Inference and hypothesis testing — AS & A Level Further Mathematics (9231) questions by topic
Confidence intervals and t-tests dominate this topic: testing whether an athlete's training times have improved from a stated mean, constructing a 98% confidence interval for rowers' pulse rates, and comparing weekday versus weekend customer spending with a two-sample test.
Find, test, state, show and explain lead the command words across 31 questions from 2023 to 2025 papers, worth from 4 to 13 marks, where choosing a t-value over a z-value for a small sample is the most decisive step. Working through each confidence-interval and test-statistic calculation with instant marking catches a wrong z-value or premature rounding before the error changes a conclusion drawn several lines later.
What examiners look for
- A recurring criticism is choosing an incorrect z-value, or opting for a z-value such as 1.645 or 1.96 when a t-distribution suits a small sample with unknown population variance.
- Examiners note using a pooled estimate for the standard deviation instead of separate sample variances, or the reverse, not pooling variance despite the question stating equal population variances.
- A common mistake used an incorrect t-value, most often 1.796, in the confidence-interval formula.
- Reports flag premature rounding, working to only 3 significant figures during the method, which caused an error in the third figure of the final test statistic.
- Strong scripts recognised a two-sample t-test with pooled variance was required and retained at least 4 significant figures throughout the working.
May–June 2025, Paper 41
The manager of a hardware store is interested in whether there is a difference in the amount spent per customer on weekdays ($x) compared to weekends ($y). Random samples of 120…
Answer this question and get it marked →A researcher is interested in whether there is a difference between two schools in students' aptitude for English. She randomly chooses 10 students from school X and 8 students of…
Answer this question and get it marked →May–June 2025, Paper 42
The manager of a hardware store is interested in whether there is a difference in the amount spent per customer on weekdays ($x) compared to weekends ($y). Random samples of 120…
Answer this question and get it marked →A researcher is interested in whether there is a difference between two schools in students' aptitude for English. She randomly chooses 10 students from school X and 8 students of…
Answer this question and get it marked →May–June 2025, Paper 43
A farmer is investigating whether using a new fertiliser will increase the yield of tomato plants. The farmer selects 40 tomato plants at random and gives them the new fertiliser.…
Answer this question and get it marked →A doctor is investigating the concentration of blood glucose in patients at risk of developing type 2 diabetes, where blood glucose is measured in appropriate units. The doctor…
Answer this question and get it marked →May–June 2025, Paper 44
A random sample of 12 observations of a normal random variable is taken. The results give unbiased estimates for the population mean and variance as 10.24 and 0.52 respectively.…
Answer this question and get it marked →Lina and Mona are two statisticians who also write songs. The 'time' of a song is the number of minutes for which it lasts. For a random sample of 10 of her songs, Lina calculates…
Answer this question and get it marked →May–June 2024, Paper 41
The times taken by members of a large cycling club to complete a cross-country circuit have a normal distribution with mean mu minutes. The times taken, x minutes, are recorded…
Answer this question and get it marked →Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by…
Answer this question and get it marked →May–June 2024, Paper 42
The times taken by members of a large cycling club to complete a cross-country circuit have a normal distribution with mean mu minutes. The times taken, x minutes, are recorded…
Answer this question and get it marked →Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by…
Answer this question and get it marked →May–June 2024, Paper 43
A rowing club has a large number of members. A random sample of 12 of these members is taken and the pulse rate, x beats per minute (bpm), of each is measured after a 30-minute…
Answer this question and get it marked →Seva is investigating the lengths of the tails of adult wallabies in two regions of Australia, X and Y. He chooses a random sample of 50 adult wallabies from region X and records…
Answer this question and get it marked →October–November 2024, Paper 41
Ellie is investigating the heights of two types of beech tree, A and B, in a certain region. She has chosen a random sample of 60 beech trees of type A in the region, recorded…
Answer this question and get it marked →Ansal is investigating the wingspans of Monarch butterflies in two different regions, X and Y. He takes a random sample of 8 Monarch butterflies from region X and records their…
Answer this question and get it marked →October–November 2024, Paper 42
A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally…
Answer this question and get it marked →Dev owns a small company which produces bottles of juice. He uses two machines, X and Y, to fill empty bottles with juice. Dev is investigating the volumes of juice in the…
Answer this question and get it marked →October–November 2024, Paper 43
Ellie is investigating the heights of two types of beech tree, A and B, in a certain region. She has chosen a random sample of 60 beech trees of type A in the region, recorded…
Answer this question and get it marked →Ansal is investigating the wingspans of Monarch butterflies in two different regions, X and Y. He takes a random sample of 8 Monarch butterflies from region X and records their…
Answer this question and get it marked →May–June 2023, Paper 41
The lengths of the leaves of a particular type of tree are normally distributed with mean mu cm. The lengths, x cm, of a random sample of 12 leaves of this type are recorded. The…
Answer this question and get it marked →The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample of size 10 is taken from the children at school P. Their individual times to…
Answer this question and get it marked →May–June 2023, Paper 42
The lengths of the leaves of a particular type of tree are normally distributed with mean mu cm. The lengths, x cm, of a random sample of 12 leaves of this type are recorded. The…
Answer this question and get it marked →The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample of size 10 is taken from the children at school P. Their individual times to…
Answer this question and get it marked →May–June 2023, Paper 43
Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, x m, for each kangaroo. He…
Answer this question and get it marked →An inspector is checking the lengths of metal rods produced by two machines, X and Y. These rods should be of the same length, but the inspector suspects that those made by…
Answer this question and get it marked →October–November 2023, Paper 41
Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve…
Answer this question and get it marked →Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a…
Answer this question and get it marked →October–November 2023, Paper 42
A factory produces small bottles of natural spring water. Two different machines, X and Y, are used to fill empty bottles with the water. A quality control engineer checks the…
Answer this question and get it marked →October–November 2023, Paper 43
Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve…
Answer this question and get it marked →Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a…
Answer this question and get it marked →Other AS & A Level Further Mathematics topics
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- Summation of series (28)
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