Answer:
The 75th percentile of this distribution is 11 .25 minutes.
Step-by-step explanation:
The random variable X is defined as the waiting time in line at an ice cream shop.
The random variable X follows a Uniform distribution with parameters a = 3 minutes and b = 14 minutes.
The probability density function of X is:
[tex]f_{X}(x)=\frac{1}{b-a};\ a<X<b;\ a<b[/tex]
The pth percentile is a data value such that at least p% of the data-set is less than or equal to this data value and at least (100-p)% of the data-set are more than or equal to this data value.
Then the 75th percentile of this distribution is:
[tex]P (X < x) = 0.75[/tex]
[tex]\int\limits^{x}_{3} {\frac{1}{14-3}} \, dx=0.75\\\\ \frac{1}{11}\ \cdot\ \int\limits^{x}_{3} {1} \, dx=0.75\\\\\frac{x-3}{11}=0.75\\\\x-3=8.25\\\\x=11.25[/tex]
Thus, the 75th percentile of this distribution is 11 .25 minutes.
