# IBDP Past Year Exam Questions – Continuous RV

**1. [M13.P2.TZ2]**

It is believed that the lifespans of Manx cats are normally distributed with a mean of 13.5 years and a variance of 9.5 years^{2}.

**(a)** Calculate the range of lifespans of Manx cats whose lifespans are within one standard deviation of the mean. [2 marks]

**(b)** Estimate the number of Manx cats in a population of 10 000 that will have a lifespan of less than 10 years. Give your answer to the nearest whole number. [3 marks]

**2. [N13.P2.TZ0]**

The duration of direct flights from London to Singapore in a particular year followed a normal distribution with mean μ and standard deviation σ . 92 % of flights took under 13 hours, while only 12 % of flights took under 12 hours 35 minutes. Find μ and σ to the nearest minute.

**3. [M13.P2.TZ1]**

Emily walks to school every day. The length of time this takes can be modelled by a

normal distribution with a mean of 11 minutes and a standard deviation of 3 minutes.

She is late if her journey takes more than 15 minutes.

**(a)** Find the probability she is late next Monday. [2 marks]

**(b)** Find the probability she is late at least once during the next week (Monday to Friday). [3 marks]

**4. [M8.P2.TZ1]**

A company produces computer microchips, which have a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.7 months.

**(a)** If a microchip is guaranteed for 84 months find the probability that it will fail before the guarantee ends. [2 marks]

**(b)** The probability that a microchip does not fail before the end of the guarantee is required to be 99 %. For how many months should it be guaranteed? [2 marks]

**(c)** A rival company produces microchips where the probability that they will fail after 84 months is 0.88. Given that the life expectancy also follows a normal distribution with standard deviation 3.7 months, find the mean. [2 marks]

**5. [M14.P2.TZ1]**

A student sits a national test and is told that the marks follow a normal distribution with mean 100. The student receives a mark of 124 and is told that he is at the 68^{th} percentile. Calculate the variance of the distribution.

**6. [M14.P2.TZ2]**

The weights, in kg, of one-year-old bear cubs are modelled by a normal distribution with mean μ and standard deviation σ .

**(a)** Given that the upper quartile weight is 21.3 kg and the lower quartile weight is 17.1 kg, calculate the value of μ and the value of σ . [4 marks]

A random sample of 100 of these bear cubs is selected.

**(b)** Find the expected number of bear cubs weighing more than 22 kg [1 mark]

**7. [M10.P2.TZ1]**

In a factory producing glasses, the weights of glasses are known to have a mean of 160 grams. It is also known that the interquartile range of the weights of glasses is 28 grams. Assuming the weights of glasses to be normally distributed, find the standard deviation of the weights of glasses.

**8. [M10.P2.TZ2]**

After being sprayed with a weedkiller, the survival time of weeds in a field is normally distributed with a mean of 15 days.

**(a)** If the probability of survival after 21 days is 0.2 , find the standard deviation of the survival time. [3 marks]

When another field is sprayed, the survival time of weeds is normally distributed with a mean of 18 days.

**(b)** If the standard deviation of the survival time is unchanged, find the probability of survival after 21 days. [2 marks]

**9. [M11.P2.TZ2]**

The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a standard deviation of 0.2 kg.

**(a)** Determine the probability that a fish which is caught weighs less than 1.4 kg. [1 mark]

**(b)** John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg. [3 marks]

**(c)** Determine the probability that a fish which is caught weighs less than 1 kg, given that it weighs less than 1.4 kg. [2 marks]

**10. [M13.P2.TZ2]**

It is believed that the lifespans of Manx cats are normally distributed with a mean of 13.5 years and a variance of 9.5 years2.

**(a)** Calculate the range of lifespans of Manx cats whose lifespans are within one standard deviation of the mean. [2 marks]

**(b)** Estimate the number of Manx cats in a population of 10 000 that will have a lifespan of less than 10 years. Give your answer to the nearest whole number. [3 marks]

**11. [M16.P2.TZ1]**

The heights of students in a single year group in a large school can be modelled by a normal distribution. It is given that 40 % of the students are shorter than 1.62 m and 25 % are taller than 1.79 m . Find the mean and standard deviation of the heights of the students.

**12. [N10.P2.TZ0]**

The weight loss, in kilograms, of people using the slimming regime SLIM3M for a period of three months is modelled by a random variable X . Experimental data showed that 67 % of the individuals using SLIM3M lost up to five kilograms and 12.4 % lost at least seven kilograms. Assuming that X follows a normal distribution, find the expected weight loss of a person who follows the SLIM3M regime for three months.

**13. [N09.P2.TZ0]**

**14. [N09.P2.TZ0]**

Tim goes to a popular restaurant that does not take any reservations for tables. It has been determined that the waiting times for a table are normally distributed with a mean of 18 minutes and standard deviation of 4 minutes.

**(a)** Tim says he will leave if he is not seated at a table within 25 minutes of arriving at the restaurant. Find the probability that Tim will leave without being seated. [2 marks]

**(b)** Tim has been waiting for 15 minutes. Find the probability that he will be seated within the next five minutes. [4 marks]

**15. [M15.P2.TZ1]**

The finishing times in a marathon race follow a normal distribution with mean 210 minutes and standard deviation 22 minutes.

**(a)** Find the probability that a runner finishes the race in under three hours. [2 marks]

The fastest 90 % of the finishers receive a certificate.

**(b)** Find the time, below which a competitor has to complete the race, in order to gain a certificate. [2 marks]