Ch 4 · 6.4 Sampling and Estimation · A Level Mathematics 9709
Topic 4 of 5 in Probability & Statistics 2 — A Level Mathematics 9709 — 54 lessons.
Introduction
No polling company asks every voter, and no factory tests every light bulb to destruction. They take a sample and use it to estimate something about the whole population. This lesson shows how to sample fairly, how much a sample mean varies, and how to give an estimate with a margin of error: a confidence interval. (Syllabus section 6.4, Paper 6.)
Key ideas — Populations and samples
The population is everything you want to know about; a sample is the part you actually measure. A sample should be random: every member of the population has an equal chance of being chosen, independently. One way: number the population, then pick numbers with a random number generator, ignoring repeats and numbers out of range.
🧑🏫 Teacher's explanation
Imagine you want to know the average height of every student in a massive school of 5,000 people. It would be exhausting and expensive to measure every single person—this entire group is your population. Instead, you grab 50 students at random to measure. This smaller group is your sample. We use the sample to make an educated guess about the population.
📖 Definitions
🔍 Worked example — Easy
A researcher wants to estimate the average weight of a species of bird. They weigh 25 birds and find a sample mean barx = 12.4g and a sample variance s^2 = 1.56g^2. State the unbiased estimates for the population mean μ and population variance σ^2.
🔍 Worked example — Standard
A sample of n=100 lightbulbs is tested for their lifespan. The sample mean is 1200 hours with a sample standard deviation of 50 hours. Using the Central Limit Theorem, find the approximate distribution of the sample mean barX.
🔍 Worked example — Exam-style
A large sample of n=400 commuters is surveyed about their travel time. 160 commuters reported being late. Find a 95\% confidence interval for the proportion p of commuters who are late. (Use z = 1.96 for the 95\% level)
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The distribution of the sample mean
The mean bar X of a random sample of size n is itself a random variable (it changes from sample to sample), with
🧑🏫 Teacher's explanation
Imagine you want to know the average height of every student in a massive school. You can't measure everyone, so you pick a group of 30 students—this is your 'sample'. If you pick them randomly, the average height of your group, barX, is your best guess for the whole school's average, μ.
📖 Definitions
🔍 Worked example — Easy
A population has a mean μ = 50 and a variance σ^2 = 100. A random sample of size n = 25 is taken. Find the expected value and variance of the sample mean barX.
🔍 Worked example — Standard
The weights of bags of flour follow a normal distribution with μ = 1.00 kg and σ = 0.02 kg. A random sample of n = 16 bags is taken. Find the probability that the sample mean weight is less than 0.99 kg.
🔍 Worked example — Exam-style
A large factory produces bolts where the diameter follows a non-normal distribution with μ = 10 mm and σ^2 = 4 mm^2. A sample of n = 100 bolts is taken. Find the probability that the sample mean diameter is between 9.9 mm and 10.1 mm.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Unbiased estimates
An estimate is unbiased if, on average over many samples, it gives the true value. From a sample:
🧑🏫 Teacher's explanation
Imagine you are trying to guess the average height of every student in a massive school. You can't measure everyone, so you pick 20 students at random. The average height of those 20 students is your 'estimate'. Because you only picked a few, your estimate might be slightly too high or slightly too low. However, if you repeated this process—picking 20 students, recording the mean, putting them back, and picking 20 more—hundreds of times, the average of all those different sample means would eventually equal the true school average. This is what we mean by an 'unbiased' estimate: it is correct 'on average'.
📖 Definitions
🔍 Worked example — Easy
A sample of 5 weights is taken from a population. The weights are 10, 12, 15, 13, and 20. Calculate the unbiased estimate of the population mean barx and the unbiased estimate of the population variance s^2.
🔍 Worked example — Standard
A sample of 10 observations has a mean of 50 and a sample variance (using n in the denominator) of 24. Calculate the unbiased estimate of the population variance s^2.
⚠️ Common errors
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🎯 Exam question
Key ideas — Confidence intervals for a mean
When the population is normal with known σ (or n is large, using s for σ):
🧑🏫 Teacher's explanation
Imagine you want to know the average height of every student in a massive school. You can't measure everyone, so you pick a group of 50 students. This group is your 'sample', and their average height is your 'estimate'. However, you know that if you picked a different group of 50, you'd get a slightly different average. You can't be 100% sure your sample average is exactly the school's true average.\n\nA confidence interval is like throwing a net around your sample average. Instead of saying 'the average is 165cm', you say 'I am 95% confident the true average is between 162cm and 168cm'. The 'confidence' isn't about the specific interval you found; it's about the reliability of the method. If you repeated this experiment 100 times, 95 of those 'nets' would successfully catch the true average height.
📖 Definitions
🔍 Worked example — Easy
A population is normally distributed with σ = 10. A sample of n = 25 is taken and the sample mean barx = 50. Calculate the 95% confidence interval for the population mean μ.
🔍 Worked example — Standard
A large sample (n=100) of lightbulb lifetimes has a mean of 1200 hours and a standard deviation s = 50 hours. Calculate the 90% confidence interval for the population mean μ.
🔍 Worked example — Exam-style
In a survey of 400 people, 120 reported they prefer tea over coffee. Calculate the 95% confidence interval for the proportion of the population that prefers tea.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Confidence intervals for a proportion
From a large sample with x successes out of n, hat p = x/n, and an approximate interval is
🧑🏫 Teacher's explanation
Imagine you want to know what percentage of students in a massive school of 5,000 people actually like broccoli. You can't ask everyone, so you ask a random group of 100 students. The percentage you find in that group, called the sample proportion hatp, is your best guess. However, you know that if you picked a *different* group of 100, you'd get a slightly different percentage.
📖 Definitions
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples
The masses (g) of 50 eggs give Σ x = 2350 and Σ x^2 = 111 600.
🔍 Worked example — Easy
A sample of 25 lightbulbs has a mean life of 1200 hours and a standard deviation of 50 hours. Assuming the population is normally distributed, find the 95% confidence interval for the population mean μ.
🔍 Worked example — Standard
The weights of a population of apples follow N(150, 10^2) grams. A random sample of 16 apples is taken. Find the probability that the sample mean barX is greater than 155 grams.
🔍 Worked example — Exam-style
In a large survey of 1000 voters, 540 support Candidate A. Find the 99% confidence interval for the proportion p of the population that supports Candidate A. (Use z=2.576)
⚠️ Common errors
Common mistakes
Practice
Answers: (1) bar x = 8.5; s^2 = tfrac19(760 - 722.5) = 4.17. (2) bar X sim N(80, 16): P(Z < -1.25) = 0.106. (3) 12.4 ± 1.645 × 2/√(60) = 12.4 ± 0.42: (11.98, 12.82). (4) 0.3 ± 2.576√(0.21/150) = 0.3 ± 0.096: (0.204, 0.396). (5) It is not random: people who shop elsewhere or at other times (e.g. those at work on Monday mornings) cannot be chosen, so the sample is likely to be biased.