Week 6

Question 1

Degrees of Freedom & Shape of Chi-Squared distribution

Answer 1

D.O.F.#: of observations that are free to vary after sample mean has been calculated

Chi-Squared distribution: the chi-squared distribution is a family of distributions, depending on degrees of freedom:

Question 2

Sample variance & Distribution

Answer 2

X₁, X₂,…Xn is a random sample size “n” from a population with mean μ and variance σ²
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The sample variance is defined as: = s² = Σ ( xi - x ) ²/ ( n - 1 )

s² varies from sample to sample, and therefore a random variable
The probability distribution of s² is the sampling distribution of the sample variance

The mean of the sampling distribution of “ “ s² is :E(s²)= σ²

The variance of the “ “ of “ “ s² is: Var(s²) = 2σ⁴/n-1

If the population distribution is Normal, then: (n-1)s²/σ² ~ X²n-1

Question 3

Estimator v/s Estimate, Point and Interval estimates
(population parameters are often unknown and need to be estimated from the sample data)

Answer 3

• Estimator: a tool that employs sample data to provide an approximation of an unknown parameter, takes the form of a random variable that varies from sample to sample
• Estimate: specific value computed from a particular sample, it takes the form of a number or realized value of the random variable
• a point estimate is a single number
• a confidence interval provides additional information about variability

Width of confidence interval: lower confidence limit < point estimate < upper confidence limit