The time it takes to make a kitchen cabinet is uniformly distributed and ranges between 125 to 200 minutes.
a. What is the probability that it takes exactly 180 minutes to make a kitchen cabinet?
b. What is the probability that it takes between 150 to 220 minutes to make a kitchen cabinet?
Solution
The probability density curve is
$$ \begin{aligned} f(x)=\left\{ \begin{array}{ll} \frac{1}{200- 125}=\frac{1}{75} , & \hbox{$125 \leq x\leq 200$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{aligned} $$
a. The the probability that it takes exactly 180 minutes to make a kitchen cabinet is
$P(X=180) = 0$,
because for continuous distribution probability at a point is zero.
b. The probability that it takes between 150 to 220 minutes to make a kitchen cabinet is
$$ \begin{aligned} P(150< X<220) &= P(150< X < 200)\\ &=\int_{150}^{200} f(x) \; dx\\ &= \frac{1}{75}\int_{150}^{200} \; dx\\ &=\frac{1}{75}\big[x\big]_{150}^{200} \\ &= \frac{1}{75}\big[ 200 -150]\\ &= 0.6667 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
