Suppose that a class in elementary statistics was allocated 15 minutes to complete a quiz. It can be assumed that the time X it takes to complete the quiz is uniformly distributed over the interval [0, 15]. Suppose a student from the class is selected at random, what is the probability that the student will take more than 10 minutes to complete the quiz?
Solution
Let $X$ denote the times to complete the quiz uniformly distributed over the interval [0,15]
Given that $X\sim U(0,15)$. The pdf of $X$ is
$$ \begin{aligned} f(x) &= \frac{1}{15-0},0\leq x\leq 15\\ &=\frac{1}{15}, 0\leq x\leq 15 \end{aligned} $$
The probability that the student will take more than 10 minutes to complete the quiz is
$$ \begin{aligned} P(X\geq 10) &= \int_{10}^{15} f(x) \; dx\\ &= \frac{1}{15}\int_{10}^{15} \; dx\\ &= \frac{1}{15}\big[x\big]_{10}^{15} \\ &= \frac{1}{15}\big[ 15-10]\\ &= 0.3333 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
