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GATE 2020 ST – Question 18

Estimation · Confidence Intervals · 1 mark · Multiple choice

A sample X₁,…,X₁₀ has Weibull density $3\theta x^2e^{-\theta x^3}$ for x>0 (zero otherwise), θ>0. Let χ²ν,a be the(1−a) quantile. Which is a90% confidence interval for θ?

  1. $(\chi^2_{20,.95}/(2\sum X_i^3),\chi^2_{20,.05}/(2\sum X_i^3))$
  2. $(\chi^2_{20,.90}/(2\sum X_i^4),\chi^2_{20,.10}/(2\sum X_i^4))$
  3. $(\chi^2_{10,.90}/(4\sum X_i^3),\chi^2_{10,.10}/(4\sum X_i^3))$
  4. $(\chi^2_{10,.95}/(3\sum X_i^2),\chi^2_{10,.05}/(3\sum X_i^2))$

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Correct answer: (A) $(\chi^2_{20,.95}/(2\sum X_i^3),\chi^2_{20,.05}/(2\sum X_i^3))$

Explanation

Transform Zi=Xi³; then Zi is exponential with rateθ. Hence2θΣXi³ is χ²20. Place that pivot between its5th and95th percentiles and divide by2ΣXi³. With the stated quantile indexing, this gives A.

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