Question

The waiting time in line at an ice cream shop has a uniform distribution between 3 and 14 minutes. What is the 75th percentile of this distribution? (Recall: The 75th percentile divides the distribution into 2 parts so that 75% of area is to the left of 75th percentile) _______ minutes Answer: (Round answer to two decimal places.)

Answer 1

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:

$f_{X}(x)=\frac{1}{b-a};\ a$

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:

$P (X < x) = 0.75$

$\int\limits^{x}_{3} {\frac{1}{14-3}} \, dx=0.75\\\\ \frac{1}{11}\ \cdot\ \int\limits^{x}_{3} {1} \, dx=0.75\\\\\frac{x-3}{11}=0.75\\\\x-3=8.25\\\\x=11.25$

Thus, the 75th percentile of this distribution is 11 .25 minutes.