Populations and Samples

Launch The tumultuous town mayor decides to trash the way the town collects trash. The mayor vows to knock on every door and talk to each household to get new trash collection ideas. Describe a situation where this approach would be a good idea and a situation where it would be a bad idea. GOOD IDEA— Small town BAD IDEA— Big town (take too long)

Focus Question Sometimes it takes too much effort to study everyone in a large group. When is it reasonable to use a small group to represent a larger group? When is it not reasonable?

Population and Samples Pretend these are all of the items you want to study.

Population and Samples A population is the complete set of items being studied.

Population and Samples A sample of a population is a part of the population. A sample is useful when you want to find out about a population, but you do not have the resources to study every member of the population. Every member in the sample is called a subject.

Population or Sample? You are investigating the lengths of all of the words in a book. Determine whether each description is a population or a sample: All of the words on a page of the book. Sample

Population or Sample? You are investigating the lengths of all of the words in a book. Determine whether each description is a population or a sample. Every other word in the book. Sample

Population or Sample? You are investigating the lengths of all of the words in a book. Determine whether each description is a population or a sample. All of the words in the book. Population

Population or Sample? You are investigating the lengths of all of the words in a book. Determine whether each description is a population or a sample. One word in the book. Sample

Population or Sample? You are studying the T-shirts being sold at a clothing store. Determine whether each description is a population or a sample. Every other T-shirt in the store. Sample

Population or Sample? You are studying the T-shirts being sold at a clothing store. Determine whether each description is a population or a sample. All of the T-shirts in the store. Population

Population or Sample? You are studying the T-shirts being sold at a clothing store. Determine whether each description is a population or a sample. All of the medium-sized T-shirts. Sample

Population or Sample? You are studying the T-shirts being sold at a clothing store. Determine whether each description is a population or a sample. One T-shirt in the store. Sample

Population or Sample? You are studying the T-shirts being sold at a clothing store. Determine whether each description is a population or a sample. All of the striped T-shirts in the store. Sample

Got It? Suppose you have a sample of 15 people. Can you tell what population you are studying? Explain. No, you cannot tell what population you are studying because there is not enough information about the situation. For example, the population could be all of the people in your town. It could be all of the people in one particular office building. It could be all of the people in the world.

Bias A bias is a tendency toward a particular perspective that is different from the overall perspective of the population.

QUESTION: Do you like sushi? BIAS TOWARD “YES” This sample has a bias toward “yes” because most of the subjects like sushi.

QUESTION: Do you like sushi? BIAS TOWARD “NO” This sample has a bias toward “no” because most of the subjects do not like sushi.

Bias You are studying the people in the United States. You want to know who spends at least two months every year within 200 miles of the ocean. Which description is your population? Everyone in Colorado, Iowa, and Kansas Everyone in Florida, Hawaii, and Maine 100 people from each state Everyone in the U.S.

You are studying the people in the United States You are studying the people in the United States. You want to know who spends at least two months every year within 200 miles of the ocean. Describe each sample as having bias or not having bias. Justify your reasoning. Colorado, Iowa, and Kansas are states that do not border the ocean, so the sample is not likely to include people who spend at least two months near the ocean. The sample has bias.

Bias You are studying the people in the United States. You want to know who spends at least two months every year within 200 miles of the ocean. Describe each sample as having bias or not having bias. Justify your reasoning. Florida, Hawaii, and Maine are states that border the ocean, so the sample is likely to include mostly people who spend at least two months near the ocean. The sample has bias.

Bias You are studying the people in the United States. You want to know who spends at least two months every year within 200 miles of the ocean. Describe each sample as having bias or not having bias. Justify your reasoning. “100 people from each state” includes people all across the United States, so the sample more accurately represents the population. The sample does not have bias.

Bias You are studying the people in the United States. You want to know who goes sledding each year. Which samples are likely to contain a bias? Justify your reasoning. Everyone in Colorado, Utah, and Vermont This sample has bias because all three states have mountains with snow in the winter. The sample is likely to include a great number of people who go sledding each year.

Bias You are studying the people in the United States. You want to know who goes sledding each year. Which samples are likely to contain a bias? Justify your reasoning. Everyone in Florida, Louisiana, and Texas This sample has bias because all three states have a warm climate and are not located in a region with much snow. This sample is likely to include many people who do not go sledding each year.

Bias You are studying the people in the United States. You want to know who goes sledding each year. Which samples are likely to contain a bias? Justify your reasoning. Everyone who owns a sled This sample has bias because people who own sleds are more likely to go sledding each year than people who do not own sleds.

Types of Samples You are investigating the percentage of males in the population below. The population:

Types of Samples In a representative sample, the number of subjects in the sample with the trait being studied is proportional to the number of members in the population with that trait.

Types of Samples In a biased sample, the number of subjects in the sample with the trait being studied is not proportional to the number of members in the population with that trait.

Making Inferences An inference is a judgment that is made by interpreting data. A valid inference is true about the population. Valid inferences can be made when they are based on data from a representative sample.

Making Inferences An inference is a judgment that is made by interpreting data. An invalid inference is false about the population, or does not follow from the data. A biased sample can lead to invalid inferences.

Valid or Invalid? Suppose it is 200 years in the future. You collect a representative sample of humans and robots. Tell whether each inference is valid or invalid. 2 out of every 8 members in the population are robots. Invalid

Valid or Invalid? Suppose it is 200 years in the future. You collect a representative sample of humans and robots. Tell whether each inference is valid or invalid. There are more humans than robots in the population. Valid

Valid or Invalid? Suppose it is 200 years in the future. You collect a representative sample of humans and robots. Tell whether each inference is valid or invalid. 25% of the population are robots. Invalid

Valid or Invalid? Suppose it is 200 years in the future. You collect a representative sample of humans and robots. Tell whether each inference is valid or invalid. 20% of the population are robots. Valid

A magazine company studies a representative sample of people who read their magazine. In the sample, there are 12 women and 8 men. Tell whether each inference about their readership is valid or invalid. 60% of the magazine’s readers are women. Valid

Valid or Invalid? A magazine company studies a representative sample of people who read their magazine. In the sample, there are 12 women and 8 men. Tell whether each inference about their readership is valid or invalid. 8 out of 12 of the magazine’s readers are men. Invalid; 8 out of every 20 people in the population are men.

Valid or Invalid? A magazine company studies a representative sample of people who read their magazine. In the sample, there are 12 women and 8 men. Tell whether each inference about their readership is valid or invalid. More men read the magazine than women. Invalid; 60% of the population are women, so there are more women who read the magazine.

Got It? Suppose you are studying the people in the United States who read newspapers. Would the sample “1 person from each state” be a representative sample made up of 50 subjects? No, it is not a representative sample because there are too few subjects in the sample. Even though it may or may not have bias, you cannot make valid inferences because the sample is too small.

Focus Question When is it reasonable to use a small group to represent a larger group? It is reasonable when the number of items in the small group with the traits being studied is proportional to the number of items with those traits in the large group. When is it not reasonable? It is not reasonable when the number of items in the small group with the traits being studied is not proportional to the number of items with those traits in the large group. Also, when the size of the small group is too small.

HW: Population vs. Sample Worksheet