 1 then bYX < 1.

2 Marks

We know that r^2 = bXY * bYX

. Since r^2 <= 1, it follows that bXY * bYX <= 1. If bXY > 1, then for the product to be less than or equal to 1, bYX must be less than 1.

9. Mention two properties of regression coefficients.

2 Marks

Regression coefficients are independent of change of origin but not scale.

The geometric mean of two regression coefficients is the correlation coefficient.

10. Define an attribute with an example.

2 Marks

Definition: A qualitative characteristic that cannot be measured quantitatively but can only be categorized. Example: Marital status (Married, Single, Divorced).

11. Define 'trial' and 'event'.

2 Marks

Trial: A performance of a random experiment. Example: Tossing a coin.

Event: A possible outcome or a collection of outcomes of a trial. Example: Getting a 'Head'.

12. Give the axiomatic definition of probability.

2 Marks

Probability P(A) is a real-valued function satisfying:

Non-negativity: P(A) >= 0.

Certainty: P(S) = 1.

Additivity: For mutually exclusive events A and B, P(A union B) = P(A) + P(B).

13. State the addition theorem of probability.

2 Marks

For any two events A and B:

P(A union B) = P(A) + P(B) - P(A intersect B)

14. If A and B are independent, P(A intersect B) = 0.25, P(A) = 0.5, find P(B).

2 Marks

For independent events, P(A intersect B) = P(A) * P(B)

.
0.25 = 0.5 * P(B) => P(B) = 0.25 / 0.5 = 0.5.

15. If A and B are mutually exclusive, what is P(A intersect B)?

2 Marks

By definition, mutually exclusive events cannot occur simultaneously, so P(A intersect B) = 0

.

SECTION-B (10 x 5 = 50 Marks)

16. (a) Types of data used in statistics with examples.

5 Marks

Data is primarily categorized into:

• Primary Data: Collected for the first time by the investigator. Example: Census surveys.
• Secondary Data: Data already collected by others. Example: Using data from a WHO report.
• Qualitative Data: Descriptive data. Example: Blood groups (A, B, AB, O).
• Quantitative Data: Numerical data. Example: Weight of students.

16. (b) Note on graphical representation of statistical data.

5 Marks Graphical representation simplifies complex data and makes comparisons easier. Key types include:

Bar Charts: Used for qualitative or discrete data.

Histograms: Used for continuous frequency distributions.

Pie Charts: Showing proportions of a whole.

Ogive: Representing cumulative frequencies.

17. (a) Represent the given data with a histogram.

5 Marks

The data provided is in an inclusive form (e.g., 15-19, 20-24). First, convert to exclusive class boundaries: 14.5-19.5, 19.5-24.5, etc.

Class Boundaries	Frequency
14.5 - 19.5	9
19.5 - 24.5	11
24.5 - 29.5	10
29.5 - 34.5	44
34.5 - 39.5	45
39.5 - 44.5	54
44.5 - 49.5	37
49.5 - 54.5	26
54.5 - 59.5	8
59.5 - 64.5	5

17. (b) 'Less than' and 'More than' cumulative frequency curves.

5 Marks

Less than Ogive: Points are plotted as (upper limit, cumulative frequency). It is a rising curve.

More than Ogive: Points are plotted as (lower limit, cumulative frequency). It is a falling curve.

Construction: 1. Prepare cumulative frequency table. 2. Plot points on a graph. 3. Join points with a smooth curve. Intersection of both curves gives the Median.

18. (a) Measures of central tendency and requisites of a good measure.

5 Marks

Measures: Arithmetic Mean (AM), Median, Mode, Geometric Mean, Harmonic Mean

.

Requisites: It should be rigidly defined, based on all observations, easy to calculate, and stable under sampling fluctuations

.

Mathematical treatment: Arithmetic Mean is the most amenable to further mathematical treatment

.

18. (b) Relationship between raw and central moments.

5 Marks

Central moments (mu_r) in terms of raw moments (mu'_r) about the origin:

mu_2 = mu'_2 - (mu'_1)^2

mu_3 = mu'_3 - 3*mu'_2*mu'_1 + 2*(mu'_1)^3

19. (a) Definitions of Range, QD, MD, SD, and Variance.

5 Marks

Range: Difference between maximum and minimum value.

Quartile Deviation (QD): (Q3 - Q1) / 2.

Mean Deviation (MD): Average of absolute deviations from a central value.

Standard Deviation (SD): Root mean square deviation from the arithmetic mean.

Variance: Square of Standard Deviation.

19. (b) Define Skewness and Kurtosis with formulas.

5 Marks

Skewness: Measure of asymmetry. Formula (Karl Pearson): (Mean - Mode) / SD.

Kurtosis: Measure of "peakedness" or flatness of a distribution. Formula: beta_2 = mu_4 / (mu_2)^2.

20. (a) Scatter Diagram and Correlation.

5 Marks

Scatter Diagram: A graph plotting pairs of (x, y) values as points

.

Correlation Idea: If points cluster along a rising line, correlation is positive. If falling, it's negative. If points are randomly scattered, correlation is zero.

20. (b) Correlation Coefficients.

5 Marks

Karl Pearson's r: Measure of linear relationship between two variables

.
Spearman's rank: Measure of relationship between ranked data. Formula: 1 - [6 * Sum(d^2) / (n(n^2 - 1))].
Range of r: [-1, +1].

21. (a) Regression lines of X on Y and Y on X.

5 Marks

Y on X: (y - y_bar) = bYX * (x - x_bar)

.
X on Y: (x - x_bar) = bXY * (y - y_bar).

21. (b) Principle of Least Squares for fitting y = a + bx.

5 Marks

Minimize the sum of squared residuals: S = Sum(yi - (a + bxi))^2

. To find 'a' and 'b', differentiate S with respect to 'a' and 'b' and equate to zero. This leads to Normal Equations:

Sum(y) = na + b*Sum(x)

Sum(xy) = a*Sum(x) + b*Sum(x^2)

22. (a) Probability Definitions.

5 Marks

Exhaustive: All possible outcomes.

Mutually Exclusive: Cannot happen at once.

Equally Likely: No outcome is preferred over others.

Independent: Occurrence of one doesn't affect the other.

22. (b) Set theoretic expressions for events.

5 Marks

Only A: A intersect B' intersect C'

All three: A intersect B intersect C

At least two: (A intersect B) union (B intersect C) union (A intersect C)

A and B but not C: A intersect B intersect C'

None: A' intersect B' intersect C'

25. (a) Bayes' Theorem and Manager Appointment Problem.

5 Marks

Bayes' Theorem: Provides the posterior probability of an event given prior evidence

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Let X, Y, Z be manager appointments. P(X)=4/9, P(Y)=2/9, P(Z)=3/9. Let B = Bonus Scheme. P(B|X)=3/10, P(B|Y)=5/10, P(B|Z)=8/10.
P(B) = (4/9 * 3/10) + (2/9 * 5/10) + (3/9 * 8/10) = (12+10+24)/90 = 46/90.
P(X|B) = [P(X)*P(B|X)] / P(B) = (12/90) / (46/90) = 12/46 = 6/23.