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

The 75th percentile of this distribution is 11 .25 minutes.

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)=(1)/(b-a);\ a<X<b;\ a<b`

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) {(1)/(14-3)} \, dx=0.75\\\\ (1)/(11)\ \cdot\ \int\limits^(x)_(3) {1} \, dx=0.75\\\\(x-3)/(11)=0.75\\\\x-3=8.25\\\\x=11.25`

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