Sampling & Estimation (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

The times, $T$ seconds, of a random sample of 8 TV adverts on Channel π are as follows.

Calculate unbiased estimates of the population mean and variance of $T$.

It is known that the population of $T$ follows a Normal distribution with a variance of 81.

Using the sample from part (a), find a 95% confidence interval for the population mean of $T$.

Write down the probability that the confidence interval contains the true population mean.

The masses, $M$ grams, of a random sample of 80 potatoes from a farm are summarised as follows.

$n=80\sum m=17764\sum m^2=4103225$

Calculate unbiased estimates of the population mean and variance.

Calculate a 90% confidence interval for the population mean.

Explain why it was necessary to use the Central Limit theorem in your answer to part (b).

The gestation period of a female kangaroo, $X$ can be modelled as a Normal distribution with a mean of 29 days and a standard deviation of 4 days.

Given that a randomly selected female kangaroo is pregnant, find the probability that the gestation period will be between 27 and 30 days.

A random sample of 16 pregnant kangaroos is taken and the mean of their gestation periods is calculated.

(i) Write down the distribution of the sample mean, $̅X$.

(ii) Calculate the probability that the sample mean of the 16 gestation periods is between 27 and 30 days.

Explain whether it was necessary to use the Central Limit theorem in the solution to part (b).

For the video game, Super Maria, it is known that the length of time, $T$ minutes, it takes a gamer to complete the final level of the game can be modelled as a Normal distribution with $T~N(57.2,5^2)$.

Find the interquartile range for the times taken to complete the final level of Super Maria.

During a Super Maria competition, gamers are randomly put into teams of 9 and each member plays the final level. The mean time for each team is calculated and prizes are given to teams whose means are in the fastest 10% of mean times.

(i) Write down the distribution of the sample mean, $̅T$.

(ii) Find, to the nearest second, the maximum mean time that would lead to a team winning a prize. Give your answer in minutes and seconds.

A spinner has 5 sectors labelled 1 to 5. The spinner is spun 150 times and there were 131 occasions when it landed on the number 3.

Calculate an approximate 92% confidence interval for $p$, the probability that the spinner lands on the number 3.

Calculate the width of the confidence interval to 3 significant figures.

Explain why the method in part (a) would not be suitable if the spinner had only been spun 15 times.

ChocoCakes sells chocolate brownies and claims they are good value for money.  Sasha, an inspector, takes a random sample of 40 brownies and records the mass, $M$ grams, of each one.  The results are summarised as follows.

$\sum m=3404\sum m^2=297409$

Calculate an unbiased estimate of the population mean of the mass of a brownie and show that an unbiased estimate for the variance is 198 g, correct to 3 significant figures.

Sasha takes another large sample of $n$ brownies from ChocoCakes and finds the sample mean, $̅M$. Sasha claims that $P(M̅>81.6)=0.9909$.

Use your estimates from (a) to

(i) Write down the distribution of $̅M$

(ii) Show that $3.5\sqrt{\frac{n}{198}}=2.36$

(iii) And hence find the value of $n$.

Explain why it was important that the sample was large.

The amount of rainfall, in millimetres, on a summer’s day in London has mean 1.6 and standard deviation 0.6.

A random sample of 40 days is taken over one summer, find the probability that the mean daily amount of rainfall for this sample is between 1.5 and 1.8 millimetres.

Explain whether it was necessary to use the Central Limit theorem in the solution to part (a).

The times, $T$ minutes, taken by athletes to run a marathon can be modelled by a normal distribution with a known standard deviation of 19 minutes. Below shows the times taken to complete a marathon by a random sample of 7 athletes.

Calculate unbiased estimates for the mean and the variance of $T$.

Calculate a 95% confidence interval for the population mean marathon time.

Find the probability that the confidence interval found in part (b) is wholly below the true value of the population mean.

A random sample of 30 athletes is now taken and a 95% confidence interval is found. Without calculation, state whether this confidence interval will be wider or narrower than the confidence interval found in part (b). Give a reason for your answer.

Jim, a pigeon researcher, records the speeds, $S$ kilometres per hour, of pigeons as they fly past a particular spot. Jim takes a random sample of 60 pigeons and the results are summarised as follows.

$n=60\sum s=6309\sum s^2=688312$

Calculate a 97% confidence interval for the population mean speed, giving the end-points correct to 1 decimal place.

Explain whether it was necessary to use the Central Limit theorem in part (a).

In total, Jim takes three random samples, each containing 60 pigeons. Jim calculates the 97% confidence intervals for the population mean, $\mu$, for each sample. Find the probability that all three of these intervals contain the true value of $\mu$.

The amount of time, measured in hours, that French bulldogs sleep in a day can be modelled using $N(13.2,3.6)$.

Find the probability that a randomly selected French bulldog sleeps for more than 12 hours in a day.

At a French bulldog fan club meeting, owners share the lengths of time that their dogs sleep. Collectively, they have 8 French bulldogs, and it can be assumed that they form a random sample.

Find the probability that the mean length of time that the 8 French bulldogs sleep is more than 12 hours in a day.

Explain whether it was necessary to use the Central Limit theorem in part (b).

The amount of time, in hours, that English bulldogs sleep in a day can also be modelled by a normal distribution with mean 10.4.  It is known from observations that there is a 10% chance that a random sample of 10 English bulldogs will have a mean of less than 10.1 for the number of hours they sleep in a day.

Find the standard deviation of the lengths of time that English bulldogs sleep in a day.

The price, $\pounds P$, for an hour tutoring session in a local town can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. A random sample of 20 one-hour tutoring sessions was taken, and the 99% confidence interval for  was calculated to be £29.42 to £41.68.

Write down the mean price of the 20 tutoring sessions in the sample.

Show that $\sigma =10.64$, correct to two decimal places.

Find the minimum sample size needed so that the width of the 99% confidence interval for $\mu$ is less than 5.

A random sample of 80 people from Politicity were asked whether they are planning to vote in the next election, $k$ people said that they were planning to vote. An $\alpha \%$ confidence interval for the population proportion planning to vote is $0.4678<p<0.6822$.

By finding the mid-point of this confidence interval, find the value of $k$.

Find the value of $\alpha$.

It is known that 59.3% of the population of Politicity voted in the last election.

Assuming everyone who plans to vote, does indeed vote, state with a reason whether there is evidence to say that the proportion of people who vote has changed.

Loompers, a candy shop, sells bags of pick ‘n’ mix sweets. The masses, $M$ grams, of these bags follow a normal distribution with variance 973. William, the owner, advertises that the mean mass of a bag of pick ‘n’ mix sweets is 300 g. Charlie, the owner of a rival shop, takes a random sample of 60 bags of pick ‘n’ mix sweets and finds that the mean mass is 289 g. Charlie claims that William is overstating the mean mass.

By calculating a 98% confidence interval for the true population mean, state with a reason whether Charlie’s claim is justified.

Would your answer to part (a) have been any different if the population of masses was not normally distributed? Give a reason for your answer.

William takes a random sample of 60 bags and calculates the confidence interval for the population mean to be 282 to 304 grams. State, with a reason, whether William used a higher or lower confidence level than Charlie’s 98%.

At Abacus High School, students are not grouped by ability, instead they are randomly assigned to a teacher. Two months before their external examination, students are given a practice paper. The headteacher uses this to measure student progress and teacher performance. It is known that each year the standard deviation of marks in this practice paper is 15.7 but the mean mark $\mu$ varies. Mr Cauchy calculates the mean for the 35 students in his class to be 75.4.

Using Mr Cauchy’s class as a random sample, calculate a 95% confidence interval for $\mu$.

Explain whether it was necessary to use the Central Limit theorem in your answer to part (a).

Find the probability that the confidence interval found in part (a) is wholly above the true value of $\mu$.

The headteacher calculates that this year’s population mean for the practice paper is 68.7 marks.

What conclusions might the Headteacher make about Mr Cauchy and his students?

The mass of a penny follows a normal distribution with mean 3.56 g and standard deviation 0.63 g.

Find the probability that a randomly selected penny weighs less than 3.9 g.

Stuart has ten random pennies in his pocket.

(i) Find the probability that the total mass of the ten pennies is less than 39 g.

(ii) Find the probability that more than one quarter of Stuart’s pennies weigh less than 3.9 g.

Stuart uses his sample to calculate a 90% confidence interval for the population mean. Given that the confidence interval is 3.48 to 3.54 grams, give the conclusion Stuart should come to about his sample.

Zodiac Soda Ltd sells watermelon flavoured soda. The volume of soda in one of the bottles, $V$ ml, is modelled using the distribution $VN(w,9^2)$. Zodiac Soda Ltd sells the soda in packs of eight bottles.

A pack is chosen at random. Find the probability the mean volume of the eight bottles is within 5 ml of $w$.

A pack is classed as insufficient if the mean volume of the eight bottles is less than 374 ml.

Given that 2.5% of all packs are insufficient, calculate the value of $w$ giving your answer to the nearest millilitre.

The owner of Zodiac Soda Ltd takes a random sample of 50 bottles and calculates a 95% and 99% confidence interval for the population mean $w$. The two intervals are 379 to 383 ml and 375 to 381 ml respectively.

State, with a reason, which interval has the 95% confidence level.

Roger has a keen fascination with rabbits that live on Easter Island, in particular the lengths of their ears. The mean length of a rabbit’s ear on Easter Island is $\mu$ cm. He takes a random sample of 40 rabbits and finds that a 90% confidence interval for $\mu$ is 8.3 to 9.9 cm.

Using the same sample, find a 99% confidence interval for $\mu$. Give the end-points correct to 1 decimal place.

Roger wants to find a confidence interval for $\mu$ with a width of 1.28 cm. Find the confidence level that Roger should use.

Roger takes 100 random samples and calculates a 95% confidence interval for each one. How many of the confidence intervals would be expected to contain the true population mean, $\mu$.

El Cumpleanyos is a business specialising in birthday party goods. Takato collects a random sample of 30 birthday candles from El Cumpleanyos and records how long each one stays lit, $T$ minutes. The results are summarised as follows.

$n=30\sum t=827\sum t^2=25374$ 

Kenta takes a different random sample of 50 birthday candles and calculates unbiased estimates for the mean and variance of $T$, shown below.

$̅t=28.1\mathrm{and}{s}_t^2=105.7$

(i) By combining the two samples into one sample of size 80, calculate unbiased estimates for the mean and variance of $T$.

(ii) Using the mean and variance in part (a)(i), calculate a 92% confidence interval for the population mean time that the candles stay lit.

Kazu takes five random samples of candles and for each sample he calculates a 92% confidence interval for the population mean. Find the probability that exactly four of the confidence intervals will contain the true value of the population mean.

A spinner has 8 sectors labelled 1 to 8. Richard spins the spinner $n$ times and it lands on the number ‘7’ 35 times. An $\alpha \%$ confidence interval for the true probability of landing on the number ‘7’ is $0.1059<p<0.1741$.

Find the value of $\alpha$.

Richard claims that the spinner has unequal sectors such that the probability of it landing on a seven is different to the probability of it landing on the other numbers.

Use the confidence interval to comment on Richard’s claim.

A biased four-sided die is rolled once and the random variable $D$ is the number showing when the die lands. The probability distribution of $D$ is shown in the table below.

Find $E\left(X\right)$ and show that $Var\left(X\right)=1$.

The die is rolled 100 times and the sum, $S$, of all the numbers it lands on is found.

By considering the mean, find an estimate for $P(S<195)$.

State, with a reason, whether it was necessary to use the Central Limit theorem in your answer to part (b).

Students at a school for superheroes have the strength of their powers measured using units called Powerpoints (PP). The strengths of the students’ powers are normally distributed with a mean $\mu$ PP and a standard deviation 15 PP. The school has thousands of students so three teachers take random samples and use them to calculate confidence intervals for the population mean $\mu$.

Jean’s sample consists of 100 students. She calculates a 95% confidence interval for $\mu$.

Calculate the width of Jean’s confidence interval.

Scott’s sample consists of 50 students. He wants the width of his confidence interval to be the same as Jean’s. Calculate the confidence level that Scott should use to calculate his confidence interval.

Ororo uses a 99% confidence level for her confidence interval. She wants the width of her confidence interval to be no bigger than Jean’s. Calculate the smallest number of students that Ororo should include in her sample.