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 $T$ is normally distributed with variance 81.

Use the sample in part (a) to calculate a 95% confidence interval for the population mean of $T$.

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

$n=80$

$\sum m=17 764$

$\sum m^2=4 103 225$

Calculate unbiased estimates of the population mean and variance of the masses of potatoes from the farm.

Calculate a 90% confidence interval for the population mean mass.

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

The gestation period, $X$ days, of female kangaroos of a certain species has a normal distribution with mean 29 and standard deviation 4.

Find the probability that the gestation period of a randomly chosen pregnant female kangaroo is between 27 and 30 days.

A random sample of 16 pregnant female kangaroos is chosen and the mean of their gestation periods is denoted by $\overline{X}$.

(i) State the distribution of $\overline{X}$, giving the values of any parameters.

(ii) Find $\text{P}(27<\overline{X}<30)$.

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

The time, $T$ minutes, taken by a gamer to complete the final level of a video game is normally distributed with mean 57.2 and standard deviation 5.

During a competition, gamers are randomly divided into teams of 9 and each gamer plays the final level. The mean time, $\overline{T}$ minutes, for each team is calculated. Prizes are awarded to the teams whose mean times are in the fastest 10% of mean times.

(i) State the distribution of $\overline{T}$, giving the values of any parameters.

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

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

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

Calculate the width of the confidence interval, giving your answer correct to 3 significant figures.

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

Sasha, an inspector, takes a random sample of 40 chocolate brownies sold by ChocoCakes and records the mass, $m$ grams, of each one. The results are summarised as follows.

$n=40$

$\sum m=3404$

$\sum m^2=297 409$

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

Sasha takes another random sample of $n$ brownies from ChocoCakes, where $n$ is large, and calculates the sample mean mass, $\overline{M}$ grams. It is given that $\text{P}(\overline{M}>81.6)=0.9909$.

Using your estimates from part (a),

(i) write down the approximate distribution of $\overline{M}$, stating the values of any parameters,

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

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

Explain why it was necessary for the sample in part (b) to be large.

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

A random sample of 40 summer days is taken. Find the probability that the mean daily rainfall for this sample is between 1.5 and 1.8 millimetres.

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

A random sample of 40 independent observations is taken from the distribution $\text{N}\left(30,5^2\right)$ and the sample mean is denoted by $\overline{Y}$.

(i) State, with a reason, whether the Central Limit Theorem is needed to model $\overline{Y}$ as a normal distribution.

(ii) Find $\text{P}\left(29<\overline{Y}<30\right)$.

The times, $T$ minutes, taken by athletes to run a marathon have a normal distribution with standard deviation 19 minutes. The times of a random sample of 7 athletes are as follows.

Calculate unbiased estimates of the population mean and 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 for the population mean is calculated. 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 records the speeds, $v{\text{ km h}}^{-1}$, of pigeons as they fly past a particular spot. For a random sample of 60 pigeons, the results are summarised as follows.

$n=60$

$\sum v=6309$

$\sum v^2=688 312$

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 your answer to part (a).

In total, Jim takes three independent random samples, each of 60 pigeons, and calculates a 97% confidence interval for the population mean speed from each sample.

Find the probability that all three of these confidence intervals contain the true value of the population mean speed.

The time, $F$ hours, that a French bulldog sleeps in a day is normally distributed with mean 13.2 and variance 3.6.

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

The members of a French bulldog club have 8 French bulldogs between them, and these can be regarded as a random sample.

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

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

The time, $E$ hours, that an English bulldog sleeps in a day is normally distributed with mean 10.4. For a random sample of 10 English bulldogs, the probability that the sample mean time is less than 10.1 hours is 0.1.

Find the standard deviation of $E$.

The price, $P$ dollars, of a one-hour tutoring session in a town is normally distributed with mean $\mu$ and standard deviation $\sigma$. A random sample of 20 one-hour tutoring sessions was taken, and a 99% confidence interval for $\mu$ was calculated to be 29.42 to 41.68.

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

Show that $\sigma =10.64$, correct to 2 decimal places.

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

A random sample of 80 people from Politicity were asked whether they intend to vote in an election, and $k$ of them said that they did. Using this sample, an approximate $\alpha \%$ confidence interval for $p$, the proportion of all people in Politicity who intend to vote, was calculated to be $0.4678<p<0.6822$.

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

Calculate the value of $\alpha$, giving your answer correct to 3 significant figures.

It is given that 59.3% of the population of Politicity voted in the previous election.

Assuming that everyone who intends to vote does so, state, with a reason, whether there is evidence to suggest that the proportion of people who vote has changed.

Loompers, a sweet shop, sells bags of sweets. The masses, $M$ grams, of the bags are normally distributed with variance 973. William, the owner, advertises that the mean mass of a bag is 300 g. Charlie, the owner of a rival shop, takes a random sample of 60 bags and finds that the sample mean mass is 289 g. Charlie claims that William is overstating the mean mass.

Calculate a 98% confidence interval for the population mean mass, and state, with a reason, whether Charlie's claim is justified.

State, with a reason, whether your answer to part (a) would have been different if the population of masses was not normally distributed.

William takes another random sample of 60 bags and calculates a confidence interval for the population mean mass to be 282 g to 304 g.

State, with a reason, whether William used a higher or lower confidence level than 98%.

Students at a school are randomly assigned to teachers. Two months before their external examination, the students sit a practice paper. It is known that each year the standard deviation of marks on this practice paper is 15.7, but the population mean mark, $\mu$, varies.

The mean mark of the 35 students in Mr Cauchy's class is 75.4.

Treating 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 finds that the population mean mark for all the students taking the practice paper this year is 68.7.

Use your confidence interval from part (a) to comment on the performance of Mr Cauchy's class.

The mass of a coin of a certain type is normally distributed with mean 3.56 g and standard deviation 0.63 g. Stuart chooses 10 of these coins at random.

Find the probability that the total mass of the 10 coins is less than 39 g.

Stuart thinks that his 10 coins come from an older batch which has a different mean mass. Using his 10 coins, he calculates a 90% confidence interval for the population mean mass of coins in this older batch, and obtains 3.61 g to 4.27 g.

Comment on whether the mean mass of 3.56 g applies to this older batch.

The volume, $V$ ml, of watermelon soda in a bottle has a normal distribution with mean $w$ and standard deviation 9. The soda is sold in packs of 8 bottles.

A pack is chosen at random. Find the probability that the mean volume of soda in the 8 bottles is within 5 ml of $w$.

A pack is classed as insufficient if the mean volume of the 8 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 the company takes a random sample of 50 bottles and calculates a 95% confidence interval and a 99% confidence interval for $w$. The two intervals are 377.5 ml to 382.5 ml and 376.7 ml to 383.3 ml.

State, with a reason, which of these is the 95% confidence interval.

A spinner has 8 sectors labelled 1 to 8. Richard spins the spinner $n$ times and it lands on the number 7 on 35 of these spins. Using these results, an approximate $\alpha \%$ confidence interval for $p$, the probability that the spinner lands on the number 7 on any spin, is calculated to be $0.1059<p<0.1741$.

Find the value of $\alpha$, giving your answer correct to the nearest integer.

Richard claims that the sectors on the spinner are of unequal size, so that the probability of landing on the number 7 is different from the probability of landing on each of the other numbers.

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

A biased four-sided die has faces marked 1, 2, 3 and 4. The random variable $D$ is the number shown when the die is rolled once. The probability distribution of $D$ is shown in the table.

Find $\text{E}\left(D\right)$ and show that $\text{Var}\left(D\right)=1$.

The die is rolled 100 times and the sum of the numbers obtained is denoted by $S$.

By considering the sample mean, find an estimate for $\text{P}(S<195)$.

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

The scores, in points, of students at a school on a skill test are normally distributed with mean $\mu$ and standard deviation 15. Three teachers take independent random samples of students and use them to calculate confidence intervals for $\mu$.

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 equal to the width of Jean's confidence interval.

Calculate the confidence level that Scott should use, giving your answer correct to 3 significant figures.

Ororo calculates a 99% confidence interval for $\mu$. She wants the width of her confidence interval to be no greater than the width of Jean's confidence interval.

Calculate the smallest number of students that Ororo should include in her sample.

The lengths of the ears of rabbits on a certain island have population mean $\mu$ cm. Roger takes a random sample of 40 rabbits from the island and finds that a 90% confidence interval for $\mu$ is 8.3 cm to 9.9 cm.

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

Using the same sample, Roger wishes to calculate a confidence interval for $\mu$ that has a width of 1.28 cm.

Find the confidence level that Roger should use.

Roger takes 100 independent random samples, each of 40 rabbits, and calculates a 95% confidence interval for $\mu$ from each sample.

Find the number of these confidence intervals that would be expected to contain $\mu$.

The time, $T$ minutes, that a birthday candle of a certain type stays lit is investigated by two employees.

Takato takes a random sample of 30 candles. His results are summarised as follows.

$n=30$

$\sum t=827$

$\sum t^2=25 374$

Kenta takes an independent random sample of 50 candles and calculates unbiased estimates of the population mean and variance of $T$.

$\overline{t}=28.1$

$s^2=105.7$

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

(ii) Using your estimates from part (a)(i), calculate a 92% confidence interval for the population mean time that a candle stays lit.

Kazu takes five independent random samples of candles, each of size 80, and uses each sample to calculate a 92% confidence interval for the population mean.

Find the probability that exactly four of these five confidence intervals contain the true value of the population mean.