Summary STK110-TUTORIAL 7A

STK110 Preparation sheet: TUT 7A memo 2023

Question 1

At the end of a rugby match, the game can be extended into injury time. The injury
time that is incurred is uniformly distributed between 0 and 10 minutes.
Let: 𝑥 the amount of injury time in minutes that is incurred at the end of a match.
𝑥̅ the average amount of injury time in minutes that is incurred at the end of a match
for 30 matches.


a. Write down the probability density function of 𝑥
1 1
, 0 𝑥 10
𝑓 𝑥 10 0 10
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

b. Calculate the probability that a game is extended with injury time between 6 and 8 minutes.




1
𝑃 6 𝑥 8 2 0.2
10

c. The injury time incurred was recorded for 30 matches. Calculate the probability that the average injury
time is between 4 and 6 minutes.
Let 𝑥̅ average injury time (in minutes) incurred at the end of the game

𝑎 𝑏 0 10
𝐸 𝑥 𝜇 5
2 2

𝑏 𝑎 10 0
𝑉𝑎𝑟 𝑥 𝜎 8.333
12 12
𝜎 √8.333 2.8867

4 5 6 5
∴𝑃 4 𝑥̅ 6 𝑃 𝑧 𝑃 1.8975 𝑧 1.8975 𝑃 1.90 𝑧 1.90
2.8867 2.8867
√30 √30
0.9713 0.0287 0.9426

d. Find the probability that the sampling error of 𝑥̅ for a random sample of 30 matches, will be more than
1 minute.
𝑃 |𝑥 𝜇| 1
𝑃 𝑥 𝜇 1 𝑃 𝑥 𝜇 1
𝑥 𝜇 1 𝑥 𝜇 1
𝑃 𝑃
𝜎 0.5270 𝜎 0.5270
𝑃 𝑍 1.90 𝑃 𝑍 1.90 2 0.0287 0.0574

e. What will the 20th percentile of average injury time be?
𝑃 𝜇
𝑃 𝑧 0.84 0.2 ∴ 20 0.84 ∴ 𝑃20 0.84 0.5270 5 ∴𝑃 4.5573
𝜎 𝑋