ISOM 3540 2022 Fall Midterm

1. Let X denote a random variable with distribution function given by

$$F_X(x)=\{\begin{array}{ll}0& \text{if }x<0\\ 0.2& \text{if }0\le x\le 4\\ a+\frac{b}{x}& \text{if }4\le x\le 8\\ 1& \text{if }x\ge 8\end{array}$$

(a) Find $a$ and $b$ in the above distribution.

(b) Determine the quantile function $Q(t)$ of $X$.

(c) Find the median of $X$.

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2. Below is the mosaic matrix for the three variables of 856 customers of a restaurant, namely Pay (the job is hourly paid or fixed salaried), Live, Alone and Tried (whether they tried a new cuisine). Describe briefly the relationship of the three variables and demonstrate roughly how we can compute Pearson residuals.

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3. In a study of the relationship between students’ performance in an examination and how long they spent on preparing the examination, we got summary results on $A$, the event that a student received grade “A”, $H$, the event that a student spent more than 20 hours in preparing the examination, and $I$, the event that a student has high IQ level. Assume that $P(A)=0.15$, $P(H\mid A)=0.8$, $P(H\mid A^c)=0.05$, $P(I\mid A)=0.6$ and $P(I\mid A^c)=0.2$. Answer the following two questions.

(a) Suppose Peter spent more than 20 hours in preparing the examination. What is the chance that he received grade “A” in the examination?

(b) Suppose Mary spent more than 20 hours in preparing the examination and she has high IQ level. What is the chance that she received grade “A’ in the examination? State your assumption in the calculation.