b) Suppose that the time it takes a data collection operator to fill out an electronic form
for a database is uniformly between 1.5 and 2.2 minutes
i. What is the probability that it will take less than two minutes to fill out the
form?
ii. Determine the cumulative distribution function of the time it takes to fill out
the form

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Let's address each part of the problem step by step. ### Given: - The time $$ X $$ to fill out the electronic form is uniformly distributed between $$ a = 1.5 $$ minutes and $$ b = 2.2 $$ minutes. - Thus, $$ X \sim \text{Uniform}(1.5, 2.2) $$. ### Part b.i: Probability that it takes less than 2 minutes For a uniform distribution, the probability that $$ X $$ is less than a specific value $$ c $$ within the interval $$[a, b]$$ is given by: $$ P(X < c) = \frac{c - a}{b - a} $$ Here, $$ c = 2 $$ minutes. $$ P(X < 2) = \frac{2 - 1.5}{2.2 - 1.5} = \frac{0.5}{0.7} = \frac{5}{7} \approx 0.7143 $$ **Answer:** The probability is $$ \frac{5}{7} $$ (approximately 71.43 %). --- ### Part b.ii: Cumulative Distribution Function (CDF) of $$ X $$ The CDF $$ F_X(x) $$ for a uniform distribution is defined as: $$ F_X(x) = \begin{cases} 0 & \text{for } x < a, \ \frac{x - a}{b - a} & \text{for } a \leq x \leq b, \ 1 & \text{for } x > b. \end{cases} $$ Substituting $$ a = 1.5 $$ and $$ b = 2.2 $$: $$ F_X(x) = \begin{cases} 0 & \text{if } x < 1.5, \ \frac{x - 1.5}{0.7} & \text{if } 1.5 \leq x \leq 2.2, \ 1 & \text{if } x > 2.2. \end{cases} $$ **Answer:** Problem b.ii Answer: The CDF Fₓ(x) is 0, for x < 1.5; (x – 1.5) ⁄ 0.7, for 1.5 ≤ x ≤ 2.2; and 1 for all x > 2.2. Thus, Fₓ(x) = 0, x < 1.5; (x−1.5)/0.7, 1.5 ≤ x ≤ 2.2; 1, x > 2.2. **Part b.i: Probability that it takes less than 2 minutes** The probability is $$ \frac{5}{7} $$ (approximately 71.43 %). **Part b.ii: Cumulative Distribution Function (CDF)** $$ F_X(x) = \begin{cases} 0 & \text{if } x < 1.5, \ \frac{x - 1.5}{0.7} & \text{if } 1.5 \leq x \leq 2.2, \ 1 & \text{if } x > 2.2. \end{cases} $$

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