AP Stats – 4.1 Sampling and Surveys

Identify the population and sample in each of the following settings The student government at a high school surveys 100 students at the school to get their opinions about a change to the bell schedule. Population is all students at the school; sample is the 100 students surveyed. The quality control manager at a bottling company selects a sample of 10 cans from the production line every hour to see whether the volume of the soda is within acceptable limits. Population is all cans produced that hour; sample is the 10 cans inspected.

How to Sample Badly To say that a sampling method is biased, it has to consistently underestimate or overestimate the value we want to know in repeated sampling. If a sampling method is unbiased, it will still produce estimates that differ from the value we want to know simply by chance. However, these estimates will be too small about half the time and too large about half the time. AP Tip – If asked to describe how the design of a study leads to bias, you must: 1) identify a problem with the design, and 2) explain how this would leave to an overestimate or an underestimate.

Check your Understanding For each of the following situations, identify the sampling method used. Then explain how the sampling method could lead to bias. A farmer brings a juice company several crates of oranges each week. A company inspector looks at 10 oranges from the top of each crate before deciding whether to buy all the oranges. Convenience sampling. This could lead the inspector to overestimate the quality of the oranges if the farmer puts the best oranges on top. The ABC program Nightline once asked whether the United Nations should continue to have its headquarters in the United States. Viewers were invited to call one telephone number to respond YES and another for NO. There was a charge for calling either number. More than 186,000 callers responded, and 67% said NO. Voluntary response sampling. In this case, those who are happy that the UN has its headquarters in the US already have what they want and so are less likely to respond. The proportion who answered NO in the sample is likely to be higher than the true proportion in the US who would answer NO.

How to Sample Well: SRS

How to Sample Well: SRS

Exam Common Errors Don’t forget to address what to do with repeated integers. Students must explicitly state that repeated integers should be ignored or say that they will generate random numbers until they get n different numbers in the specified range. When working with a table of random digits, it is very important that each label have the same number of digits. Ex – need 50 labels, use 01-50 not 1-50.

Example with using Table D The management company of a local mall with 21 stores plans to survey 3 of them to determine the hours they would like to stay open during the holiday season. Use Table D at line 101 to select an SRS of 3 stores.

Other Random Sampling Methods Make sure you can explain how to choose a variable for stratification and why stratified random samples are sometimes better than simple random samples. The best variables to use for stratification are those that would most accurately predict the response. Ex – Social Security. Age would be a good variable. Income as well. However, gender would not be very useful. Efficiency is the primary benefit of cluster sampling. It is perfectly valid if the clusters are chosen at random. In a stratified sample, we divide the population into strata and take “some from all”; in a cluster sample, we divide the population into clusters and take “all from some.”

Ex – Sampling at a School Assembly This is a GREAT example! Make sure you read and work through this one! Why would it not be a good idea to use horizontal rows as the clusters? Only 4 clusters will be selected, and it is possible that an entire grade could be left out of the sample. Why would it not be a good idea to use only the students in the aisle seats? These may be the last students to the assembly, so we could overestimate the opinions of the less responsible students.

Check your Understanding

Inference for Sampling The dotplot is centered at 0.70. This means that there is no bias – the sample proportion isn’t consistently less than 0.70 or consistently greater than 0.70. Using an unbiased sampling method doesn’t guarantee perfect estimates however! Margin of error compensates for the variability that results from taking a random sample from a population. It does not account for a mistake made during the data collection process.

Sample Surveys – What can go Wrong? Undercoverage and Nonresponse happen AFTER the sample has been chosen. Be sure that you speculate on the direction of the bias. For example, if a survey about unemployment was conducted over the phone, nonresponse will lead to bias. It is likely that the estimated proportion of unemployed will be too high because people who are unemployed are more likely to be at home and available for the survey. AP Tip: Students often lose credit when describing what can go wrong with sample surveys because they use the wrong terminology. To be safe, just clearly describe the problem and its consequences in the context of the question.

More things that can go wrong… Response bias – when people misrepresent themselves. Wording of questions – the most important influence on the answers given to a sample survey.

Check your Understanding