What is statistics? Statistics is the science of dealing with data. Data is any type of info packaged in numerical form. Common examples: Political polls, Health/medical studies

Some Basic Definitions Population: collection of individuals or objects we want to study statistically “What is the population to which the statistical statement applies?” N-value: how many individuals/objects there are in the population

Example Study: What percentage of the M&Ms in the jar are blue? Population: all of the M&Ms in the jar N-value: 4392

Census Census: the process of collecting data by going through every member of the population Our example: Count all M&Ms in the jar, count all of the blue ones, find percentage. Drawbacks: Expensive Too much work Almost impossible for large populations

Census vs Survey Census: the process of collecting data by going through every member of the population Survey: process of collecting data only from some members of the population (and use that data to draw conclusions & make inferences about the entire population) Poll: data collection done by asking questions

Use samples! Sample: a subgroup of the population chosen to provide the data Sampling: the act of selecting a sample Finding a good sample is EXTREMELY DIFFICULT!!!! Sampling frame: the actual subset of the population from which the sample will be drawn

Example Study: What percentage of our class likes cheeseburgers? Population: all members of our class N-value: 20 Sampling frame: all of the women in our class A Sample: all of the women in our class who are present today

Sampling frames make a difference! CNN/USA Today/ Gallup Poll, Nov 2004: If the election for Congress were being held today, which party’s candidate would you vote for in your district? Asked of 1866 registered voters nationwide: 49% for Dem, 47% for Rep, 4% undecided Asked of 1573 likely voters nationwide: 50% for Rep, 46% for Dem, 3% undecided Differences: sampling frames for each was different…..sampling frame for the second poll more representative of people who actually voted, and closely predicted actual results. However, it’s much easier to get a list of registered voters as opposed to likely voters

Representative Samples When a population is highly homogeneous, a very small sample may be representative Ex: blood samples, thoroughly mixed cake batter, etc More heterogeneous populations -> more difficult to find representative samples

Are these samples representative? Question: What is the average time it takes a UNL student to walk to class? Samples: All students living in dorms All students who use city buses All students in the Union at noon All students currently taking math classes

1936 Literary Digest Poll US presidential election: Alfred Landon (R) vs incumbent Franklin D Roosevelt (D) Sampling frame included: Every person listed in a telephone directory anywhere in the US Every person on a magazine subscription list Every person listed on the roster of a club or professional association List of 10 million people created to whom mock ballots were mailed

1936 Literary Digest Poll Poll predicted Landon with 57% of vote vs Roosevelt’s 43% Reality: 62% for Roosevelt and 38% for Landon What went wrong?! Think about the sample. Representative? Biased? During the depression, those people with phones, magazine subscriptions, club memberships were RICH

Bias Selection bias: when the choice of the sample has a built-in tendency to exclude a particular group or characteristic within the population Literary Digest poll only had 24% response rate Low response rate -> nonresponse bias (selection bias) People selected themselves out of the survey. Always low response rate for mail surveys. Also, people more passionate about a topic are more likely to respond.

Lots of different kinds of bias Leading-question bias: Are you in favor of paying higher taxes to bail the federal government out of its disastrous economic policies and its mismanagement of the federal budget? Question order bias Afraid to answer bias: Have you ever cheated on your income taxes?

Morals Bigger samples aren’t necessarily better samples! Watch out for different types of bias! A representative sample is key!

Lots of Sampling Methods Convenience sampling: selection of individuals included in the sample is dictated by what is easiest or cheapest Notoriously bad! Ex: Want to know the average score on the last quiz? Sample: Look at the scores of the people sitting next to you. Ex: Want to know how people feel about making the switch to the Big Ten? Sample: Set up a table outside of your house for people to come by and fill out questionnaire

Quota sampling Quota sampling: the sample should have so many women, so many men, so many Christians, so many Muslims, so many urban-dwellers, so many rural farmers, etc The proportions in each category in the sample should be the same as those in the population

Example of quota sampling Intro to Stats has 120 students 40 freshman 30 sophomores 30 juniors 20 seniors To fill out questionnaire, prof selects 24 freshman 18 sophomores 18 juniors 12 seniors

1948 US Presidential Election Gallup poll used detailed quota sampling Sample size: 3250 people Prediction vs reality: Thomas Dewey: 49.5% / 44.5% Harry Truman: 44.5% / 49.9% What went wrong? Missing criterion wrt the categories considered for quota. Interviewers we free to choose whom to interview -> selection bias

Simple Random Sampling SRS: all members of the population have an equal chance at being included in the sample How were previous examples not SRS? Examples of methods: Pull names from a hat Flip a coin Random number generator

Stratified Sampling Break the sampling frame into categories (strata), then randomly choose a sample from these strata Those chosen strata are subdivided into substrata, and a random sample taken. Subdivide again and take a random sample, etc End up with clusters, but usually reliable

Stratified Sampling Example

Now survey these houses!

More Definitions Statistic: Numerical information drawn from a sample Parameter: unknown measure (numerical info) from the population Hopefully, the statistic will be close to the parameter so conclusions made about the sample will be true for the whole population.

Error and Bias Sampling error: the difference between the parameter (estimated) and the statistic Sampling error attributed to: Chance error Sampling variability: different samples give different results Sampling bias: bad sample chosen

Sample Size Population size = N Sample size = n Sampling proportion = n/N Modern public opinion polls: 1000 ≤ n ≤ 1500

Capture-Recapture Used to estimate the N-value Steps: Choose a sample of size , tag the members, and release. After some time, capture a new sample of size and take an exact head count of tagged individuals. Call that number k. The N-value is approximately Proportion is sample approximately the proportion in the population

Small fish in a big pond A pond of fish! Capture = 200 fish. Tag them. Capture = 150 fish. Notice that k = 21 of these fish have tags. There are approximately N ≈ (200*150)/21 ≈ 1428 fish

CORRELATION DOES NOT IMPLY CAUSATION!!!!!!!!!! Clinical Studies Try to study cause and effect, whereas surveys just observe and report CORRELATION DOES NOT IMPLY CAUSATION!!!!!!!!!!

Alar Scare Alar: chemical used by apple growers 1973: mice exposed to active chemicals in Alar at 8 times greater than the max tolerated dosage A child would have to eat 200,000 apples per day to get that dosage Alar doesn’t really cause cancer, but no longer used. Washington State apple industry lost $375 million.

Clinical studies Concerned with determining whether a single variable or treatment (vaccine, drug, therapy, etc) can cause a certain effect (disease, symptom, cure, etc) Confounding variables: all other possible contributing causes that could produce the same effect First step: isolate the treatment under investigation from confounding variables

Controlled Study Subjects are divided into two different groups: Treatment group: consists of subjects receiving the actual treatment Control group: consists of subjects that are not receiving any treatment (for comparison only) Randomized controlled study: subjects are assigned to the treatment group or control group randomly....hopefully groups are representative samples

Placebos Placebo: fake treatment intended to look like the real treatment Controlled placebo study: controlled study in which control group is given a placebo Placebo effect: just the idea of getting treatment can produce positive results

Don’t tell them about the placebo! Blind study: neither the members of the treatment group nor the members of the control group know to which of the two groups they belong Double-blind study: the scientists conducting the study don’t know either

Homework Read Chapter 13 Answer the questions on the Vocabulary worksheet Exercises beginning on page 515: 1-4, 13, 17-25, 30-32, 45-48, 57-60, 70