BSc CSIT (TU) Science Statistics II (BSc CSIT, STA210) Question Paper 2077 Nepal

Sampling

Sampling is the statistical process of selecting a subset (a sample) of individuals or items from a larger group (the population) in order to estimate characteristics of the whole population. It is used because studying the entire population (a census) is often costly, time-consuming, or practically impossible.

Sampling methods are broadly divided into probability and non-probability sampling.

A. Probability Sampling

Every unit of the population has a known, non-zero chance of selection. Results can be generalized and sampling error can be estimated.

1. Simple Random Sampling

Every unit has an equal chance of selection (lottery method or random numbers).

• Merits: Unbiased; easy to analyse; sampling error measurable.
• Demerits: Needs a complete sampling frame; may not represent small subgroups; expensive for widely scattered populations.

2. Stratified Random Sampling

Population divided into homogeneous strata, then random samples drawn from each.

• Merits: Greater precision; ensures representation of every subgroup.
• Demerits: Requires prior knowledge of strata; complex; faulty stratification reduces efficiency.

3. Systematic Sampling

Every $k^{th}$ unit is selected after a random start, where $k = N/n$.

• Merits: Simple, quick, evenly spread over the frame.
• Demerits: Biased if the list has a hidden periodic pattern.

4. Cluster / Multistage Sampling

Population divided into clusters; some clusters are selected and all (or sampled) units within them studied.

• Merits: Economical; no full frame of units needed; suited to geographically spread populations.
• Demerits: Higher sampling error; less precise than other methods.

B. Non-Probability Sampling

Units are selected on a non-random basis; probability of selection is unknown, so sampling error cannot be measured.

1. Convenience Sampling

Units chosen because they are easy to reach.

• Merits: Fast and cheap. Demerits: Highly biased, not generalizable.

2. Judgement (Purposive) Sampling

Expert chooses units believed to be representative.

• Merits: Useful for small/specialized studies. Demerits: Subjective; depends on investigator's judgement.

3. Quota Sampling

Units selected to fill fixed quotas for sub-groups.

• Merits: Quick; ensures representation of groups. Demerits: Selection within quota is biased.

4. Snowball Sampling

Existing respondents recruit further respondents.

• Merits: Good for hidden/rare populations. Demerits: Strong selection bias.

Conclusion

Probability sampling is preferred when accuracy and generalization are required, whereas non-probability sampling is used when speed, cost, or accessibility dominate.

Analysis of Variance (ANOVA)

ANOVA, developed by R. A. Fisher, is a technique used to test the equality of means of three or more populations simultaneously by partitioning the total variation in the data into components attributable to different sources. It compares the variance between groups with the variance within groups using the $F$-statistic.

Assumptions: observations are independent, drawn from normal populations, and the populations have equal variances (homogeneity).

One-Way ANOVA Procedure

A single factor with $k$ treatments (groups) and $N$ total observations is studied.

Step 1 — Hypotheses

$$H_0: \mu_1 = \mu_2 = \dots = \mu_k \quad\text{vs}\quad H_1: \text{at least one mean differs}$$

Step 2 — Grand total and correction factor

$$T = \sum x_{ij}, \qquad CF = \frac{T^2}{N}$$

Step 3 — Sum of squares

$$SST = \sum x_{ij}^2 - CF \quad (\text{Total})$$ $$SSB = \sum_{j} \frac{T_j^2}{n_j} - CF \quad (\text{Between treatments})$$ $$SSE = SST - SSB \quad (\text{Within / Error})$$

where $T_j$ is the total of the $j^{th}$ group having $n_j$ observations.

Step 4 — Degrees of freedom Between $= k-1$, Error $= N-k$, Total $= N-1$.

Step 5 — Mean squares and F-ratio

$$MSB = \frac{SSB}{k-1}, \qquad MSE = \frac{SSE}{N-k}, \qquad F = \frac{MSB}{MSE}$$

ANOVA Table

Step 6 — Decision: Compare calculated $F$ with the table value $F_{\alpha,(k-1,N-k)}$. If $F_{cal} > F_{tab}$, reject $H_0$ and conclude that the treatment means differ significantly.

Theory of Estimation

Estimation is the branch of statistical inference concerned with using sample data to assign numerical values (estimates) to the unknown parameters of a population (e.g., mean $\mu$, variance $\sigma^2$, proportion $P$). A sample statistic used for this purpose is called an estimator, and a particular numerical value it takes is an estimate.

Estimation is of two types: point estimation and interval estimation.

Point Estimation vs Interval Estimation

In interval estimation the interval is called a confidence interval and the probability $(1-\alpha)$ that it contains the parameter is the confidence coefficient.

Properties of a Good Estimator

1. Unbiasedness: The expected value of the estimator equals the parameter, $E(\hat{\theta}) = \theta$. (e.g., $E(\bar{x}) = \mu$.)
2. Consistency: As the sample size $n \to \infty$, the estimator converges to the true parameter value.
3. Efficiency: Among unbiased estimators, the one with the smallest variance is the most efficient.
4. Sufficiency: A sufficient estimator uses all the information in the sample relevant to the parameter, leaving nothing more to be gained from the data.

An ideal estimator is unbiased, consistent, efficient, and sufficient.

Karl Pearson's Coefficient of Correlation

Karl Pearson's coefficient of correlation, denoted $r$, measures the degree and direction of the linear relationship between two quantitative variables $X$ and $Y$. It is defined as the ratio of the covariance of the variables to the product of their standard deviations:

$$r = \frac{\text{Cov}(X,Y)}{\sigma_X \,\sigma_Y} = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x})^2}\,\sqrt{\sum (y-\bar{y})^2}}$$

Properties

1. Range: $r$ always lies between $-1$ and $+1$, i.e. $-1 \le r \le +1$.
2. Direction: $r > 0$ indicates positive correlation, $r < 0$ negative correlation, and $r = 0$ no linear correlation.
3. Unit-free: $r$ is a pure number, independent of the units of measurement.
4. Independent of change of origin and scale: correlation is unaffected by adding/subtracting a constant or multiplying/dividing by a positive constant.
5. Symmetric: $r_{xy} = r_{yx}$.
6. It is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{xy}\cdot b_{yx}}$, taking the sign of the regression coefficients.

Regression Coefficients

In a linear regression between two variables $X$ and $Y$, the regression coefficient is the slope of the regression line and measures the average change in the dependent variable for a unit change in the independent variable.

• Regression coefficient of $Y$ on $X$:

$$b_{yx} = r\,\frac{\sigma_y}{\sigma_x} = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2}$$

• Regression coefficient of $X$ on $Y$:

$$b_{xy} = r\,\frac{\sigma_x}{\sigma_y} = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sum(y-\bar{y})^2}$$

Properties

1. The correlation coefficient is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{yx}\cdot b_{xy}}$.
2. Both regression coefficients have the same sign, which is also the sign of $r$.
3. The product of the two regression coefficients cannot exceed 1: $b_{yx}\cdot b_{xy} = r^2 \le 1$.
4. If one regression coefficient is greater than 1, the other must be less than 1.
5. Regression coefficients are independent of change of origin but not of change of scale.
6. The arithmetic mean of the two regression coefficients is greater than or equal to $r$ (when $r>0$).

Sampling Distribution

If all possible samples of a fixed size $n$ are drawn from a population and a statistic (such as the mean $\bar{x}$, proportion, or variance) is computed for each sample, the probability distribution of that statistic over all such samples is called the sampling distribution of the statistic.

For example, the sampling distribution of the mean $\bar{x}$ describes how sample means vary from sample to sample. By the Central Limit Theorem, for large $n$ this distribution is approximately normal with

$$E(\bar{x}) = \mu, \qquad \text{Var}(\bar{x}) = \frac{\sigma^2}{n}.$$

Standard Error (S.E.)

The standard error is the standard deviation of the sampling distribution of a statistic. It measures the variability of the statistic due to sampling and is a key indicator of the precision/reliability of an estimate.

• S.E. of the mean: $\;\text{S.E.}(\bar{x}) = \dfrac{\sigma}{\sqrt{n}}$
• S.E. of a proportion: $\;\text{S.E.}(p) = \sqrt{\dfrac{PQ}{n}}$

Uses of S.E.: it is used to construct confidence intervals, to test hypotheses (test statistic = (estimate − parameter)/S.E.), and to judge accuracy. A smaller standard error (larger $n$) indicates a more reliable estimate.

Confidence Interval for a Population Mean

A confidence interval (CI) is a range of values, computed from a sample, that is expected to contain the unknown population mean $\mu$ with a stated probability $(1-\alpha)$, called the confidence level (e.g., 95%).

Case 1: Population variance $\sigma^2$ known (or large sample, $n \ge 30$)

Use the standard normal ($Z$) distribution:

$$\bar{x} \pm Z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$$

where $\bar{x}$ is the sample mean and $Z_{\alpha/2}$ is the critical value (e.g., 1.96 for 95%, 2.58 for 99%).

Case 2: Population variance unknown and small sample ($n < 30$)

Replace $\sigma$ by the sample standard deviation $s$ and use the $t$-distribution with $(n-1)$ degrees of freedom:

$$\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}$$

Steps

1. Compute the sample mean $\bar{x}$ (and $s$ if needed).
2. Fix the confidence level and obtain the critical value $Z_{\alpha/2}$ or $t_{\alpha/2,n-1}$.
3. Compute the standard error $\sigma/\sqrt{n}$ (or $s/\sqrt{n}$).
4. Compute the margin of error $E = (\text{critical value}) \times (\text{S.E.})$.
5. The interval is $(\bar{x} - E,\ \bar{x} + E)$.

Interpretation: A 95% CI means that if the sampling were repeated many times, about 95% of such intervals would contain the true mean $\mu$.

F-test for Equality of Two Population Variances

The F-test is used to test whether two normal populations have equal variances, based on the ratio of two independent sample variances. It is the basis for ANOVA and for testing the homogeneity assumption.

Hypotheses

$$H_0: \sigma_1^2 = \sigma_2^2 \qquad \text{vs} \qquad H_1: \sigma_1^2 \neq \sigma_2^2$$

Test Statistic

Given two independent samples of sizes $n_1$ and $n_2$ with unbiased sample variances

$$s_1^2 = \frac{\sum(x_1-\bar{x}_1)^2}{n_1-1}, \qquad s_2^2 = \frac{\sum(x_2-\bar{x}_2)^2}{n_2-1},$$

the statistic is

$$F = \frac{s_1^2}{s_2^2}, \quad \text{with } s_1^2 > s_2^2 \ (\text{larger variance in numerator}),$$

which follows the $F$-distribution with $(n_1-1, n_2-1)$ degrees of freedom.

Decision Rule

Compare $F_{cal}$ with the table value $F_{\alpha,(n_1-1,\,n_2-1)}$.

• If $F_{cal} > F_{tab}$ → reject $H_0$ (variances differ significantly).
• If $F_{cal} \le F_{tab}$ → accept $H_0$ (no significant difference).

Assumptions: both samples are random, independent, and drawn from normal populations.

Index Numbers

An index number is a statistical measure that expresses the relative change in the level of a variable (or group of variables) such as price, quantity, or value, over time or between places, with respect to a fixed base period (taken as 100). They are often called economic barometers.

A price index measures the relative change in the prices of a basket of commodities between the base year (prices $p_0$, quantities $q_0$) and the current year (prices $p_1$, quantities $q_1$).

Laspeyres' Price Index

Uses base-year quantities ($q_0$) as weights:

$$P_{01}^{L} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$$

• Merit: Requires only base-year weights, so easy to compute over time.
• Demerit: Ignores changes in consumption pattern; tends to overestimate the rise in prices.

Paasche's Price Index

Uses current-year quantities ($q_1$) as weights:

$$P_{01}^{P} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$$

• Merit: Reflects current consumption pattern.
• Demerit: Current-year weights must be collected every period (costly); tends to underestimate the rise in prices.

Note: Fisher's ideal index is the geometric mean of the two, $P^F = \sqrt{P^L \times P^P}$.

Addition and Multiplication Theorems of Probability

Addition Theorem

The addition theorem gives the probability of the union of events (occurrence of at least one event).

For any two events $A$ and $B$:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

If $A$ and $B$ are mutually exclusive ($A \cap B = \varnothing$):

$$P(A \cup B) = P(A) + P(B)$$

Example: Drawing one card from a deck of 52, $P(\text{King or Queen}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$ (mutually exclusive).

Multiplication Theorem

The multiplication theorem gives the probability of the joint occurrence (intersection) of events.

For any two events:

$$P(A \cap B) = P(A)\cdot P(B\mid A) = P(B)\cdot P(A\mid B)$$

If $A$ and $B$ are independent:

$$P(A \cap B) = P(A)\cdot P(B)$$

Example: Tossing two fair coins, $P(\text{both heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ (independent events).

Summary: the addition theorem deals with "OR" (union) of events, while the multiplication theorem deals with "AND" (intersection) of events.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of occurrences of a rare event in a fixed interval of time, space, or area, when the events occur independently and at a constant average rate $\lambda$.

A random variable $X$ follows a Poisson distribution if its probability mass function is

$$P(X = x) = \frac{e^{-\lambda}\,\lambda^{x}}{x!}, \qquad x = 0, 1, 2, \dots; \ \lambda > 0,$$

where $\lambda$ is the average number of occurrences (the parameter) and $e \approx 2.718$.

It is obtained as a limiting case of the binomial distribution when $n \to \infty$, $p \to 0$, with $np = \lambda$ finite.

Mean and Variance

A distinctive property is that the mean equals the variance:

$$\text{Mean} = \lambda, \qquad \text{Variance} = \lambda.$$

Applications

1. Number of telephone calls received at an exchange per minute.
2. Number of printing/typing errors per page of a book.
3. Number of accidents at a junction per day.
4. Number of defective items in a large batch (rare defects).
5. Number of customers arriving at a counter in a given time (queueing theory).
6. Number of radioactive particle emissions per unit time.

Random Variable

A random variable is a real-valued function that assigns a numerical value to each outcome of a random experiment (each point in the sample space). It is usually denoted by capital letters $X, Y, Z$.

Example: In tossing two coins, if $X$ = number of heads, then $X$ takes values $0, 1, 2$.

Random variables are of two types: discrete and continuous.

Discrete vs Continuous Random Variables

Examples:

• Discrete: number of children in a family $(0, 1, 2, \dots)$.
• Continuous: the exact height of a student (e.g., any value such as 165.3 cm).