The heights, \(H\) centimetres, of a random sample of 8 one-year-old plants of a certain variety were measured, with the following results.

\[104\quad96\quad110\quad122\quad96\quad101\quad99\quad113\]

(a) Calculate unbiased estimates of the population mean and variance of \(H\). [3]

It is now given that the true value of the population variance is 85.0.

(b) Stating a necessary assumption, find a \(99\%\) confidence interval for the population mean of \(H\), giving the limits of the interval correct to 1 decimal place. [4]

Ten random values of a variable \(X\) are given below.

\[8.1\quad8.3\quad9.5\quad10.1\quad7.9\quad8.7\quad9.1\quad8.8\quad9.4\quad9.0\]

Calculate unbiased estimates of the population mean and variance of \(X\). [3]

A five-sided spinner has edges marked 1, 2, 3, 4, 5. Ayesha believes that the spinner is biased so that it lands on five more often than it should. She spins the spinner 200 times, and it lands on five on 60 spins.

Use Ayesha’s result to calculate an approximate \(94\%\) confidence interval for the probability that the spinner lands on five on one spin. [3]

Eric throws darts at a target. The probability that he will hit the centre of the target on one throw is \(p\).

(a) Eric takes 100 throws and hits the centre \(x\) times. He uses this result to find an approximate \(95\%\) confidence interval for \(p\). The width of this interval is \(0.165\) correct to 3 significant figures. Find the two possible values of \(x\). [4]

Eric takes some more throws. He uses his results to calculate two approximate \(a\%\) confidence intervals for \(p\), where \(a>90\). It is given that the probability that exactly one of these two intervals contains the true value of \(p\) is \(0.1128\).

(b) Find the value of \(a\). [3]

A researcher chose a random sample of 150 adults in Dombridge for a survey about dog ownership.

The number of adults in the survey who owned a dog was \(a\). An approximate \(94\%\) confidence interval for the population proportion, \(p\), of all the adults in Dombridge who owned a dog was calculated. The width of the interval was \(0.13241\), correct to 5 decimal places.

(a) Calculate the two possible values of \(a\). [5]

Two approximate \(94\%\) confidence intervals were found.

(b) Given that at least one of these intervals contains the true value of \(p\), find the probability that both intervals contain the true value of \(p\). [3]

The survey included a question about the distance, \(X\) kilometres, that the adults walked with their dog each day. The results were summarised as follows: \[n=150,\qquad\sum x=462,\qquad\sum x^2=1485.\]

(c) Find unbiased estimates of the mean and variance of \(X\). [3]

The masses of a certain species of animal are known to be normally distributed with standard deviation \(\sigma\) kg. A researcher obtains the masses of a random sample of \(n\) animals of this species and uses these masses to find two confidence intervals (\(\alpha\)% and 90%) for the population mean. The width of the \(\alpha\)% confidence interval is \(1.414\times\) the width of the 90% confidence interval.

(a) Find \(\alpha\). [3]

(b) Find the probability that the 90% confidence interval contains the population mean given that the \(\alpha\)% confidence interval contains the population mean. [1]

The times, in minutes, taken by students to complete a test have mean \(\mu\) and standard deviation \(\sigma\). The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for \(\mu\).

(a) Given that the end points of the 95% confidence interval are 31.02 and 33.98, correct to 4 significant figures, calculate the value of \(\sigma\). [3]

(b) The calculation of the confidence interval required the use of the Central Limit theorem. Explain why it is valid to use the Central Limit theorem in this case. [1]

A researcher calculates a number, \(r\), of 95% confidence intervals for \(\mu\).

(c) Find the largest value of \(r\) such that the probability that all \(r\) confidence intervals contain the true value of \(\mu\) is greater than 0.5. [4]

A student notes the length, \(t\) minutes, of certain lectures. The results for a random sample of 80 lectures are summarised as follows. \[n=80,\qquad\sum t=6430,\qquad\sum t^2=519740.\]

(a) Calculate a 96% confidence interval for the population mean length of the lectures. [6]

(b) The method used in part (a) is valid because the sample size was large. Explain why the method would not be valid if the sample size were small. [1]

The random variable \(X\) has the distribution \(\operatorname{Bin}(8,\tfrac34)\). A random sample of 100 values of \(X\) is chosen, and the sample mean, \(\bar X\), is found.

(a) Find \(\mathrm P(\bar X>6.2)\). You are not expected to use a continuity correction. [6]

(b) State why the Central Limit Theorem was needed in the calculation in part (a). [1]

The height of a certain species of plant is denoted by \(H\) cm. The heights of a random sample of 100 plants were measured, and the following results were found.

The mean, \(\bar h\), for the sample was 80.2. An unbiased estimate of the population variance of \(H\) was 15.6.

Calculate the value of \(\sum h^2\). [3]

A biased spinner has four sides. Each side is of a different colour: yellow, red, green or black. The probability, \(p\), that the spinner will land on red is unknown. The spinner was spun 200 times, and the proportion, \(a\), of times that it landed on red was noted. This proportion was used to calculate an approximate 90% confidence interval for \(p\). The width of this confidence interval was 0.1066 correct to 4 significant figures.

Find the two possible values of \(a\). [4]

A machine dispenses coffee into cups. The volume, \(V\) cm\(^3\), of coffee in a cup was measured for a random sample of 150 cups. The results were summarised as follows. \[\sum v=46350,\qquad\sum v^2=14410800.\]

(a(i)) Calculate unbiased estimates of the population mean, \(\mu\), and population variance, \(\sigma^2\). [3]

(a(ii)) Calculate a 95% confidence interval for \(\mu\). [3]

Another random sample of \(n\) cups of coffee is taken, where \(100<n<120\). A 95% confidence interval for \(\mu\) is calculated using this sample. You may assume that, for large samples, unbiased estimates of \(\sigma^2\) are very similar.

(b) Without calculation, state whether this confidence interval would be wider or narrower than the confidence interval found in part (a)(ii). Give a reason for your answer. [2]

Emma needs to choose one person at random from three people, P, Q and R. She plans to throw two fair coins and note the number, \(n\), of heads. If \(n\) is 0, she will choose P. If \(n\) is 1, she will choose Q. If \(n\) is 2, she will choose R.

(a) By considering probabilities, show that the choice made by this method is not random. [2]

Later, Emma has to choose two people at random from three people.

(b) Describe how Emma could use a single throw of a fair six-sided dice to make this random choice. [2]

A random sample of 200 values of a random variable \(X\) gives the following results. \[n=200,\qquad\sum(x-2)=60,\qquad\sum(x-2)^2=20.\]

(a) Find a 95% confidence interval for the population mean of \(X\). [6]

(b) State which part of your solution to part (a) makes use of the Central Limit Theorem. [1]

Maroulla’s calculator can generate random numbers between 0.000 and 0.999 inclusive, correct to 3 significant figures. She plans to use her calculator to choose a sample of members from the 851 members in her health club. She numbers the members from 1 to 851. Then she uses her calculator to generate some random numbers. She multiplies each random number by 851 and rounds up to the next whole number to give the number of a member in the sample. This is called a ‘member number’.

(a) Maroulla’s first random number is 0.401.

(a) Find the member number that is produced by this random number. [1]

(b) Find all possible random numbers, correct to 3 decimal places, that would produce the following member numbers.

(b(i)) A member number of 680. [1]

(b(ii)) A member number of 850. [1]

(c) Explain briefly how your answers to part (b) show that Maroulla’s method does not produce a random sample. [1]

Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students.

Nikki’s friend says, “This survey is about sports facilities, so you should choose a sample of students from the school sports teams.”

(a) State, with a reason, whether you agree with Nikki’s friend. [1]

Nikki chooses an appropriate random sample of 60 students. She finds that 45 of these students think that the sports facilities are good.

(b) Calculate an approximate 95% confidence interval for the proportion of students who think that the sports facilities are good. [3]

For a different investigation, Nikki uses another large random sample to calculate a 99% confidence interval and an \(x\)% confidence interval. The width of the 99% confidence interval is double the width of the \(x\)% confidence interval.

(c) Calculate the value of \(x\). [4]

The heights of a certain species of deer have standard deviation 0.35 m. A zoologist takes a random sample of 150 deer and obtains a mean height of 1.42 m.

(a) Calculate a 96% confidence interval for the population mean height. [3]

(b) Bubay says that 96% of deer of this species are likely to have heights within this confidence interval. Explain briefly whether he is correct. [1]

The times, \(T\) minutes, taken by a random sample of 75 students to complete a test satisfy \(\sum t=230\) and \(\sum t^2=930\).

(a) Calculate unbiased estimates of the population mean and variance of T. [3]

Assume these estimates are the true population mean and variance. Another random sample of 75 students is taken and its mean is denoted by \(\overline T\).

(b) Find \(a\) such that \(P(\overline T>a)=0.234\). [3]

The lengths of a random sample of 50 roads were measured. A 95% confidence interval for the mean length, in metres, of all roads in the region is [245,263].

(a) Find the mean length of the roads in the sample. [1]

(b) Estimate the standard deviation of road lengths in the region. [2]

(c) Road lengths are now given to be normally distributed. State, with a reason, whether this changes your calculation in (b). [1]

A population is normally distributed with mean 35 and standard deviation 8.1. A random sample of size 140 has mean \(\overline X\).

(a) Find \(P(\overline X>36)\). [3]

(b) Given \(P(\overline X<a)=0.986\), find \(a\). [3]

A daily train journey time is normally distributed with standard deviation 5.8 minutes. The mean of a random sample of 20 journey times is 81.5 minutes.

(a) Calculate a 98% confidence interval for the population mean journey time. [3]

(b) A student says: “The times for 98% of these journeys are likely to be within the confidence interval.” Explain briefly whether this is true. [1]

Two independent 98% confidence intervals are found.

(c) Given that at least one contains the population mean, find the probability that both contain it. [2]

Henri numbers 804 college students from 1 to 804 and uses random calculator digits. The first 20 digits are 5 6 7 1 0 9 8 4 3 1 0 9 6 6 5 0 2 1 7 6. His first two student numbers are 567 and 109.

(a) Find the next two student numbers in the sample. [2]

He samples 30 students and records their weekly social-media time \(X\) hours: \(\sum x=610,\ \sum x^2=12405\).

(b) Calculate an unbiased estimate of the mean and show that an unbiased estimate of the variance is less than 0.1. [3]

(c) A friend thinks Henri has probably made a mistake calculating \(\sum x\) or \(\sum x^2\). Use (b) to comment. [1]

A random sample of 50 college students contains 18 with exactly one brother. An approximate \(a\%\) confidence interval for the population proportion \(p\) has lower limit 0.244 to 3 significant figures. Find \(a\) to the nearest integer. [4]

A random sample of 150 leaves has mean width 3.12 cm. Its approximate 95% confidence interval for the population mean width is [3.01,3.23] cm.

(a) Estimate the population standard deviation. [3]

(b) Explain whether the central limit theorem was necessary in (a). [1]

Lengths \(X\) cm of 100 sampled insects satisfy \(\sum x=36.8,\ \sum x^2=17.34\).

(a) Calculate unbiased estimates of the population mean and variance. [3]

(b) State a necessary condition for these estimates to be reliable. [1]

A random sample of 250 Barapet residents contains 78 BETEC phone owners.

(a) Calculate an approximate 98% confidence interval for the population proportion owning a BETEC phone. [3]

(b) Manjit claims that more than 40% of residents own one. Use (a) to comment. [1]

A random variable \(X\) has the distribution \(N(410,400)\).

Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405. [3]

In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha\%\) confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487, correct to 3 significant figures.

Find the value of \(\alpha\) correct to the nearest integer. [4]

The length, in minutes, of mathematics lectures at a certain college has mean \(\mu\) and standard deviation 8.3.

(a) The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95% confidence interval for \(\mu\). [3]

The length, in minutes, of history lectures at the college has mean \(m\) and standard deviation \(s\).

(b) Using a random sample of 100 history lectures, a 95% confidence interval for \(m\) was found to have width 2.8 minutes. Find the value of \(s\). [2]

A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.

(a) The mean time for a random sample of \(n\) of these journeys was found. A 94% confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places. Find the value of \(n\). [3]

(b) A passenger noted the times for 50 randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time. [1]

(c) A researcher took 4 random samples and a 94% confidence interval for the population mean was found from each sample. Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean. [2]

A sample of 5 randomly selected values of a variable \(X\) is as follows: \[1,\quad2,\quad6,\quad1,\quad a,\] where \(a>0\).

Given that an unbiased estimate of the variance of \(X\) calculated from this sample is \(\frac{11}{2}\), find the value of \(a\). [3]

In a survey of 200 randomly chosen students from a certain college, 23% of the students said that they owned a car.

Calculate an approximate 93% confidence interval for the proportion of students from the college who own a car. [3]

A club has 264 members, numbered from 1 to 264. Donash wants to choose a random sample of members for a survey. In order to choose the members for the sample he uses his calculator to generate random digits. His first 20 random digits are as follows.

\[10612\qquad11801\qquad21473\qquad22759\]

(a) The numbers of the first two members in the sample are 106 and 121. Write down the numbers of the next two members in the sample. [2]

(b) To obtain the numbers for members after the 4th member, Donash starts with the second random digit, 0, and obtains the numbers 061 and 211. Explain why this method will not produce a random sample. [1]

In a random sample of 100 students at Luciana’s college, \(x\) students said that they liked exams. Luciana used this result to find an approximate 90% confidence interval for the proportion, \(p\), of all students at her college who liked exams. Her confidence interval had width 0.15792.

(a) Find the two possible values of \(x\). [4]

Suzma independently took another random sample and found another approximate 90% confidence interval for \(p\).

(b) Find the probability that neither of the two confidence intervals contains the true value of \(p\). [1]

Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.

(a) Calculate an approximate 98% confidence interval for the proportion, \(p\), of students at Anita’s college who watch Bunch. [3]

Carlos says that the confidence interval found in (a) is not useful because it is too wide.

(b) Without calculation, explain briefly how Carlos can use the results of Anita’s survey to find a narrower confidence interval for \(p\). [1]

The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below.

\[5.9\quad6.5\quad6.7\quad5.9\quad6.9\quad6.0\quad6.4\quad6.2\quad5.8\quad5.8\]

Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety. [3]

A builders’ merchant sells stones of different sizes.

(a) The masses of size A stones have standard deviation 6 grams. The mean mass of a random sample of 200 size A stones is 45 grams. Find a 95% confidence interval for the population mean mass of size A stones. [3]

(b) The masses of size B stones have standard deviation 11 grams. Using a random sample of size 200, an \(\alpha\%\) confidence interval for the population mean mass is found to have width 4 grams. Find \(\alpha\). [4]

Each of a random sample of 80 adults gave an estimate, \(h\) metres, of the height of a particular building. The results were summarised as follows. \[n=80,\qquad\sum h=2048,\qquad\sum h^2=52760.\]

(a) Calculate unbiased estimates of the population mean and variance. [3]

(b) Using this sample, the upper boundary of an \(\alpha\%\) confidence interval for the population mean is 26.0. Find the value of \(\alpha\). [4]

\(X\) is a random variable with distribution \(B(10,0.2)\). A random sample of 160 values of \(X\) is taken.

(a) Find the approximate distribution of the sample mean, including the values of the parameters. [3]

(b) Hence find the probability that the sample mean is less than 1.8. [3]

The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. \[n=200,\qquad\sum x=2520,\qquad\sum x^2=31852.\]

(a) Calculate a 95% confidence interval for the population mean diameter. [6]

Jean chose 40 random samples and used each sample to calculate a 95% confidence interval for the population mean diameter.

(b) How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter? [1]

(a) A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was \(72.3\text{ m}\) and an unbiased estimate of the population variance was \(64.3\text{ m}^2\). Find a 92% confidence interval for the population mean length of throws by this athlete. [3]

(b) A discus thrower wishes to calculate a 92% confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method. [1]

\(X\) is a random variable with distribution \(\operatorname{Po}(2.90)\). A random sample of 100 values of \(X\) is taken.

Find the probability that the sample mean is less than 2.88. [5]

The number of characters in emails sent by a particular company is modelled by the distribution \(N(1250,480^2)\).

Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300. [3]

A random sample of 5 values of a variable \(X\) is given below. \[2\quad3\quad3\quad5\quad a\]

(a) Find an expression, in terms of \(a\), for the mean of these values. [1]

It is given that an unbiased estimate of the population variance of \(X\), using these values, is 4. It is also given that \(a\) is positive.

(b) Find and simplify a quadratic equation in terms of \(a\) and hence find the value of \(a\). [3]

The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.

\[200\quad201\quad198\quad202\quad200\quad199\quad199\quad201\quad197\quad202\quad200\quad199\]

(a) Find unbiased estimates of the population mean and variance. [3]

(b) Give a statistical reason why these estimates may not be reliable. [1]

A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355<p<0.445\).

Find the confidence level of this confidence interval. Give your answer correct to the nearest integer. [5]

It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(N(12.5,10.24)\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\overline H\), of the heights.

(a) State the distribution of \(\overline H\), giving the values of any parameters. [2]

(b) Find \(P(12<\overline H<13)\). [3]

A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha\)% confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162.

Calculate the value of \(\alpha\) correct to 2 significant figures. [5]

Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at their college.

(a) Andy suggests that they go to the music building during the lunch hour and choose six students at random from the students who are there.

(a) Give a reason why this method is unsatisfactory. [1]

(b) Jessica decides to use another method. She numbers all the students in the college from 1 to 256. Then she uses her calculator and generates the following random numbers.

\[204393\quad162007\quad204028\quad587119\quad207395\]

From these numbers she obtains six student numbers. The first three of her student numbers are 204, 162 and 7.

(b) Continue Jessica’s method to obtain the next three student numbers. [2]

The probability that a certain spinner lands on red on any spin is \(p\). The spinner is spun 140 times and it lands on red 35 times.

(a) Find an approximate 96% confidence interval for \(p\). [3]

From three further experiments, Jack finds a 90% confidence interval, a 95% confidence interval and a 99% confidence interval for \(p\).

(b) Find the probability that exactly two of these confidence intervals contain the true value of \(p\). [3]

The time, in minutes, taken by students to complete a test has the distribution \(N(125,36)\).

(a) Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. [3]

(b) Explain with a reason whether it was necessary to use the Central Limit theorem in the solution to part (a). [1]

100 randomly chosen adults each throw a ball once. The length, \(l\) metres, of each throw is recorded. The results are summarised below. \[n=100,\qquad\sum l=3820,\qquad\sum l^2=182200.\]

Calculate a 94% confidence interval for the population mean length of throws by adults. [6]

The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. \[n=100,\qquad\sum h=7570,\qquad\sum h^2=588050.\]

(a) Find unbiased estimates of the population mean and variance. [3]

(b) Calculate a 99% confidence interval for the mean height of animals of this species. [3]

Four random samples were taken and a 99% confidence interval for the population mean, \(\mu\), was found from each sample.

(c) Find the probability that all four of these confidence intervals contain the true value of \(\mu\). [2]

The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. \[n=90,\qquad\sum m=4509,\qquad\sum m^2=225950.\]

(a) Find unbiased estimates of the population mean and variance. [3]

(b) Calculate a 98% confidence interval for the population mean. [3]

(c) Explain why it was necessary to use the Central Limit theorem in answering part (b). [1]

(d) Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean. [2]

A construction company notes the time, \(t\) days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. \[\sum t=4820,\qquad\sum t^2=392050.\]

(a) Calculate a 98% confidence interval for the population mean time. [6]

(b) Explain why it was necessary to use the Central Limit theorem in part (a). [1]

In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

(a) It was found that 102 adults in the sample travel by bus. Find an approximate 90% confidence interval for the proportion of all the adults in Fromleigh who travel by bus. [3]

The survey included a question about the amount, \(x\) dollars, spent on travel per year. The results are summarised as follows: \[n=250,\qquad\sum x=50\,460,\qquad\sum x^2=19\,854\,200.\]

(b) Find unbiased estimates of the population mean and variance of the amount spent per year on travel. [3]

A councillor wanted to select a random sample of houses in Fromleigh. He planned to select the first house on each of the 143 streets in Fromleigh.

(c) Explain why this would not provide a random sample. [1]

A six-sided die has faces marked 1, 2, 3, 4, 5, 6. When the die is thrown 300 times it shows a six on 56 throws.

(a) Calculate an approximate 96% confidence interval for the probability that the die shows a six on one throw. [3]

Maroulla claims that the die is biased.

(b) Use your answer to part (a) to comment on this claim. [1]

The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are \[5.2,\quad4.8,\quad5.5,\quad6.1,\quad4.8,\quad3.9,\quad4.4.\]

(a) Calculate unbiased estimates of the population mean and variance of \(X\). [3]

It is now given that the true population variance of \(X\) is 0.55, and that \(X\) has a normal distribution.

(b) Find a 95% confidence interval for the population mean of \(X\). [3]

A random sample of 100 values of a variable \(X\) is taken. The values are summarised by \[n=100,\qquad\sum x=1556,\qquad\sum x^2=29004.\]

Calculate unbiased estimates of the population mean and variance of \(X\). [3]

Sunita has a six-sided die with faces marked 1, 2, 3, 4, 5, 6. The probability of a six on any throw is \(p\). She throws the die 500 times and obtains 70 sixes.

(a) Calculate an approximate 99% confidence interval for \(p\). [4]

(b) Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief. [1]

Sunita uses the result of her 500 throws to calculate an \(\alpha\%\) confidence interval for \(p\). The interval has width 0.04.

(c) Find \(\alpha\). [5]

Lengths of a species of lizard are normally distributed with standard deviation 3.2 cm. A naturalist measures a random sample of 100 lizards and obtains an \(\alpha\%\) confidence interval for the population mean. The total width of the interval is 1.25 cm.

Find \(\alpha\). [5]