UGC NET Management Unit - VIII Syllabus

Under what conditions does the Binomial distribution approximate a Poisson distribution?

When n is large, p is very small, and np = λ remains finite (rare events with many trials).

Give two typical examples of phenomena modelled by the Poisson distribution.

Number of calls arriving at a switchboard per hour; number of defects per unit; number of accidents per day (rare, independent events over a fixed interval).

What type of variable does the Poisson distribution describe — discrete or continuous?

Discrete; it gives the probability of a count of events (0, 1, 2, …) in a fixed interval of time or space.

List the key properties of the Normal distribution curve.

Bell-shaped and symmetric about the mean; mean = median = mode; asymptotic to the x-axis; total area under the curve = 1; defined by two parameters, μ and σ.

State the empirical (68-95-99.7) rule for the Normal distribution.

About 68% of values lie within μ ± 1σ, about 95% within μ ± 2σ, and about 99.7% within μ ± 3σ.

What is the standard normal distribution and its parameters?

A normal distribution with mean μ = 0 and standard deviation σ = 1; values are denoted by Z.

How is a normal variable X converted to a standard normal variable Z?

Z = (X − μ) / σ, the standardization (z-score) formula.

What is the relationship between mean and standard deviation determining the position and spread of a normal curve?

The mean μ fixes the centre (location) of the curve; the standard deviation σ determines its width/spread (larger σ means a flatter, wider curve).

For a normal distribution, what is the value of skewness and kurtosis?

Skewness = 0 (perfectly symmetric) and it is mesokurtic with kurtosis coefficient β2 = 3.

State the probability density function of the Exponential distribution.

f(x) = λe^(−λx) for x ≥ 0 (and 0 otherwise), where λ > 0 is the rate parameter.

What are the mean and variance of the Exponential distribution?

Mean = 1/λ and Variance = 1/λ²; the standard deviation also equals 1/λ.

What special property makes the Exponential distribution unique among continuous distributions?

The memoryless (lack of memory) property: P(X > s + t | X > s) = P(X > t).

What does the Exponential distribution typically model, and how does it relate to the Poisson distribution?

It models the waiting time between events; if events occur as a Poisson process with rate λ, the inter-arrival times follow an Exponential distribution with parameter λ.

Differentiate a population from a sample.

A population is the entire set of units under study; a sample is a representative subset of the population selected for actual observation.

Distinguish between a parameter and a statistic.

A parameter is a numerical characteristic of a population (e.g., μ, σ); a statistic is a numerical characteristic computed from a sample (e.g., x̄, s).

What is the difference between probability (random) sampling and non-probability sampling?

In probability sampling every unit has a known, non-zero chance of selection (allows estimation of sampling error); in non-probability sampling selection is based on judgement/convenience, with no known selection probability.

Name the four main types of probability sampling.

Simple random sampling, Stratified sampling, Systematic sampling, and Cluster sampling.

Describe stratified random sampling.

The population is divided into homogeneous sub-groups (strata), and random samples are drawn from each stratum, improving representation and precision.

What is systematic sampling?

Selecting every kth unit from an ordered list after a random start, where k = N/n is the sampling interval.

List common non-probability sampling methods.

Convenience sampling, Judgement (purposive) sampling, Quota sampling, and Snowball sampling.

What is sampling error?

The difference between a sample statistic and the true population parameter, arising because only a sample (not the whole population) is studied; it decreases as sample size increases.

State the standard error of the sample mean.

SE(x̄) = σ / √n, the standard deviation of the sampling distribution of the mean.

What does the Central Limit Theorem state?

For a sufficiently large sample size (n ≥ 30), the sampling distribution of the sample mean is approximately normal regardless of the population's shape, with mean μ and standard error σ/√n.

List the standard steps in a hypothesis testing procedure.

1) State H0 and H1; 2) Choose significance level α; 3) Select the appropriate test statistic; 4) Determine the critical/rejection region; 5) Compute the test statistic; 6) Compare and decide to reject or fail to reject H0; 7) Draw the conclusion.

Distinguish between a Type I and a Type II error.

Type I error (α): rejecting a true null hypothesis. Type II error (β): failing to reject a false null hypothesis.

When is a Z-test used rather than a t-test?

A Z-test is used when the sample is large (n ≥ 30) and/or the population standard deviation (σ) is known; the t-test is used for small samples (n < 30) when σ is unknown.

What is the t-test used for and what distribution does it follow?

The t-test compares means for small samples when σ is unknown; it follows Student's t-distribution, which depends on degrees of freedom and is flatter with heavier tails than the normal curve.

What is the F-test used for and how is the F-statistic defined?

The F-test compares two variances or is used in ANOVA to test equality of several means; F = larger variance / smaller variance (S1²/S2²), always ≥ 1, and follows the F-distribution with two degrees of freedom.

What are the main uses of the Chi-square (χ²) test?

Test of goodness of fit, test of independence of attributes (association in a contingency table), and test for a population variance; it is a non-parametric test using frequency data.

State the Chi-square test statistic formula and its key requirement.

χ² = Σ[(O − E)² / E], where O = observed and E = expected frequencies; it requires data in frequencies and expected frequency of each cell generally ≥ 5.