STK110 Preparation sheet: TUT 7A 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
b. Calculate the probability that a game is extended with injury time between 6 and 8 minutes.
c. The injury time incurred was recorded for 30 matches. Calculate the probability that the average injury
time is between 4 and 6 minutes.
d. Find the probability that the sampling error of 𝑥̅ for a random sample of 30 matches will be more than 1
minute.
e. What will the 20th percentile of average injury time be?
Question 2
A claim has been made that the optimal playing age for a rugby player is 27 years.
After this age, their playing abilities will start to
Let 𝑥 = age of a rugby player (in years) deteriorate.
Suppose that the ages of rugby players are normally distributed with a mean of
25.9 years and standard deviation of 1.54 years.
𝑥̅ = average age for a sample of rugby players (in years)
a. What is the probability that the age of a randomly selected player will be less than the optimal age of 27
years?
b. What must a player’s age be for his age to be in the top 1% of all player’s ages?
c. Find the standard error for a random sample of 25 players if the population average optimal
playing age, 𝜇, is estimated by the sample mean 𝑥̅ ?
d. What is the probability that the average age for a sample of 25 randomly selected players will be more
than the optimal age of 27 years?
e. What is the minimum average age for a sample of 30 randomly selected players to be in the upper
10%?
Preliminary Memo
Q1b 0.2 Q2a 0.7611
Q1c 0.9426 Q2b 29.49
Q1d 0.0574 Q2c 0.308
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
b. Calculate the probability that a game is extended with injury time between 6 and 8 minutes.
c. The injury time incurred was recorded for 30 matches. Calculate the probability that the average injury
time is between 4 and 6 minutes.
d. Find the probability that the sampling error of 𝑥̅ for a random sample of 30 matches will be more than 1
minute.
e. What will the 20th percentile of average injury time be?
Question 2
A claim has been made that the optimal playing age for a rugby player is 27 years.
After this age, their playing abilities will start to
Let 𝑥 = age of a rugby player (in years) deteriorate.
Suppose that the ages of rugby players are normally distributed with a mean of
25.9 years and standard deviation of 1.54 years.
𝑥̅ = average age for a sample of rugby players (in years)
a. What is the probability that the age of a randomly selected player will be less than the optimal age of 27
years?
b. What must a player’s age be for his age to be in the top 1% of all player’s ages?
c. Find the standard error for a random sample of 25 players if the population average optimal
playing age, 𝜇, is estimated by the sample mean 𝑥̅ ?
d. What is the probability that the average age for a sample of 25 randomly selected players will be more
than the optimal age of 27 years?
e. What is the minimum average age for a sample of 30 randomly selected players to be in the upper
10%?
Preliminary Memo
Q1b 0.2 Q2a 0.7611
Q1c 0.9426 Q2b 29.49
Q1d 0.0574 Q2c 0.308
