8.10 Sampling

Distinguish: a (a) survey, (b) census, (c) population, (d) sample. Give one-sentence definitions.

A school has 600 students. A class of 30 is surveyed. Identify the population and the sample.

A researcher gives each student a number and uses a random-number generator to pick 30. What sampling method is this?

A sample of 40 out of 500 pupils is taken. What percentage is the sample of the population?

In a sample of 40 students, 5 were left-handed. Estimate the number of left-handed students in a school of 600.

A survey question reads "Do you agree that the canteen serves poor food?" Why is this biased?

A survey of 100 pet owners finds 65% have dogs. Estimate how many dog owners there are in a community of 800 pet owners.

A teacher surveys only students who attend the library to ask about reading habits. State one source of bias.

In a sample of 50 cars, 12 are red. Estimate the percentage of red cars in the population.

A teacher takes the first 30 students arriving at school as a sample. Identify a potential bias.

A school has Year 7: 200, Year 8: 180, Year 9: 220. A stratified sample of 60 is to be taken. How many from each year?

A poll of 200 voters finds 110 support candidate A. Estimate the percentage of supporters in the wider population, and comment on how confident this estimate is.

A researcher wants to learn about Year 8 lunch preferences. Suggest a sampling plan that avoids bias.

Identify whether each scenario is a census, a sample or biased: (a) UK government population census every 10 years. (b) Surveying every 10th student exiting the school. (c) Asking the football team about favourite sport.

A class of 30 students is surveyed about study time. The mean is 7.4 hours/week. Is this representative of all Y8 students? Suggest two reasons it might not be.

Improve this question: "How often do you eat the disgusting cafeteria food?" Provide a fairer version.

A sample of 40 students has 18 left-handed. The expected proportion is 10%. Comment on whether the sample is consistent with this.

A teacher conducts a survey on a school trip. List two reasons why this sample is unlikely to represent the whole school.

Identify the most representative sampling method for a school survey: (a) volunteers in the library; (b) random class draw; (c) every 10th student exiting; (d) stratified across year-groups.

A data set has the values 152, 148, 1500, 155 (heights in cm). Clean it and explain.

A school survey on phone use has 800 students. A stratified random sample of 80 is taken: Year 7 (200), Year 8 (180), Year 9 (220), Year 10 (100), Year 11 (100). How many from each year?

A sample of 30 students reports an average of 6.5 h homework per week with range 4. Estimate the school-wide average (school of 600 students) and discuss the confidence in this estimate.

For a survey about lunch preferences in a school of 1000 students, you want results accurate to ±3%. Approximately how large a sample is needed?

A school surveys 120 students about whether they bring their own lunch. 78 say yes. (a) What percentage in the sample? (b) Estimate the count in a year-group of 600. (c) Comment on accuracy.

A survey question is "Do you prefer cats over dogs?" State two ways this question is poorly designed and improve it.

A teacher considers two sampling plans for a school of 600. (i) 60 random students from the whole school; (ii) the first 60 students arriving at school. State which is better and why.

A doctor wants to estimate the average daily steps for the local town (~50 000 people). Suggest two methods, with pros and cons.

A school survey contains the question: "How many hours of TV do you watch a week? (a) 0–5, (b) 6–10, (c) 10–15, (d) more than 15." Identify the problem with the response options.

Sample with replacement vs without replacement. A class of 30 students has names placed in a hat. Five draws are made (with replacement). On average, how many distinct names are drawn?

In an opinion poll, respondents are asked: "How satisfied are you with the school cafeteria? Very satisfied / Satisfied / Neutral / Dissatisfied / Very dissatisfied." Identify whether this is biased.

A school of 800 students will be surveyed about transport. Year 7: 200, Year 8: 200, Year 9: 200, Year 10: 100, Year 11: 100. A stratified sample of 80 is required, but you want to ensure ≥ 5 from Year 11. How many from each year?

A sample of 50 dogs at a vet has mean weight 18 kg, range 30 kg. The vet treats 1000 dogs/year. Estimate the population mean and discuss the spread.

A polling firm surveys 1000 voters and reports candidate A at 51% support. Margin of error ±3%. What does this mean for confidence?

A pet-owner survey has 60% response rate (out of 500 sent). 200 responded "yes, owns a pet". Estimate the percentage of pet owners in the surveyed population.

A student conducts a survey at the school gate asking about phone use. Identify three types of bias possible and how to reduce each.

A stratified sample of 100 from a population of 1000 is split: 600 women and 400 men. After sampling, the survey finds 60 women say "yes" and 30 men. Estimate the overall population proportion of "yes" responses.

A class of 28 students completes a maths test. The teacher's class mean is 16/20 with range 7. The teacher reports "the school average is 16/20". Critique this claim.

A survey of 200 students yields a 95% confidence interval of "30–40% prefer pizza". Interpret this interval correctly. What does **not** it mean?

Design an unbiased question to estimate the proportion of students in a school who exercise daily, and state the sampling method.

A teacher cleans a dataset of 200 marks. She notices 15 zeros, 1 value of 100, and 5 missing entries. Outline a cleaning strategy.