Question

the round off errors when measuring the distance that a long jumper has jumped is uniformly distributed between 0 and 4.5 mm. round values to 4 decimal places when possible.
a. the mean of this distribution is
b. the standard deviation is
c. the probability that the round off error for a jumpers distance is exactly 2 is p(x = 2) =
d. the probability that the round off error for the distance that a long jumper has jumped is between 0.2 and 2.8 mm is p(0.2 < x < 2.8) =
e. the probability that the jumps round off error is greater than 1 is p(x > 1) =
f. p(x > 1.7 | x > 0.6) =
g. find the 18th percentile.
h. find the maximum for the lower quartile.
hint:
written hint ?
helpful videos: probability +, conditional probability + conditional probability + conditional probability + percentiles +

Explanation:

Step1: Recall mean formula for uniform distribution

For a uniform distribution $X\sim U(a,b)$, the mean $\mu=\frac{a + b}{2}$. Here $a = 0$ and $b=4.5$, so $\mu=\frac{0 + 4.5}{2}=2.2500$.

Step2: Recall standard - deviation formula for uniform distribution

The standard deviation $\sigma=\sqrt{\frac{(b - a)^2}{12}}$. Substituting $a = 0$ and $b = 4.5$, we get $\sigma=\sqrt{\frac{(4.5-0)^2}{12}}=\sqrt{\frac{20.25}{12}}\approx1.2990$.

Step3: Probability at a single point for continuous distribution

For a continuous distribution, the probability at a single point is 0. So $P(X = 2)=0.0000$.

Step4: Calculate probability for an interval

The probability density function of $X\sim U(a,b)$ is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$. Here $f(x)=\frac{1}{4.5-0}=\frac{1}{4.5}$. Then $P(0.2

Step5: Calculate right - tailed probability

$P(X>1)=\frac{4.5 - 1}{4.5}=\frac{3.5}{4.5}\approx0.7778$.

Step6: Calculate conditional probability

By the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. For $A:X > 1.7$ and $B:X>0.6$, $P(X>1.7|X > 0.6)=\frac{P(X>1.7)}{P(X>0.6)}=\frac{\frac{4.5 - 1.7}{4.5}}{\frac{4.5 - 0.6}{4.5}}=\frac{4.5 - 1.7}{4.5 - 0.6}=\frac{2.8}{3.9}\approx0.7179$.

Step7: Find the 18th percentile

The $k$th percentile $x_k$ of a uniform distribution $U(a,b)$ is given by $x_k=a+(b - a)\frac{k}{100}$. For the 18th percentile ($k = 18$), $x_{18}=0+(4.5-0)\frac{18}{100}=0.8100$.

Step8: Find the lower quartile

The lower quartile is the 25th percentile. Using the formula $x_k=a+(b - a)\frac{k}{100}$ with $k = 25$, we get $x_{25}=0+(4.5-0)\frac{25}{100}=1.1250$.

Answer:

a. 2.2500
b. 1.2990
c. 0.0000
d. 0.5778
e. 0.7778
f. 0.7179
g. 0.8100
h. 1.1250