Question

A cake weighing one kilogram is cut into two pieces, and each piece is weighed separately. Denote the measured weights of the two pieces by x and y . Assume that the errors in obtaining x and y are independent and normally distributed with mean zero and the same (unknown) variance. Devise a test for the hypothesis that the true weights of the two pieces are equal.

Answer 1

Answer:

Step-by-step explanation:

Arithmetic, geometric and harmonic progressions.  Trigonometry.  Two dimensional

coordinate geometry:  Straight lines, circles, parabolas, ellipses and hyperbolas.

Elementary set theory.  Functions and relations.  Elementary combinatorics:  Per-

mutations and combinations, Binomial and multinomial theorem.

Theory of equations.

Complex numbers and De Moivre’s theorem.

Vectors  and  vector  spaces.   Algebra  of  matrices.   Determinant,  rank,  trace  and

inverse of a matrix.  Solutions of linear equations.  Eigenvalues and eigenvectors of

matrices.

Limits and continuity of functions of one variable.  Differentiation.  Leibnitz for-

mula.  Applications of differential calculus, maxima and minima.  Taylor’s theorem.

Indefinite integral.  Fundamental theorem of calculus.  Riemann integration and prop-

erties.  Improper integrals.

Statistics and Probability

Notions  of  sample  space  and  probability.   Combinatorial  probability.   Conditional

probability and independence.  Bayes Theorem.  Random variables and expectations.

Moments and moment generating functions.  Standard univariate discrete and con-

tinuous distributions.  Distribution of functions of a random variable.  Distribution of

order statistics.  Joint probability distributions.  Marginal and conditional probability

distributions.  Multinomial distribution.  Bivariate normal and multivariate normal

distributions.

Sampling distributions of statistics.  Statement and applications of Weak law of

large numbers and Central limit theorem.

Descriptive statistical measures.  Contingency tables and measures of association.

Product  moment  and  other  types  of  correlation.   Partial  and  multiple  correlation.

Simple and multiple linear regression.

Elementary theory of estimation (unbiasedness, minimum variance, sufficiency).

Methods  of  estimation  (maximum  likelihood  method,  method  of  moments).   Tests

of hypotheses (basic concepts and simple applications of Neyman-Pearson Lemma).

Confidence intervals.  Inference related to regression.  ANOVA. Elements of nonpara-

metric inference.

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Basic