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Statistical inference — Question 19

QCAA 2021, Paper 2 · 7 marks

Q19 · 2021 · Technology-activeComplex unfamiliar7 marks

QUESTION 19 (7 marks)

Consider the following information for a continuous random variable XX: E(X)=μ=∫−∞∞xp(x) dxE(X)=\mu=\int_{-\infty}^{\infty}xp(x)\,dx and Var⁡(X)=∫−∞∞(x−μ)2p(x) dx\operatorname{Var}(X)=\int_{-\infty}^{\infty}(x-\mu)^2p(x)\,dx. The waiting time (minutes) until workers at a certain call centre receive their nnth phone call, where n∈Z+n\in\mathbb Z^+, is a random variable TT with probability density function f(t)={kntn−1(n−1)!e−t/3,t≥0,0,otherwise,f(t)=\begin{cases}\dfrac{k^nt^{n-1}}{(n-1)!}e^{-t/3},&t\ge0,\\0,&\text{otherwise,}\end{cases} where kk is a positive constant. The waiting time until workers receive their 5th call is collected from a random sample of 80 workers. Determine the probability that the mean waiting time from this sample is more than 16 minutes.
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