How are Weibull parameters η (eta) and β (beta) estimated from life data?

• A. Using a Weibull probability plot or maximum likelihood estimation; on the plot, slope gives β and intercept yields η.
• B. Using a straight-line fit to lifetimes in chronological order; slope gives η, intercept yields β.
• C. By counting failures at specific times and averaging, which yields β only.
• D. By computing the average life and magnifying.

Answer: (A)

Estimating Weibull parameters relies on transforming the data into a form that shows a straight-line relationship when the data follow a Weibull distribution. The Weibull distribution has a scale η and a shape β, and a common practice is to use either a Weibull probability plot or maximum likelihood estimation.

On a Weibull probability plot, you compute the empirical failure probabilities F(t) for your failure times t, then plot ln(-ln(1−F(t))) versus ln t. If the data follow a Weibull model, the points fall on a straight line. The slope of that line is the shape parameter β, and the intercept is ln η, so you recover η as η = exp(intercept). This graphical method gives intuitive estimates and can incorporate censored data as needed in practice.

Maximum likelihood estimation is the other standard method: you write the likelihood of the observed lifetimes (and any censored observations) under the Weibull distribution, then find η and β that maximize this likelihood, typically with numerical optimization.

The other options aren’t correct because fitting lifetimes in chronological order isn’t the right transform for the Weibull model, averaging failures yields only a rough summary rather than both parameters, and simply computing an average life with magnification doesn’t provide the proper parameter estimates.