Statistical Sciences / Statistical Methods & Experimental Designs / Full Subject

140 Qs Overview

Statistical Methods & Experimental Designs

Official Statistical Sciences Statistical Methods & Experimental Designs complete subject practice test — All questions across all syllabus chapters.

Q.76

ICAR 2023Shift 1 MCQ (Single Correct) +4 −1

The point of inflexion of Normal (mu, sigma^2) curve are :

प्रसामान्य (mu, sigma^2) वक्र के नति परिवर्तन बिंदु (Points of inflexion) कहाँ स्थित होते हैं?

mu +- sigma mu +- sigma mu +- 2sigma mu +- 2sigma mu +- 3sigma mu +- 3sigma mu +- 0.3sigma mu +- 0.3sigma

Mathematical derivation of points of inflexion:
The probability density function of normal distribution is:

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)$$

Correct: Setting the second derivative $f''(x) = 0$ yields:
$$f''(x) = f(x) \left[ \frac{(x - \mu)^2 - \sigma^2}{\sigma^4} \right] = 0$$
$$(x - \mu)^2 = \sigma^2 \implies x = \mu \pm \sigma$$

Hence, the points of inflexion where the curve changes curvature from convex to concave are precisely at $\mu \pm \sigma$.

Incorrect: $\mu \pm 2\sigma$ encompasses approximately 95.45% of the distribution.
Incorrect: $\mu \pm 3\sigma$ encompasses 99.73% of the distribution.
Incorrect: $\mu \pm 0.3\sigma$ is not an inflexion coordinate.

प्रसामान्य वक्र के नति परिवर्तन बिंदुओं की गणितीय व्युत्पत्ति:
प्रसामान्य बंटन का प्रायिकता घनत्व फलन (pdf) $f(x)$ होता है।

सही: नति परिवर्तन बिंदु (Points of inflexion) ज्ञात करने हेतु द्वितीय अवकलज को शून्य के बराबर रखने पर:
$$f''(x) = 0 \implies (x - \mu)^2 - \sigma^2 = 0 \implies x = \mu \pm \sigma$$

अतः प्रसामान्य वक्र के नति परिवर्तन बिंदु $\mu \pm \sigma$ पर स्थित होते हैं जहाँ वक्र की वक्रता बदलती है।

गलत: $\mu \pm 2\sigma$ 95.45% क्षेत्रफल को समाहित करता है।
गलत: $\mu \pm 3\sigma$ 99.73% क्षेत्रफल को समाहित करता है।
गलत: यह एक अमान्य मान है।

Explanation not matching this question? क्या व्याख्या इस प्रश्न से मेल नहीं खा रही? Report

Q.77

ICAR 2023Shift 1 MCQ (Single Correct) +4 −1

In a normal curve, 99% of the observations are included in the range :

प्रसामान्य वक्र में, 99% प्रेक्षण किस परास (Range) में सम्मिलित होते हैं?

x_bar +- 0.67 x_bar +- 0.67 x_bar +- 1.96 x_bar +- 1.96 x_bar +- 2.0 x_bar +- 2.0 x_bar +- 2.58 x_bar +- 2.58

Q.78

ICAR 2023Shift 1 MCQ (Single Correct) +4 −1

In the standard normal distribution, $P(Z \le 0)$ is equal to:

मानक प्रसामान्य बंटन में, $P(Z \le 0)$ का मान किसके बराबर होता है?

$1$ $1$ $0.5$ $0.5$ $> 0$ $> 0$ $< 1$ $< 1$

Q.79

ICAR 2023Shift 1 MCQ (Single Correct) +4 −1

Let $\bar{X}$ be the mean of a random sample of size $5$ from a normal distribution with $\mu = 0$ and variance $\sigma^2 = 125$. Determine $c$ so that $P\{\bar{X} < c\} = 0.90$:

मान लीजिए $\bar{X}$ सामान्य बंटन (जहाँ $\mu = 0$ और प्रसरण $\sigma^2 = 125$ है) से आकार $5$ के एक यादृच्छिक प्रतिदर्श का माध्य है। $c$ का मान ज्ञात कीजिए ताकि $P\{\bar{X} < c\} = 0.90$ हो:

$6.10$ $6.10$ $6.20$ $6.20$ $6.30$ $6.30$ $6.40$ $6.40$

Q.80

ICAR 2023Shift 1 MCQ (Single Correct) +4 −1

A random variable $X$ is normally distributed with zero mean and unit variance. The variance of $X^2$ is:

एक यादृच्छिक चर $X$ शून्य माध्य और इकाई प्रसरण के साथ प्रसामान्य रूप से बंटित है। $X^2$ का प्रसरण क्या होगा?

$0$ $0$ $1$ $1$ $2$ $2$ $4$ $4$

General Agriculture, Economics & Extension

8 Topics

Agronomy, Agro-Climatic Zones, Cropping Systems, Tillage and Soil Fertility Management Major Plant Diseases, Insect Pests, Weeds and Integrated Pest Management (IPM) Plant Physiology, Mineral Nutrition, Genetics and Plant Breeding Fundamentals Historical Milestones, Agricultural Research Infrastructure and ICAR Institutes Micro & Macro Economics: Demand, Supply, Production Costs and Market Structures Agricultural Marketing, Market Intermediaries, Price Policy, MSP and Buffer Stocking Agricultural Finance, Credit Institutions (NABARD, RRBs, Commercial Banks) and KCC Extension Education Principles, Rural Development Programs and Community Reconstruction

Mathematics

9 Topics

Differential Calculus: Limits, Continuity, Differentiability, Successive Differentiation and Expansion of Functions Sequences and Series: Arithmetic, Geometric, Harmonic Progressions and Infinite Series Convergence Matrices and Determinants: Matrix Operations, Rank, Inverse, Systems of Linear Equations and Eigenvalues Sets, Relations, Functions, Mathematical Induction and Theory of Equations Integral Calculus: Definite and Indefinite Integrals, Reduction Formulas and Areas Under Curves Differential Equations: First Order Ordinary Differential Equations, Linear Differential Equations with Constant Coefficients Applications of Derivatives: Tangents, Normals, Maxima-Minima, Mean Value Theorems and Indeterminate Forms Numerical Methods: Interpolation, Numerical Differentiation, Integration (Trapezoidal, Simpson) and Root Finding Coordinate Geometry: Straight Lines, Circles, Parabola, Ellipse and Hyperbola in Two Dimensions