VCE Methods Probability Application Task 4

Question 1 [2018 Exam 2 Section B Q4]

Doctors are studying the resting heart rate of adults in two neighbouring towns: Mathsland and Statsville. Resting heart rate is measured in beats per minute (bpm).
The resting heart rate of adults in Mathsland is known to be normally distributed with a mean of 68 bpm and a standard deviation of 8 bpm.

a. Find the probability that a randomly selected Mathsland adult has a resting heart rate between 60 bpm and 90 bpm. Give your answer correct to three decimal places. 1 mark

The doctors consider a person to have a slow heart rate if the person's resting heart rate is less than 60 bpm. The probability that a randomly chosen Mathsland adult has a slow heart rate is 0.1587.
It is known that 29% of Mathsland adults play sport regularly.
It is also known that 9% of Mathsland adults play sport regularly and have a slow heart rate.
Let \(S\) be the event that a randomly selected Mathsland adult plays sport regularly and let \(H\) be the event that a randomly selected Mathsland adult has a slow heart rate.

b.

i. Find \(\Pr(H|S)\), correct to three decimal places. 1 mark

ii. Are the events \(H\) and \(S\) independent? Justify your answer. 1 mark

c.

i. Find the probability that a random sample of 16 Mathsland adults will contain exactly one person with a slow heart rate. Give your answer correct to three decimal places. 2 marks

ii. For random samples of 16 Mathsland adults, \(\hat{P}\) is the random variable that represents the proportion of people who have a slow heart rate.
Find the probability that \(\hat{P}\) is greater than 10%, correct to three decimal places. 2 marks

iii. For random samples of \(n\) Mathsland adults, \(\hat{P}_n\) is the random variable that represents the proportion of people who have a slow heart rate.
Find the least value of \(n\) for which \(\Pr(\hat{P}_n > \frac{1}{n}) > 0.99\). 2 marks

The doctors took a large random sample of adults from the population of Statsville and calculated an approximate 95% confidence interval for the proportion of Statsville adults who have a slow heart rate. The confidence interval they obtained was (0.102, 0.145).

d.

i. Determine the sample proportion used in the calculation of this confidence interval. 1 mark

ii. Explain why this confidence interval suggests that the proportion of adults with a slow heart rate in Statsville could be different from the proportion in Mathsland. 1 mark

Every year at Mathsland Secondary College, students hike to the top of a hill that rises behind the school.
The time taken by a randomly selected student to reach the top of the hill has the probability density function \(M\) with the rule

\( M(t) = \begin{cases} \frac{3}{50} \left(\frac{t}{50}\right)^2 e^{-\left(\frac{t}{50}\right)^3} & t \ge 0 \\ 0 & t < 0 \end{cases} \)

where \(t\) is given in minutes.

e. Find the expected time, in minutes, for a randomly selected student from Mathsland Secondary College to reach the top of the hill. Give your answer correct to one decimal place. 2 marks

Students who take less than 15 minutes to get to the top of the hill are categorised as 'elite'.

f. Find the probability that a randomly selected student from Mathsland Secondary College is categorised as elite. Give your answer correct to four decimal places. 1 mark

g. The Year 12 students at Mathsland Secondary College make up \(\frac{1}{7}\) of the total number of students at the school. Of the Year 12 students at Mathsland Secondary College, 5% are categorised as elite.
Find the probability that a randomly selected non-Year 12 student at Mathsland Secondary College is categorised as elite. Give your answer correct to four decimal places. 2 marks