Explanation

• The variance of a Chi-square distribution with k degrees of freedom is 2k.
• This means that as the degrees of freedom (k) increase, the variance of the Chi-square distribution also increases.
• It indicating greater spread or dispersion in the data.

Explanation

• The Central Limit Theorem (CLT) states that, given certain conditions, the sampling distribution of the sample mean (x̄) will be approximately normal, even if the underlying population distribution is not normal.
• This is true for large sample sizes (typically n > 30).

Explanation

Since X is a uniform variate between 0 and 3, the probability density function (pdf) of X is:

f(x) = 1/3 for 0 ≤ x ≤ 3

To find E(X²), we need to calculate the expected value of X², which is:

E(X²) = ∫x²f(x)dx = ∫(x²/3)dx from 0 to 3

Evaluating the integral, we get:

E(X²) = (1/3)∫x²dx from 0 to 3 = (1/3)[(3³)/3] = 3

So, the expected value of X² is 3.

Explanation

• Regression analysis is a statistical technique used to establish a relationship between two variables (independent and dependent) to predict the value of one variable based on the other.
• It helps in building a prediction model that can forecast the outcome.

Explanation

• When several unbiased samples are drawn from the same population, the sampling technique is Random Sampling.
• In random sampling, each member of the population has an equal chance of being selected, and the samples are drawn independently.
• This allows for the calculation of sampling distributions and the estimation of population parameters.

Explanation

The basic principle of experimental design that is NOT listed is: Confounding.

Explanation

• A Type-II error means accepting an incorrect hypothesis (or failing to reject a false null hypothesis).
• This means that the test or experiment has failed to detect a significant effect or difference when it actually exists.

Explanation

• In the Cramér-Rao inequality, var(T) is the minimum variance bound, which represents the lowest possible variance of an unbiased estimator T.
• This bound is a fundamental limit in estimation theory and is used to evaluate the efficiency of estimators.

Explanation

The product of the first 'n' natural numbers is represented by the formula:

n! (n factorial) = 1 × 2 × 3 × ... × (n-2) × (n-1) × n

This is denoted by the symbol "n!" and is read as "n factorial".

Explanation

• The coefficient of correlation (ρ) is a measure of the linear relationship between two variables.
• It is a dimensionless quantity, which means that it is unaffected by changes in the scale or units of the variables.