Question

The round off errors when measuring the distance that a long jumper has jumped is uniformlydistributed between 0 and 4.1 mm. Round values to 4 decimal places when possible.a. The mean of this distribution isb. The standard deviation isC. The probability that the round off error for a jumper's distance is exactly 2.6 isP(x = 2.6)d. The probability that the round off error for the distance that a long jumper has jumped isbetween 0.7 and 1.2 mm isP(0.7 <3 <1.2)e. The probability that the jump's round off error is greater than 2.72 isP(x > 2.72) =f. P(x > 1.7 x > 0.7) =g. Find the 43rd percentile.h. Find the minimum for the upper quarter.

Answer 1

The area of the mean for a unformly distribution is gievn below

`\text{Mean}=(a+b)/(2)`

Given that a is the minimum or lowest value and b is the maximum or highest value

From the question given,

a= 0; b=4.1

The mean of this distribution is as shown below

`\begin{gathered} \text{Mean}=(0+4.1)/(2) \\ \text{Mean}=(4.1)/(2) \\ \text{Mean}=2.05 \end{gathered}`

Hence the mean of this distribution is 2.05

The standard deviation of the uniformly distribution is given by the formula below:

`s_d=\sqrt[]{((b-a)^2)/(12)}`

The standard deviation is as shown below:

`\begin{gathered} s_d=\sqrt[]{((4.1-0)^2)/(12)} \\ s_d=\sqrt[]{(4.1^2)/(12)} \\ s_d=\sqrt[]{(16.81)/(12)} \\ s_d=\sqrt[]{1.400833} \\ s_d=1.183568 \\ s_d=1.1836(\text{correct to 4 decimal place)} \end{gathered}`

Hence, the standard deviation is 1.1836 to the nearest 4 decimal place

The probability of exactly 2.6 is 0. This is because the probability of finding an exact number of a uniform distribution is 0

c. The probability that the round off error for a jumper's distance is exactly 2.6 is

P(x = 2.6) = 0

The probability that the round off error for the distance that a long jumper has jumped is

between 0.7 and 1.2 mm P(0.7 <3 <1.2) is as shown below:

e. Hence, the probability that the jump's round off error is greater than 2.72 is P(x > 2.72) = 0.3366

P(x > 1.7 x > 0.7) is as shown below:

`\begin{gathered} P(x>1.7\text{ /x>0.7)=}(4.1-1.7)/(4.1-0.7) \\ =(2.4)/(3.4) \\ =0.705882 \\ \approx0.7059(4\text{ decimal place)} \end{gathered}`

f. Hence, P(x > 1.7 / x > 0.7) = 0.7059 correct to 4 decimal place

The 43rd percentile is

`\begin{gathered} 43\text{ \% of the the maximum value} \\ =(43)/(100)*4.1 \\ =0.43*4.1 \\ =1.763 \\ 1.7630(4\text{ decimal place)} \end{gathered}`

g. Hence, the 43rd percile is 1.763

The minimum for the upper quarter is as shown below

`\begin{gathered} (3)/(4)of4.1 \\ =0.75*4.1 \\ =3.075 \end{gathered}`

h. Hence, the minimum for the upper quarter is 3.075