Question

The round off error when measuring the distance that a long jumper has jumped is uniformly distributed between 0 and 5 mm. Round all answers to two decimal places. A. The mean of this distribution is ___ B. The standard deviation is____ C. The probability that the round off error for a jumper's distance is exactly 2.5 mm is P(x = 2.5) = _____ D. The probability that the round off error for the distance that a long jumper has jumped is between 2 and 4 mm is P(2 < x < 4) = _____ E. The probability that the jump's roundoff error is greater than 1 is P(x > 1) = _____ F. P(x > 4 | x > 2) = = probability that x is greater than 4, given that it is greater than 2 = ______ G. Find the 60th percentile. H. Find the minimum for the upper quartile.

Answer 1

Answer:

Solution to all the parts are shown below.            

Step-by-step explanation:

We are given the following information in the question:

The distance of a long jumper is uniformly distributed between 0 and 5 mm.

Probability Distribution:

$f(x) = \displaystyle\frac{1}{b-a} = \frac{1}{5}$

$A) \text{ Mean} = \displaystyle\frac{a+b}{2} = \frac{0+5}{2} = 2.5$

$B) \text{ Standard Deviation} = \sqrt{V(x)} = \sqrt{\displaystyle\frac{(b-a)^2}{12}} = \sqrt{\displaystyle\frac{25}{12}} = 1.44\\\\C) P(x = 25) = 0\\\text{Since, in a uniform distribuion probability of a fixed point is always zero}\\\\D) P(2<x<4)\\=\displaystyle\int^4_2\frac{1}{5}dx = \frac{4-2}{5} = \frac{2}{5}\\\\\\E) P(x>1)\\=\displaystyle\int^5_1\frac{1}{5}dx = \frac{5-1}{5} = \frac{4}{5}\\\\\\F) P(x>4|x>2)\\\\=\displaystyle\frac{P(x>4)\cap P(x>2)}{P(x>2} = \frac{P(x>2)}{P(x>2)} = 1$

$G) \text{ 60th percentile}\\P(x<k) = 0.60 \Rightarrow k\times \displaystyle\frac{1}{5} = 0.60\\\\k = 0.60\times 5 = 3$

$H)\text{ Upper Quartile} = \text{ 75th percentile} \\P(x<k) = 0.75 \Rightarrow k\times \displaystyle\frac{1}{5} = 0.75\\\\k = 0.75\times 5 = 3.75$