The round off errors when measuring the distance that a long jumper has jumped is uniformly distributed between 0 and 5.3 mm. Round values to 4 decimal places when possible.

Question

The round off errors when measuring the distance that a long jumper has jumped is uniformly distributed between 0 and 5.3 mm. Round values to 4 decimal places when possible.

1. The mean of this distribution is 
2. The standard deviation is 
3. The probability that the round off error for a jumper's distance is exactly 2.7 is P(x = 2.7) = 
4. The probability that the round off error for the distance that a long jumper has jumped is between 1.3 and 2.6 mm is P(1.3 < x < 2.6) = 
5. The probability that the jump's round off error is greater than 3.36 is P(x &gt 3.36) = 
6. P(x > 3.8 | x > 1.5) = 
7. Find the 8th percentile. 
8. Find the maximum for the lower quartile. 

Answer 1

Hi Anni,

#1. Mean for continuous uniform distribution= (a + b)/2, where a and b are the minimum and maximum. So:

mu= (0+5.3)/2

mu= 2.6500

#2. Standard deviation for continuous uniform distribution:

sigma= sqrt[(b-a)²/12]

Again, a and b are the minimum and maximum, so:

sigma= sqrt[(5.3-0)²/12]

sigma = 1.5300

#3 In a continuous uniform distribution, i.e. one with infinite possible values--you can have 2.71, 2.715, etc.--P(any exact value)=0

P(X=2.7)=0

#4 Formula for this is:

P = (d-c)/(b-a)

a=minimum

b=maximum

c=lower bound of desired interval

d=higher bound of desired interval

P= (2.6-1.3)/(5.3-0)

P = 0.2453

#5. Same formula. P= (d -c) / (b-a)

a=0

b=5.3

c=3.36

d=5.3

P = (5.3-3.36)/(5.3-0)

P = 0.3660

#6 This is conditional probability. That notation means the probability that x is greater than 3.8 GIVEN that x is greater than 1.5.

Formula for this is P(A given B) = P(A and B)/P(B)

A= X>3.8

B= X>1.5

P(A)= same formula as above

P= (d-c)/(b-a)

a=0

b=5.3

c=3.8

d=5.3

P(A) = (5.3-3.8)/(5.3-0)

P(A)=0.2830

Keep this in mind. We can now get P(B) the same way, but change c to 1.5:

P(B) = (5.3-1.5/(5.3-0)

P(B) = 0.7170

Now, we know that if a value exceeds 3.8, it exceeds 1.5, so P(A and B)=0.2830

P(A given B) = 0.2830/0.7170 = 0.3947

P = 0.3947

#7. Formula is X = percentile * total length = pL

p=0.08 (8th percentile as decimal)

L=5.3

X= 0.08*5.3

X = 0.4240

#8. First or lower quartile is equivalent to 25th percentile, so use the same formula as above:

X=p*L

p=0.25

L=5.3

X= (0.25*5.3)

X= 1.3250

I hope this helps.